Introduction to Biostatistics: Second Edition

INTRODUCTION TO

BIOS TATIS TIC S SECOND EDITION

Robert R. Sokal and F. James Rohlf State University of New York at Stony Brook

D O V E R P U B L I C A T I O N S , INC. Mineola, New York

Copyright C o p y r i g h t © 1969, 1973. 1981. 1987 b y R o b e r t R . S o k a l and F. J a m e s R o h l f All rights reserved.

Bibliographical

Note

T h i s D o v e r e d i t i o n , first p u b l i s h e d in 2009, is a n u n a b r i d g e d r e p u b l i c a t i o n of t h e w o r k originally p u b l i s h e d in 1969 by W . H . F r e e m a n a n d C o m p a n y , N e w Y o r k . T h e a u t h o r s h a v e p r e p a r e d a new P r e f a c e f o r this e d i t i o n .

Library

of Congress

Cataloging-in-Publication

Data

S o k a l , R o b e r t R. I n t r o d u c t i o n t o Biostatistics / R o b e r t R. S o k a l a n d F. J a m e s R o h l f . D o v e r ed. p. c m . O r i g i n a l l y p u b l i s h e d : 2 n d ed. N e w Y o r k : W . H . F r e e m a n , 1969. I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s a n d index. I S B N - 1 3 : 978-0-486-46961-4 I S B N - 1 0 : 0-486-46961-1 I. B i o m e t r y . I. R o h l f , F. J a m e s , 1936- II. Title. Q H 3 2 3 . 5 . S 6 3 3 2009 570.1 '5195 dc22 2008048052

M a n u f a c t u r e d in the U n i t e d S t a l e s of A m e r i c a D o v e r P u b l i c a t i o n s , Inc., 31 Fast 2nd Street, M i n e o l a , N . Y . 1 1501

to Julie and Janice

Contents

PREFACE TO THE DOVER EDITION PREFACE 1.

2.

3.

xiii

INTRODUCTION

1

1.1

Some

1.2

The development

1.3

The statistical

definitions

1 of bioslatistics frame

2.1

Samples

2.2

Variables

2

of mind

D A T A IN B i O S T A T l S T I C S

4

6

and populations

7

in biostatisties

8

and precision

of data

2.3

Accuracy

2.4

Derived

2.5

Frequency

2.6

The handling

variables

distributions of data

3.1

The arithmetic

mean

3.2

Other

31

3.3

The median

3.4

The mode

33

3.5

The range

34

3.6

The standard

3.7

Sample

3.Ν

Practical

methods

deviation

39

means

10

13

D E S C R I P T I V E STATISTICS

3.9

xi

14 24

27 28

32

deviation

statistics

The coefficient

36

and parameters

37

for computing

mean

of variation

43

and

standard

V1U

4.

5.

6.

7.

8.

CONTENTS

I N T R O D U C T I O N TO PROBABILITY DISTRIBUTIONS: T H E B I N O M I A L A N D P O I S S O N D I S T R I B U T I O N S 46 4.1

Probability,

4.2

The

binomial

random

4.3

The

Poisson

sampling,

distribution

and hypothesis

testing

distribution

63

THE N O R M A L PROBABILITY DISTRIBUTION 5.1

Frequency

distributions

5.2

Derivation

of the normal

distribution

76

5.3

Properties

of the normal

distribution

78

5.4

Applications

of continuous

5.5

Departures

of the normal from

methods

ESTIMATION A N D HYPOTHESIS TESTING 6.1

Distribution

and

variance

of means

94

6.2

Distribution

and

variance

of other

statistics

6.3

Introduction

6.4

Student's

6.5

Confidence

limits

t distribution

6.6

The

6.7

Confidence

chi-square

6.8

Introduction

6.9

Tests

6.10

Testing

93

101

103

on sample

distribution

statistics

114

testing

hypotheses

115

employing

the t distribution

!!,,: σ2 = al

the hypothesis

109

112

for variances

lo hypothesis

of simple

85

106

based

limits

75

82

Graphic

to confidence

74

variables

distribution

normality:

limits

The

variances

7.2

The

F distribution

of samples

hypothesis

7.3

The

7.4

Heterogeneity

7.5

Partitioning

7.6

Model

I anova

7.7

Model

II anora

and their

126

129

INTRODUCTION TO ANALYSIS O F VARIANCE 7.1

48

54

means

133

134

138 / / „ : σ] =

among

143

sample

the total

sum

means

of squares

143 and degrees

of freedom

154 157

SINGLE-CLASSIFICATION ANALYSIS O F VARIANCE H.l

Computational

H.2

Equal

Λ.J

Unequal

X.4

Two groups

168


Comparisons

among

means:

Planned

H.6

Comparisons

among

means:

Unplanned

η

formulas

160

161

162 η

165

comparisons comparisons

173 179

150

CONTENTS

9.

10.

ix

TWO-WAY ANALYSIS O F VARIANCE 9.1

Two-way

anova

9.2

Two-way

anova:

9.3

Two-way

anova

12.

13.

replication

186

Significance without

testing

197

replication

199

ASSUMPTIONS O F ANALYSIS O F VARIANCE 10.1

11.

with

185

The

assumptions

10.2

Transformations

10.3

Nonparametric

REGRESSION

of anova 216 methods

in lieu of anova

11.1

Introduction Models

to regression

in regression

11.3

The

11.4

More

than

11.5

Tests

of significance

linear

uses

equation

one value

of

The

11.7

Residuals

11.8

A nonparametric

CORRELATION

231 233

regression

11.6

in regression

value

of X

250

257

and transformations test for

in regression

regression

259

263

267

Correlation

12.2

The

12.3

Significance

12.4

Applications

12.5

Kendall's

and regression

product-moment tests

268

correlation

Tests

13.2

Single-classification

284

of rank

for goodness

13.3

Tests

correlation

goodness

Mathematical

A2

Statistical

BIBLIOGRAPHY

appendix tables

349

320

of fit tests

Two-way

314

AI

286

294

of Jit: introduction

of independence:

270

280

of correlation coefficient

13.1

coefficient

in correlation

ANALYSIS O F FREQUENCIES

353

235

Y for each

of regression

12.1

INDEX

220

230

11.2

APPENDIXES

211

212

314

tables

295 301 305

243

Preface to the Dover Edition

We are pleased and honored to see the re-issue of the second edition of our Introduction to Biostatistics by Dover Publications. On reviewing the copy, we find there is little in it that needs changing for an introductory textbook of biostatistics for an advanced undergraduate or beginning graduate student. The book furnishes an introduction to most of the statistical topics such students are likely to encounter in their courses and readings in the biological and biomedical sciences. The reader may wonder what we would change if we were to write this book anew. Because of the vast changes that have taken place in modalities of computation in the last twenty years, we would deemphasize computational formulas that were designed for pre-computer desk calculators (an age before spreadsheets and comprehensive statistical computer programs) and refocus the reader's attention to structural formulas that not only explain the nature of a given statistic, but are also less prone to rounding error in calculations performed by computers. In this spirit, we would omit the equation (3.8) on page 39 and draw the readers' attention to equation (3.7) instead. Similarly, we would use structural formulas in Boxes 3.1 and 3.2 on pages 41 and 42, respectively; on page 161 and in Box 8.1 on pages 163/164, as well as in Box 12.1 on pages 278/279. Secondly, we would put more emphasis on permutation tests and resampling methods. Permutation tests and bootstrap estimates are now quite practical. We have found this approach to be not only easier for students to understand but in many cases preferable to the traditional parametric methods that are emphasized in this book.

Robert R. Sokal F. James Rohlf November 2008

Preface

T h e favorable reception that the first edition of this b o o k received f r o m teachers a n d s t u d e n t s e n c o u r a g e d us to p r e p a r e a second edition. In this revised edition, we provide a t h o r o u g h f o u n d a t i o n in biological statistics for the u n d e r g r a d u a t e student w h o has a minimal knowledge of m a t h e m a t i c s . W e intend Introduction to Biostatistics to be used in c o m p r e h e n s i v e biostatistics courses, but it can also be a d a p t e d for short courses in medical a n d professional schools; thus, we include examples f r o m the health-related sciences. We have extracted most of this text f r o m the more-inclusive second edition of our o w n Biometry. W e believe t h a t the p r o v e n pedagogic features of that book, such as its informal style, will be valuable here. We have modified some of the features f r o m Biometry, for example, in Introduction to Biostatistics we provide detailed outlines for statistical c o m p u tations but we place less e m p h a s i s on the c o m p u t a t i o n s themselves. Why? Students in m a n y u n d e r g r a d u a t e courses are not motivated to a n d have few o p p o r t u n i t i e s to p e r f o r m lengthy c o m p u t a t i o n s with biological research m a terial; also, such c o m p u t a t i o n s can easily be m a d e on electronic calculators a n d m i c r o c o m p u t e r s . T h u s , we rely on the course instructor t o advise students on the best c o m p u t a t i o n a l p r o c e d u r e s to follow. We present material in a sequence that progresses from descriptive statistics to f u n d a m e n t a l d i s t r i b u t i o n s and the testing of elementary statistical hypotheses; we (hen proceed immediately to the analysis of variance and the familiar t test

XIV

PREFACE

(which is treated as a special case of the analysis of variance a n d relegated to several sections of the book). W e d o this deliberately for two reasons: (1) since t o d a y ' s biologists all need a t h o r o u g h f o u n d a t i o n in the analysis of variance, s t u d e n t s should b e c o m e a c q u a i n t e d with the subject early in the course; a n d (2) if analysis of variance is u n d e r s t o o d early, the need to use the f distribution is reduced. (One would still w a n t to use it for the setting of confidence limits a n d in a few o t h e r special situations.) All t tests can be carried out directly as analyses of variance, a n d the a m o u n t of c o m p u t a t i o n of these analyses of variance is generally equivalent to t h a t of t tests. This larger second edition includes the K o l g o r o v - S m i r n o v two-sample test, n o n p a r a m e t r i c regression, stem-and-leaf d i a g r a m s , h a n g i n g h i s t o g r a m s , a n d the B o n f e r r o n i m e t h o d of multiple c o m p a r i s o n s . W e have rewritten t h e c h a p t e r on the analysis of frequencies in terms of the G statistic rather t h a n χ 2 , because the f o r m e r h a s been shown t o have m o r e desirable statistical properties. Also, because of t h e availability of l o g a r i t h m functions on calculators, the c o m p u t a t i o n of the G statistic is n o w easier t h a n that of the earlier chi-square test. T h u s , we reorient the c h a p t e r to e m p h a s i z e log-likeiihood-ratio tests. We have also a d d e d new h o m e w o r k exercises. We call special, d o u b l e - n u m b e r e d tables "boxes." T h e y can be used as convenient guides for c o m p u t a t i o n because they s h o w the c o m p u t a t i o n a l m e t h o d s for solving various types of biostatistical problems. They usually c o n t a i n all the steps necessary t o solve a p r o b l e m — f r o m the initial setup to the final result. T h u s , s t u d e n t s familiar with material in the b o o k can use them as quick s u m m a r y reminders of a technique. We found in teaching this course that we w a n t e d s t u d e n t s to be able to refer to the material n o w in these boxes. W e discovered that we could not cover even half as m u c h of o u r subject if we had to put this material on the blackboard d u r i n g the lecture, a n d so we m a d e u p and distributed boxe^ a n d asked s t u d e n t s to refer to them d u r i n g the lecture. I n s t r u c t o r s w h o use this b o o k m a y wish to use the boxes in a similar m a n n e r . We e m p h a s i z e the practical a p p l i c a t i o n s of statistics to biology in this book; thus, we deliberately keep discussions of statistical theory to a m i n i m u m . Derivations are given for s o m e f o r m u l a s , but these arc consigned to Appendix A l , where they should be studied a n d reworked by the student. Statistical tables to which the reader can refer when w o r k i n g t h r o u g h the m e t h o d s discussed in this b o o k are found in A p p e n d i x A2. We a r e grateful to K.. R. Gabriel, R. C. Lewontin, a n d M. K a b a y for their extensive c o m m e n t s on t h e second edition of Biometry and to M. D. M o r g a n , E. R u s s e k - C o h e n , a n d M . Singh for c o m m e n t s on an early d r a f t of this book. We also a p p r e c i a t e the w o r k of o u r secretaries, Resa C h a p e y a n d Cheryl Daly, with p r e p a r i n g the m a n u s c r i p t s , a n d of D o n n a D i G i o v a n n i , Patricia Rohlf, a n d B a r b a r a T h o m s o n with p r o o f r e a d i n g . Robert R. Sokal F. J a m e s Rohlf

INTRODUCTION TO

BIOSTATISTICS

CHAPTER

Introduction

This c h a p t e r sets the stage for your study of biostatistics. In Section 1.1, we define the field itself. We then cast a necessarily brief glance at its historical development in Section 1.2. T h e n in Section 1.3 we conclude the c h a p t e r with a discussion of the a t t i t u d e s that the person trained in statistics brings to biological research.

I.I Some definitions Wc shall define biostatistics as the application of statistical methods to the solution of biological problems. T h e biological p r o b l e m s of this definition a r e those arising in the basic biological sciences as well as in such applied areas as the health-related sciences a n d the agricultural sciences. Biostatistics is also called biological statistics o r biometry. T h e definition of biostatistics leaves us s o m e w h a t u p in the air—"statistics" has not been defined. Statistics is a scicnce well k n o w n by n a m e even to the l a y m a n . T h e n u m b e r of definitions you can find for it is limited only by the n u m b e r of b o o k s you wish to consult. We might define statistics in its m o d e r n

2

CHAS'TER 1 / INTRODUCTION

sense as the scientific study of numerical data based on natural phenomena. All p a r t s of this definition a r e i m p o r t a n t a n d deserve emphasis: Scientific study: Statistics m u s t meet t h e c o m m o n l y accepted criteria of validity of scientific evidence. W e m u s t always be objective in p r e s e n t a t i o n a n d e v a l u a t i o n of d a t a a n d a d h e r e t o the general ethical code of scientific m e t h o d ology, or we m a y find t h a t t h e old saying t h a t "figures never lie, only statisticians d o " applies to us. Data: Statistics generally deals with p o p u l a t i o n s or g r o u p s of individuals; hence it deals with quantities of i n f o r m a t i o n , not with a single datum. T h u s , t h e m e a s u r e m e n t of a single a n i m a l or the response f r o m a single biochemical test will generally not be of interest. Numerical: Unless d a t a of a study c a n be quantified in one way o r a n o t h e r , they will not be a m e n a b l e to statistical analysis. N u m e r i c a l d a t a can be m e a s u r e m e n t s (the length or w i d t h of a s t r u c t u r e or t h e a m o u n t of a chemical in a b o d y fluid, for example) o r c o u n t s (such as t h e n u m b e r of bristles or teeth). Natural phenomena: W e use this term in a wide sense to m e a n not only all t h o s e events in a n i m a t e a n d i n a n i m a t e n a t u r e that take place outside the c o n t r o l of h u m a n beings, but also those evoked by scientists a n d partly u n d e r their control, as in experiments. Different biologists will c o n c e r n themselves with different levels of n a t u r a l p h e n o m e n a ; o t h e r k i n d s of scientists, with yet different ones. But all would agree t h a t the chirping of crickets, the n u m b e r of peas in a pod, and the age of a w o m a n at m e n o p a u s e are n a t u r a l p h e n o m e n a . T h e h e a r t b e a t of rats in response to adrenalin, the m u t a t i o n rate in maize after irradiation, or t h e incidence o r m o r b i d i t y in patients treated with a vaccine m a y still be considered n a t u r a l , even t h o u g h scientists have interfered with t h e p h e n o m e n o n t h r o u g h their intervention. T h e average biologist w o u l d n o t c o n sider the n u m b e r of stereo sets b o u g h t by p e r s o n s in different states in a given year to be a n a t u r a l p h e n o m e n o n . Sociologists o r h u m a n ecologists, however, might so consider it a n d deem it w o r t h y of study. T h e qualification " n a t u r a l p h e n o m e n a " is included in the definition of statistics mostly to m a k e certain that the p h e n o m e n a studied are not a r b i t r a r y ones t h a t are entirely u n d e r the will a n d c o n t r o l of the researcher, such as the n u m b e r of animals e m p l o y e d in an experiment. T h e w o r d "statistics" is also used in a n o t h e r , t h o u g h related, way. It can be the plural of the n o u n statistic, which refers t o any one of m a n y c o m p u t e d or estimated statistical quantities, such as the m e a n , the s t a n d a r d deviation, o r the correlation coefficient. Each o n e of these is a statistic.

1.2 The development of biostatistics M o d e r n statistics a p p e a r s to have developed f r o m t w o sources as far back as the seventeenth century. T h e first s o u r c e was political science; a form of statistics developed as a quantitive description of the v a r i o u s aspects of the affairs of a g o v e r n m e n t or state (hence the term "statistics"). This subject also became k n o w n as political arithmetic. T a x e s a n d insurance caused people to b e c o m e

1.2 / THE DEVELOPMENT OF BIOSTATISTICS

3

interested in p r o b l e m s of censuses, longevity, a n d mortality. Such c o n s i d e r a t i o n s a s s u m e d increasing i m p o r t a n c e , especially in E n g l a n d as the c o u n t r y p r o s p e r e d d u r i n g the d e v e l o p m e n t of its empire. J o h n G r a u n t ( 1 6 2 0 - 1 6 7 4 ) a n d William Petty (1623-1687) were early students of vital statistics, a n d o t h e r s followed in their footsteps. At a b o u t the s a m e time, the s e c o n d s o u r c e of m o d e r n statistics developed: the m a t h e m a t i c a l t h e o r y of probability engendered by t h e interest in games of c h a n c e a m o n g the leisure classes of the time. I m p o r t a n t c o n t r i b u t i o n s to this theory were m a d e by Blaise Pascal (1623-1662) a n d Pierre de F e r m a t (1601-1665), b o t h F r e n c h m e n . J a c q u e s Bernoulli (1654-1705), a Swiss, laid the f o u n d a t i o n of m o d e r n probability t h e o r y in /Irs Conjectandi. A b r a h a m de M o i v r e (1667-1754), a F r e n c h m a n living in E n g l a n d , was the first to c o m b i n e the statistics of his d a y with probability t h e o r y in w o r k i n g o u t a n n u i t y values a n d t o a p p r o x i m a t e the i m p o r t a n t n o r m a l distribution t h r o u g h the expansion of the binomial. A later stimulus for the d e v e l o p m e n t of statistics came f r o m the science of a s t r o n o m y , in which m a n y individual o b s e r v a t i o n s h a d to be digested into a c o h e r e n t theory. M a n y of the f a m o u s a s t r o n o m e r s a n d m a t h e m a t i c i a n s of the eighteenth century, such as Pierre Simon Laplace ( 1 7 4 9 - 1 8 2 7 ) in F r a n c e a n d K a r l Friedrich G a u s s ( 1 7 7 7 - 1 8 5 5 ) in G e r m a n y , were a m o n g the leaders in this field. T h e latter's lasting c o n t r i b u t i o n to statistics is the d e v e l o p m e n t of the m e t h o d of least squares. P e r h a p s the earliest i m p o r t a n t figure in biostatistic t h o u g h t was A d o l p h e Quetelet (1796-1874), a Belgian a s t r o n o m e r a n d m a t h e m a t i c i a n , w h o in his work c o m b i n e d the t h e o r y a n d practical m e t h o d s of statistics a n d applied t h e m to p r o b l e m s of biology, medicine, a n d sociology. Francis G a l t o n (1822-1911), a cousin of C h a r l e s D a r w i n , h a s been called the father of biostatistics a n d eugenics. T h e i n a d e q u a c y of D a r w i n ' s genetic theories stimulated G a l t o n to try to solve the p r o b l e m s of heredity. G a l t o n ' s m a j o r c o n t r i b u t i o n to biology was his application of statistical m e t h o d o l o g y to the analysis of biological variation, particularly t h r o u g h the analysis of variability and t h r o u g h his study of regression a n d correlation in biological m e a s u r e m e n t s . His hope of unraveling the laws of genetics t h r o u g h these p r o c e d u r e s was in vain. He started with the most difficult material a n d with the w r o n g a s s u m p t i o n s . However, his m e t h o d o l o g y has become the f o u n d a t i o n for the application of statistics to biology. Karl P e a r s o n (1857-1936), at University College, L o n d o n , b e c a m e interested in the application of statistical m e t h o d s t o biology, particularly in the d e m o n s t r a t i o n of n a t u r a l selection. P e a r s o n ' s interest came a b o u t t h r o u g h the influence of W. F. R. W c l d o n (1860-1906), a zoologist at t h e s a m e institution. Weldon, incidentally, is credited with coining the term " b i o m e t r y " for the type of studies he and P e a r s o n pursued. P e a r s o n continued in the tradition of G a l t o n a n d laid the f o u n d a t i o n for m u c h of descriptive a n d correlational statistics. T h e d o m i n a n t figure in statistics and biometry in the twentieth century has been R o n a l d A. Fisher (1890 1962). His m a n y c o n t r i b u t i o n s to statistical theory will become o b v i o u s even to the cursory reader of this b o o k .

4

CHAPTER 1 /

INTRODUCTION

Statistics t o d a y is a b r o a d a n d extremely active field w h o s e a p p l i c a t i o n s t o u c h a l m o s t every science a n d even the humanities. New a p p l i c a t i o n s for statistics are c o n s t a n t l y being f o u n d , a n d n o o n e can predict f r o m w h a t b r a n c h of statistics new a p p l i c a t i o n s to biology will be m a d e .

1.3 The statistical frame of mind A brief perusal of a l m o s t a n y biological j o u r n a l reveals h o w pervasive the use of statistics has b e c o m e in the biological sciences. W h y h a s there been such a m a r k e d increase in the use of statistics in biology? Apparently, because biologists h a v e f o u n d t h a t the interplay of biological causal a n d response variables d o e s n o t fit the classic m o l d of n i n e t e e n t h - c e n t u r y physical science. In t h a t century, biologists such as R o b e r t M a y e r , H e r m a n n von H e l m h o l t z , a n d o t h e r s tried t o d e m o n s t r a t e t h a t biological processes were n o t h i n g but physicochemical p h e n o m e n a . In so doing, they helped create the impression t h a t the experim e n t a l m e t h o d s a n d n a t u r a l philosophy t h a t h a d led to such d r a m a t i c p r o g r e s s in the physical sciences should be imitated fully in biology. M a n y biologists, even to this day, have retained the tradition of strictly mechanistic a n d deterministic concepts of t h i n k i n g (while physicists, interestingly e n o u g h , as their science has b e c o m e m o r e refined, have begun t o resort t o statistical approaches). In biology, most p h e n o m e n a are affected by m a n y causal factors, u n c o n t r o l l a b l e in their variation a n d often unidentifiable. Statistics is needed to m e a s u r e such variable p h e n o m e n a , to d e t e r m i n e the e r r o r of m e a s u r e m e n t , a n d to ascertain the reality of m i n u t e but i m p o r t a n t differences. A m i s u n d e r s t a n d i n g of these principles and relationships h a s given rise t o the a t t i t u d e of some biologists t h a t if differences induced by an experiment, or observed by nature, are not clear on plain inspection (and therefore a r e in need of statistical analysis), they arc not w o r t h investigating. There are few legitimate fields of inquiry, however, in which, f r o m the n a t u r e of the p h e n o m e n a studied, statistical investigation is unnecessary. Statistical thinking is not really different f r o m o r d i n a r y disciplined scientific thinking, in which wc try to q u a n t i f y o u r observations. In statistics we express o u r degree of belief or disbelief as a p r o b a b i l i t y rather than as a vague, general s t a t e m e n t . F o r example, a statement that individuals of species A a r e larger t h a n those of specics Β or that w o m e n suffer m o r e often f r o m disease X t h a n d o m e n is of a kind c o m m o n l y m a d e by biological and medical scientists. Such s t a t e m e n t s can a n d should be m o r e precisely expressed in q u a n t i t a t i v e form. In m a n y ways the h u m a n mind is a r e m a r k a b l e statistical machine, a b s o r b ing m a n y facts f r o m the outside world, digesting these, a n d regurgitating them in simple s u m m a r y form. F r o m o u r experience we k n o w certain events to o c c u r frequently, o t h e r s rarely. " M a n s m o k i n g cigarette" is a frequently observed event, " M a n slipping on b a n a n a peel," rare. W e k n o w f r o m experience t h a t J a p a n e s e arc on the average shorter than Englishmen a n d that E g y p t i a n s are on the average d a r k e r t h a n Swedes. We associate t h u n d e r with lightning a l m o s t always, flics with g a r b a g e cans in the s u m m e r frequently, but s n o w with the

1.3 / THE STATISTICAL FRAME OF MIND

5

s o u t h e r n C a l i f o r n i a n desert extremely rarely. All such k n o w l e d g e comes to us as a result of experience, b o t h o u r o w n a n d that of others, which we learn a b o u t by direct c o m m u n i c a t i o n or t h r o u g h reading. All these facts have been processed by that r e m a r k a b l e c o m p u t e r , t h e h u m a n brain, which furnishes an abstract. This a b s t r a c t is constantly u n d e r revision, a n d t h o u g h occasionally faulty a n d biased, it is o n the whole astonishingly s o u n d ; it is o u r k n o w l e d g e of the m o m e n t . A l t h o u g h statistics arose t o satisfy the needs of scientific research, the develo p m e n t of its m e t h o d o l o g y in t u r n affected the sciences in which statistics is applied. T h u s , t h r o u g h positive feedback, statistics, created t o serve the needs of n a t u r a l science, h a s itself affected the c o n t e n t a n d m e t h o d s of t h e biological sciences. T o cite a n example: Analysis of variance has h a d a t r e m e n d o u s effect in influencing the types of experiments researchers carry out. T h e whole field of quantitative genetics, o n e of whose p r o b l e m s is the s e p a r a t i o n of e n v i r o n m e n t a l f r o m genetic effects, d e p e n d s u p o n the analysis of variance for its realization, and m a n y of the c o n c e p t s of q u a n t i t a t i v e genetics have b e e n directly built a r o u n d the designs inherent in the analysis of variance.

CHAPTER

Data in

Biostatistics

In Section 2.1 we explain the statistical m e a n i n g of the terms " s a m p l e " a n d " p o p u l a t i o n , " which we shall be using t h r o u g h o u t this book. Then, in Section 2.2, we c o m e to the types of o b s e r v a t i o n s t h a t we o b t a i n f r o m biological research material; we shall see h o w these c o r r e s p o n d to the different kinds of variables u p o n which we perform (he various c o m p u t a t i o n s in the rest of this b o o k . In Section 2.3 we discuss the degree of accuracy necessary for recording d a t a a n d the p r o c e d u r e for r o u n d i n g off figures. We shall then be ready to consider in Section 2.4 certain k i n d s of derived d a t a frequently used in biological science a m o n g them ratios a n d indices a n d the peculiar problems of accuracy a n d d i s t r i b u t i o n they present us. K n o w i n g how to a r r a n g e d a t a in frequency distrib u t i o n s is i m p o r t a n t because such a r r a n g e m e n t s give an overall impression of the general p a t t e r n of the variation present in a s a m p l e a n d also facilitate f u r t h e r c o m p u t a t i o n a l procedures. F r e q u e n c y distributions, as well as the p r e s e n t a t i o n of numerical d a t a , a r e discussed in Section 2.5. In Section 2.6 we briefly describe the c o m p u t a t i o n a l h a n d l i n g of d a t a .

2.1 / SAMPLES AND POPULATIONS

7

2.1 Samples and populations We shall n o w define a n u m b e r of i m p o r t a n t terms necessary for an unders t a n d i n g of biological d a t a . T h e data in biostatistics are generally based on individual observations. T h e y are observations or measurements taken on the smallest sampling unit. These smallest s a m p l i n g units frequently, b u t not necessarily, are also individuals in the o r d i n a r y biological sense. If we m e a s u r e weight in 100 rats, then the weight of each rat is an individual observation; t h e h u n d r e d rat weights together represent the sample of observations, defined as a collection of individual observations selected by a specified procedure. In this instance, one individual o b s e r v a t i o n (an item) is based on o n e individual in a biological s e n s e — t h a t is, o n e rat. However, if we h a d studied weight in a single rat over a period of time, the s a m p l e of individual o b s e r v a t i o n s w o u l d be the weights recorded on one rat at successive times. If we wish to m e a s u r e t e m p e r a t u r e in a study of ant colonies, where each colony is a basic s a m p l i n g unit, each t e m p e r a t u r e reading for o n e colony is an individual observation, a n d the sample of o b s e r v a t i o n s is the t e m p e r a t u r e s for all the colonies considered. If we consider an estimate of the D N A c o n t e n t of a single m a m m a l i a n sperm cell to be an individual o b s e r v a t i o n , the s a m p l e of o b s e r v a t i o n s may be the estimates of D N A c o n t e n t of all the sperm cells studied in o n e individual m a m m a l . W e have carefully avoided so far specifying what particular variable was being studied, because the terms "individual o b s e r v a t i o n " a n d " s a m p l e of observations" as used a b o v e define only the s t r u c t u r e but not the n a t u r e of the d a t a in a study. T h e actual property m e a s u r e d by the individual o b s e r v a t i o n s is the character, or variable. T h e m o r e c o m m o n term employed in general statistics is "variable." H o w e v e r , in biology the word " c h a r a c t e r " is frequently used synonymously. M o r e t h a n one variable can be measured on each smallest sampling unit. T h u s , in a g r o u p of 25 mice we might m e a s u r e the blood pH and the e r y t h r o c y t e c o u n t . Each m o u s e (a biological individual) is the smallest sampling unit, blood p H a n d red cell c o u n t would be the t w o variables studied, the ρ Η readings a n d cell c o u n t s are individual observations, a n d two samples of 25 o b s e r v a t i o n s (on ρ Η a n d on e r y t h r o c y t e c o u n t ) would result. O r we might speak of a bivariate sample of 25 observations, each referring to a />H reading paired with an e r y t h r o c y t e c o u n t . Next we define population. T h e biological definition of this term is well k n o w n . It refers to all the individuals of a given species ( p e r h a p s of a given life-history stage or sex) f o u n d in a circumscribed area at a given time. In statistics, p o p u l a t i o n always m e a n s the totality of individual observations about which inferences are to he made, existing anywhere in the world or at least within a definitely specified sampling area limited in space and time. If you take five men a n d study the n u m b e r of leucocytes in their peripheral blood and you are prepared to d r a w conclusions a b o u t all men from this s a m p l e of five, then the p o p u l a t i o n f r o m which the sample has been d r a w n represents the leucocyte c o u n t s of all extant males of the species Homo sapiens. If, on the other hand, you restrict yourself to a m o r e narrowly specified sample, such as live male

CHAPTER 2 / DATA IN BIOSTATISTICS

8

Chinese, aged 20, a n d you are restricling y o u r conclusions to this p a r t i c u l a r g r o u p , then the p o p u l a t i o n f r o m which you a r e s a m p l i n g will be leucocyte n u m b e r s of all Chinese males of age 20. A c o m m o n misuse of statistical m e t h o d s is to fail to define the statistical p o p u l a t i o n a b o u t which inferences can be m a d e . A report on the analysis of a s a m p l e f r o m a restricted p o p u l a t i o n should not imply that the results hold in general. T h e p o p u l a t i o n in this statistical sense is sometimes referred t o as the universe. A p o p u l a t i o n m a y represent variables of a concrete collection of objects or creatures, such as the tail lengths of all the white mice in the world, the leucocyte c o u n t s of all the Chinese m e n in the world of age 20, or the D N A c o n t e n t of all the h a m s t e r sperm cells in existence: or it m a y represent the o u t c o m e s of experiments, such as all the h e a r t b e a t frequencies p r o d u c e d in guinea pigs by injections of adrenalin. In cases of the first kind the p o p u l a t i o n is generally finite. A l t h o u g h in practice it would be impossible to collect, count, a n d e x a m i n e all h a m s t e r sperm cells, all Chinese men of age 20, or all white mice in the world, these p o p u l a t i o n s a r e in fact finite. Certain smaller p o p u l a t i o n s , such as all the w h o o p i n g cranes in N o r t h America or all the recorded cases of a rare but easily d i a g n o s e d disease X, m a y well lie within reach of a total census. By c o n t r a s t , an experiment can be repeated an infinite n u m b e r of times (at least in theory). A given experiment, such as the a d m i n i s t r a t i o n of adrenalin t o guinea pigs, could be repealed as long as the e x p e r i m e n t e r could o b t a i n material a n d his or her health and patience held out. T h e s a m p l e of experiments actually perf o r m e d is a sample f r o m an infinite n u m b e r that could be p e r f o r m e d . S o m e of the statistical m e t h o d s to be developed later m a k e a distinction between s a m p l i n g from finite a n d f r o m infinite p o p u l a t i o n s . However, t h o u g h p o p u l a t i o n s are theoretically finite in most applications in biology, they are generally so much larger than samples d r a w n from them that they can be c o n sidered de facto infinite-sized populations. 2.2 Variables in biostatisties Each biological discipline has its own set of variables, which may include conventional m o r p h o l o g i c a l m e a s u r e m e n t s ; c o n c e n t r a t i o n s of chemicals in b o d y fluids; rates of certain biological processes; frequencies of certain events, as in genetics, epidemiology, a n d radiation biology; physical readings of optical or electronic machinery used in biological research; and m a n y more. We have already referred to biological variables in a general way, but we have not yet defined them. We shall define a variable as a properly with respect to which

individuals

in a sample

d i f f e r in some

ascertainable

way.

If t h e

property

does not differ within a s a m p l e at h a n d or at least a m o n g the samples being studied, it c a n n o t be of statistical interest. Length, height, weight, n u m b e r of teeth, vitamin ( ' c o n t e n t , and genotypes are examples of variables in o r d i n a r y , genetically and phcnotypically diverse g r o u p s of organisms. W a r m - b l o o d e d n e s s in a g r o u p of m a m m a l s is not, since m a m m a l s are all alike in this regard.

2 . 2 / VARIABLES IN BIOSTATISTICS

9

a l t h o u g h b o d y t e m p e r a t u r e of individual m a m m a l s would, of course, be a variable. We c a n divide variables as follows:

Variables

Measurement variables Continuous variables Discontinuous variables Ranked variables Attributes Measurement variables are those measurements and counts that are expressed numerically. M e a s u r e m e n t variables are of t w o kinds. T h e first kind consists of continuous variables, which at least theoretically can assume an infinite n u m b e r of values between a n y t w o fixed points. F o r example, between the t w o length m e a s u r e m e n t s 1.5 a n d 1.6 cm there are an infinite n u m b e r of lengths that could be m e a s u r e d if o n e were so inclined a n d h a d a precise e n o u g h m e t h o d of calibration. Any given reading of a c o n t i n u o u s variable, such as a length of 1.57 m m , is therefore an a p p r o x i m a t i o n to the exact reading, which in practice is u n k n o w a b l e . M a n y of the variables studied in biology arc c o n t i n u o u s variables. Examples are lengths, areas, volumes, weights, angles, temperatures, periods of time, percentages, c o n c e n t r a t i o n s , a n d rates. C o n t r a s t e d with c o n t i n u o u s variables are the discontinuous variables, also k n o w n as meristic or discrete variables. These are variables that have only certain fixed numerical values, with no intermediate values possible in between. T h u s the n u m b e r of segments in a certain insect a p p e n d a g e may be 4 or 5 or 6 but never 5l or 4.3. Examples of d i s c o n t i n u o u s variables are n u m b e r s of a given s t r u c t u r e (such as segments, bristles, leel h, or glands), n u m b e r s of offspring, n u m b e r s of colonics of m i c r o o r g a n i s m s or animals, or n u m b e r s of plants in a given q u a d r a t . Some variables c a n n o t be m e a s u r e d but at least can be ordered or r a n k e d by their m a g n i t u d e . T h u s , in an experiment one might record the rank o r d e r of emergence o f t e n p u p a e without specifying the exact time at which each p u p a emerged. In such cases we code the d a t a as a ranked variable. I he o r d e r of emergence. Special m e t h o d s for dealing with such variables have been developed, and several arc furnished in this book. By expressing a variable as a series of ranks, such as 1,2, 3, 4. 5, we d o not imply that the difference in m a g n i t u d e between, say, r a n k s I and 2 is identical lo or even p r o p o r t i o n a l to the difference between r a n k s 2 a n d 3. Variables that c a n n o t be measured but must be expressed qualitatively are called attributes, or nominal variables. These are all properties, such as black or white, p r e g n a n t or not p r e g n a n t , d e a d or alive, male or female. W h e n such attributes are c o m b i n e d with frequencies, they can be treated statistically. Of 80 mice, we may, for instance, state that four were black, t w o agouti, and the

CHAPTER 2 / DATA IN BIOSTATISTICS

10

rest gray. W h e n a t t r i b u t e s are c o m b i n e d with frequencies into tables suitable for statistical analysis, they are referred to as enumeration data. T h u s the e n u m e r a t i o n d a t a on color in mice w o u l d be a r r a n g e d as follows:

Color

Frequency

Black

4

Agouti

2

Gray

74

T o t a l n u m b e r of m i c e

80

In s o m e cases a t t r i b u t e s c a n be c h a n g e d into m e a s u r e m e n t variables if this is desired. T h u s colors c a n be c h a n g e d into wavelengths o r c o l o r - c h a r t values. C e r t a i n o t h e r a t t r i b u t e s that can be r a n k e d o r ordered can be c o d e d t o bec o m e r a n k e d variables. F o r example, three a t t r i b u t e s referring to a s t r u c t u r e as " p o o r l y developed," "well developed," a n d " h y p e r t r o p h i e d " could be c o d e d 1, 2, a n d 3. A term that has not yet been explained is variate. In this b o o k we shall use it as a single reading, score, or o b s e r v a t i o n of a given variable. T h u s , if we have m e a s u r e m e n t s of the length of the tails of five mice, tail length will be a c o n t i n u o u s variable, a n d each of the five readings of length will be a variate. In this text we identify variables by capital letters, the most c o m m o n s y m b o l being Y. T h u s V may s t a n d for tail length of mice. A variate will refer t o a given length m e a s u r e m e n t ; Yt is the m e a s u r e m e n t of tail length of the /'th mouse, a n d y 4 is the m e a s u r e m e n t of tail length of the f o u r t h m o u s e in our sample.

2.3 Accuracy and precision of data " A c c u r a c y " and "precision" are used s y n o n y m o u s l y in everyday speech, but in statistics we define them m o r e rigorously. Accuracy is the closeness of a measured or computed

value

to its true

value.

Precision

is the

closeness

of repeated

measure-

ments. A biased but sensitive scale might yield inaccurate but precise weight. By chance, an insensitive scale might result in an a c c u r a t e reading, which would, however, be imprecise, since a repeated weighing would be unlikely to yield an equally accurate weight. Unless there is bias in a m e a s u r i n g i n s t r u m e n t , precision will lead to accuracy. We need therefore mainly be concerned with the former. Precise variates arc usually, but not necessarily, whole n u m b e r s . T h u s , when we count four eggs in a nest, there is no d o u b t a b o u t the exact n u m b e r of eggs in the nest if we have c o u n t e d correctly; it is 4, not 3 or 5, and clearly it could not be 4 plus or minus a fractional part. Meristic, or d i s c o n t i n u o u s , variables are generally m e a s u r e d as exact n u m b e r s . Seemingly, c o n t i n u o u s variables derived from mcristic ones can u n d e r certain c o n d i t i o n s also be exact n u m b e r s . F o r instance, ratios between exact n u m b e r s arc themselves also exact. If in a c o l o n y of a n i m a l s there are 18 females and 12 males, the ratio of females to males (a

11

2.3 / ACCURACY AND PRECISION OF DATA

M o s t c o n t i n u o u s variables, however, are a p p r o x i m a t e . W e m e a n by this that the exact value of the single m e a s u r e m e n t , the variate, is u n k n o w n a n d p r o b a b l y u n k n o w a b l e . T h e last digit of the m e a s u r e m e n t stated should imply precision; t h a t is, it should indicate t h e limits on the m e a s u r e m e n t scale between which we believe the true m e a s u r e m e n t to lie. T h u s , a length m e a s u r e m e n t of 12.3 m m implies t h a t the true length of the structure lies s o m e w h e r e between 12.25 a n d 12.35 m m . Exactly where between these implied limits the real length is we d o not k n o w . But where w o u l d a true m e a s u r e m e n t of 12.25 fall? W o u l d it not equally likely fall in either of the t w o classes 12.2 a n d 12.3—clearly an unsatisfactory state of affairs? Such an a r g u m e n t is correct, b u t w h e n we record a n u m b e r as either 12.2 or 12.3, we imply t h a t the decision w h e t h e r to put it into the higher or lower class h a s already been taken. This decision was not taken arbitrarily, b u t p r e s u m a b l y was based o n the best available m e a s u r e m e n t . If the scale of m e a s u r e m e n t is so precise t h a t a value of 12.25 would clearly have been recognized, then the m e a s u r e m e n t should have been recorded originally to four significant figures. Implied limits, therefore, always carry one more figure beyond the last significant one measured by the observer. Hence, it follows t h a t if we record the m e a s u r e m e n t as 12.32, we a r e implying that the true value lies between 12.315 a n d 12.325. Unless this is w h a t we m e a n , there would be n o p o i n t in a d d i n g the last decimal figure to o u r original measurements. If we d o a d d a n o t h e r figure, we must imply a n increase in precision. W e see, therefore, t h a t accuracy a n d precision in n u m b e r s are not a b s o l u t e concepts, but are relative. Assuming there is n o bias, a n u m b e r b e c o m e s increasingly m o r e a c c u r a t e as we are able to write m o r e significant figures for it (increase its precision). T o illustrate this concept of the relativity of accuracy, consider the following three n u m b e r s :

Implied 193 192.8 192.76

192.5 192.75 192.755

limits 193.5 192.85 192.765

We m a y imagine these n u m b e r s t o be recorded m e a s u r e m e n t s of the same structure. Let us a s s u m e that we h a d e x t r a m u n d a n e knowledge that the true length of the given s t r u c t u r e was 192.758 units. If t h a t were so, the three m e a s u r e m e n t s would increase in accuracy f r o m the t o p d o w n , as the interval between their implied limits decreased. You will n o t e that the implied limits of the t o p m o s t m e a s u r e m e n t a r e wider than those of the o n e below it, which in turn are wider t h a n those of the third m e a s u r e m e n t . Meristic variates, t h o u g h ordinarily exact, may be recorded a p p r o x i m a t e l y when large n u m b e r s are involved. T h u s w h e n c o u n t s are reported to the nearest t h o u s a n d , a c o u n t of 36,000 insects in a cubic meter of soil, for example, implies that the true n u m b e r varies s o m e w h e r e f r o m 35,500 to 36,500 insects. T o h o w m a n y significant figures should we record m e a s u r e m e n t s ? If we array t l l r » f «» m r-vl i> K i ;

οr

/-»f t-v-i <» n m l ι ι / ί rv Γι-ί-> ι-νι 1 U . i otrv ο 11 η ο i

ιη/ίι\;1

1a

I l-> α

l
CHAPTER 2 / DATA IN BIOSTATISTICS

12

one, an easy rule to remember is that the number of unit steps from the smallest to the largest measurement in an array should usually be between 30 a n d 300. Thus, if we are measuring a series of shells to the nearest millimeter a n d the largest is 8 m m and the smallest is 4 m m wide, there are only four unit steps between the largest a n d the smallest measurement. Hence, we should measure our shells to one m o r e significant decimal place. Then the two extreme measurements might be 8.2 m m a n d 4.1 mm, with 41 unit steps between them (counting the last significant digit as the unit); this would be an a d e q u a t e n u m b e r of unit steps. T h e reason for such a rule is that an error of 1 in the last significant digit of a reading of 4 m m would constitute an inadmissible error of 25%, but an e r r o r of 1 in the last digit of 4.1 is less t h a n 2.5%. Similarly, if we measured the height of the tallest of a series of plants as 173.2 cm a n d that of the shortest of these plants as 26.6 cm, the difference between these limits would comprise 1466 unit steps (of 0.1 cm), which are far too many. It would therefore be advisable to record the heights to the nearest centimeter, as follows: 173 cm for the tallest and 27 cm for the shortest. This would yield 146 unit steps. Using the rule we have stated for the n u m b e r of unit steps, we shall record two or three digits for most measurements. The last digit should always be significant; that is, it should imply a range for the true measurement of from half a "unit step" below to half a "unit step" above the recorded score, as illustrated earlier. This applies to all digits, zero included. Zeros should therefore not be written at the end of a p p r o x i m a t e n u m bers to the right of the decimal point unless they are meant to be significant digits. T h u s 7.80 must imply the limits 7.795 to 7.805. If 7.75 to 7.85 is implied, the measurement should be recorded as 7.8. When the n u m b e r of significant digits is to be reduced, we carry out the process of rounding off numbers. The rules for r o u n d i n g off are very simple. A digit to be rounded off is not changed if it is followed by a digit less than 5. If the digit to be rounded off is followed by a digit greater than 5 or by 5 followed by other nonzero digits, it is increased by 1. When the digit to be rounded off is followed by a 5 standing alone or a 5 followed by zeros, it is unchanged if it is even but increased by 1 if it is odd. T h e reason for this last rule is that when such numbers are summed in a long series, we should have as m a n y digits raised as arc being lowered, on the average; these changes should therefore balance out. Practice the above rules by r o u n d i n g off the following n u m b e r s to the indicated n u m b e r of significant digits:

Number 26.58 133.71 37 0.03725 0.03715 18,316 17.3476

Significant

digits

2 5 3 3 2 3

desired

Answer 27 133.71 0.0372 0.0372

8.000 17.3

2 . 4 / DERIVED VARIABLES

13

M o s t pocket calculators or larger c o m p u t e r s r o u n d off their displays using a different rule: they increase t h e preceding digit when the following digit is a 5 s t a n d i n g alone o r with trailing zeros. H o w e v e r , since m o s t of the m a c h i n e s usable for statistics also retain eight or ten significant figures internally, the a c c u m u l a t i o n of r o u n d i n g e r r o r s is minimized. Incidentally, if t w o calculators give answers with slight differences in the final (least significant) digits, suspect a different n u m b e r of significant digits in m e m o r y as a cause of t h e disagreement.

2.4 Derived variables T h e m a j o r i t y of variables in biometric w o r k are o b s e r v a t i o n s r e c o r d e d as direct m e a s u r e m e n t s or c o u n t s of biological material o r as readings that are the o u t p u t of various types of instruments. However, there is a n i m p o r t a n t class of variables in biological research t h a t we m a y call the derived or computed variables. These are generally based on t w o o r m o r e independently m e a s u r e d variables whose relations are expressed in a certain way. We are referring to ratios, percentages, concentrations, indices, rates, a n d the like. A ratio expresses as a single value the relation that t w o variables have, o n e to the other. In its simplest form, a ratio is expressed as in 64:24, which m a y represent the n u m b e r of wild-type versus m u t a n t individuals, the n u m b e r of males versus females, a c o u n t of parasitized individuals versus those not p a r a sitized, a n d so on. T h e s e examples imply ratios based on counts. A ratio based on a c o n t i n u o u s variable might be similarly expressed as 1.2:1.8, which m a y represent the ratio of width t o length in a sclerite of an insect o r the ratio between the c o n c e n t r a t i o n s of t w o minerals contained in w a t e r or soil. Ratios m a y also be expressed as fractions; thus, the t w o ratios a b o v e could be expressed as f | a n d f ^ - . However, for c o m p u t a t i o n a l p u r p o s e s it is m o r e useful to express the ratio as a quotient. T h e two ratios cited would therefore be 2.666 . . . and 0.666 . . . , respectively. These are pure n u m b e r s , not expressed in m e a s u r e m e n t units of any kind. It is this form for ratios that we shall consider further. Percentages are also a type of ratio. Ratios, percentages, a n d c o n c e n t r a t i o n s are basic quantities in m u c h biological research, widely used and generally familiar. An index is the ratio of the value of one variable to the value of a so-called standard one. A well-known example of an index in this sense is the cephalic index in physical a n t h r o p o l o g y . Conceived in the wide sense, an index could be the average of t w o m e a s u r e m e n t s — e i t h e r simply, such as {(length of A + length of β), or in weighted fashion, such as ^ [ ( 2 χ length of A) + length of B\. Rates are i m p o r t a n t in m a n y experimental fields of biology. T h e a m o u n t of a s u b s t a n c e liberated per unit weight or volume of biological material, weight gain per unit time, reproductive rates per unit p o p u l a t i o n size a n d time (birth rates), a n d d e a t h rates would fall in this category. T h e use of ratios a n d percentages is deeply ingrained in scientific t h o u g h t . Often ratios m a y be the only m e a n i n g f u l way to interpret and u n d e r s t a n d certain types of biological problems. If the biological process being investigated

14

CHAPTER 2 / DATA IN BIOSTATISTICS

o p e r a t e s o n the ratio of the variables studied, o n e must e x a m i n e this r a t i o to u n d e r s t a n d the process. T h u s , Sinnott a n d H a m m o n d (1935) f o u n d t h a t inheritance of the shapes of squashes of the species Cucurbita pepo could be interpreted t h r o u g h a form index based on a length-width ratio, b u t n o t t h r o u g h the i n d e p e n d e n t d i m e n s i o n s of shape. By similar m e t h o d s of investigation, we should be able to find selection affecting b o d y p r o p o r t i o n s to exist in t h e evolution of almost any o r g a n i s m . T h e r e are several d i s a d v a n t a g e s to using ratios. First, they are relatively inaccurate. Let us return to the ratio m e n t i o n e d a b o v e a n d recall f r o m t h e previous section that a m e a s u r e m e n t of 1.2 implies a true r a n g e of m e a s u r e m e n t of the variable f r o m 1.15 to 1.25; similarly, a m e a s u r e m e n t of 1.8 implies a r a n g e f r o m 1.75 to 1.85. We realize, therefore, that the true ratio m a y vary a n y w h e r e f r o m -f^J- to -Hi", or f r o m 0.622 t o 0.714. W e n o t e a possible m a x i m a l e r r o r of 4.2% if 1.2 is an original m e a s u r e m e n t : (1.25 — 1.2)/1.2; the c o r r e s p o n d i n g maximal e r r o r for the r a t i o is 7.0%: (0.714 - 0.667)/0.667. F u r t h e r m o r e , the best estimate of a ratio is n o t usually the m i d p o i n t between its possible ranges. T h u s , in o u r example the m i d p o i n t between the implied limits is 0.668 a n d the r a t i o based on 4τ§~ is 0.666 . . . ; while this is only a slight difference, the discrepancy m a y be greater in o t h e r instances. A second d i s a d v a n t a g e to ratios a n d percentages is that they m a y not be a p p r o x i m a t e l y n o r m a l l y distributed (see C h a p t e r 5) as required by m a n y statistical tests. This difficulty can frequently be o v e r c o m e by t r a n s f o r m a t i o n of the variable (as discussed in C h a p t e r 10). A third d i s a d v a n t a g e of ratios is t h a t in using them o n e loses i n f o r m a t i o n a b o u t the relationships between the t w o variables except for the i n f o r m a t i o n a b o u t the ratio itself.

2.5 Frequency distributions If we were to sample a p o p u l a t i o n of birth weights of infants, we could represent each sampled m e a s u r e m e n t by a point a l o n g an axis d e n o t i n g m a g n i t u d e of birth weight. This is illustrated in Figure 2.1 A, for a s a m p l e of 25 birth weights. If we s a m p l e repeatedly from the p o p u l a t i o n a n d o b t a i n 100 birth weights, we shall p r o b a b l y have to place some of these points on t o p of o t h e r points in o r d e r to record them all correctly (Figure 2.1 B). As we c o n t i n u e s a m p l i n g additional h u n d r e d s a n d t h o u s a n d s of birth weights (Figure 2.1C a n d D), the assemblage of points will c o n t i n u e to increase in size but will a s s u m e a fairly definite shape. The outline of the m o u n d of p o i n t s a p p r o x i m a t e s the distribution of the variable. R e m e m b e r thai a c o n t i n u o u s variable such as birth weight can a s s u m e an infinity of values between a n y t w o p o i n l s on the abscissa. T h e refinement of o u r m e a s u r e m e n t s will d e t e r m i n e how fine the n u m b e r of recorded divisions between any t w o p o i n t s a l o n g the axis will be. T h e distribution of a variable is of c o n s i d e r a b l e biological interest. If we find thai the disl ribution is asymmetrical a n d d r a w n out in one direction, it tells us that there is. perhaps, selection that causes o r g a n i s m s to fall preferentially in o n e of the tails of the distribution, or possibly that the scale of m e a s u r e m e n t

15

2.5 / FREQUENCY DISTRIBUTIONS

/

10 l·

25

I III· I . ll.l I.

0 10

100 .

I. ...il..i .i.llililillilHi hull Li. 11 ι

,ι,ι

30 r

20

10 500

i I

0

L

70

60

50

2000

40

30

20

10

n i " l ·• -t.l

• ll. 11 111

0 60

70

80

90

100

110

120

130

140

150

160

Birth w e i g h t (oz) f i g u r k 2.1 S a m p l i n g from ;i p o p u l a t i o n of birth weights of i n f a n t s (a c o n t i n u o u s variable). Λ. Λ s a m p l e of 25. Β. Λ s a m p l e of KM). C. A s a m p l e of 500. D. Λ s a m p l e of 2(XX).

CHAPTER 2 / DATA IN BIOSTATISTICS

16 200

£

150 -

JJ

2 £ I

-

_

FIGURE 2 . 2

'""

B a r d i a g r a m . F r e q u e n c y of t h e sedge Car ex in 500 q u a d r a t s . D a t a f r o m T a b l e 2.2; orginally f r o m A r c h i b a l d (1950).

flacca

0

1 2

3

4

5

(i

7

S

N u m b e r of p l a n t s q u a d r a t

chosen is such as to bring a b o u t a distortion of the distribution. If, in a s a m p l e of i m m a t u r e insects, we discover that the m e a s u r e m e n t s are b i m o d a l l y distributed (with t w o peaks), this would indicate that the p o p u l a t i o n is d i m o r p h i c . This m e a n s that different species or races m a y have become intermingled in o u r sample. O r the d i m o r p h i s m could have arisen f r o m the presence of b o t h sexes or of different instars. T h e r e are several characteristic shapes of frequency distributions. T h e most c o m m o n is the symmetrical bell shape ( a p p r o x i m a t e d by the b o t t o m g r a p h in Figure 2.1), which is the s h a p e of the n o r m a l frequency distribution discussed in C h a p t e r 5. T h e r e a r e also skewed d i s t r i b u t i o n s (drawn out m o r e at o n e tail than the other), L-shaped d i s t r i b u t i o n s as in Figure 2.2, U-shaped distributions, a n d others, all of which impart significant i n f o r m a t i o n a b o u t the relationships they represent. We shall have m o r e to say a b o u t the implications of various types of distributions in later c h a p t e r s a n d sections. After researchers have obtained d a t a in a given study, they must a r r a n g e the d a t a in a form suitable for c o m p u t a t i o n a n d interpretation. We m a y a s s u m e that variates arc r a n d o m l y ordered initially or are in the o r d e r in which the m e a s u r e m e n t s have been taken. A simple a r r a n g e m e n t would be an array of the d a t a by o r d e r of m a g n i t u d e . T h u s , for example, the variates 7, 6, 5, 7, 8, 9, 6, 7, 4, 6, 7 could be arrayed in o r d e r of decreasing m a g n i t u d e as follows: 9. 8, 7, 7, 7, 7, 6, 6, 6, 5, 4. W h e r e there are some variates of the same value, such as the 6's a n d 7's in this Fictitious example, a time-saving device might immediately have occurred to you namely, to list a frequency for each of the recurring variates; thus: 9, 8, 7(4 χ ). 6(3 χ ), 5, 4. Such a s h o r t h a n d n o t a t i o n is o n e way to represent a frcqucncy distribution, which is simply an a r r a n g e m e n t of the classes of variates with the frequency of each class indicated. C o n v e n t i o n a l l y , a frequency distribution is stated in t a b u l a r form; for our example, this is d o n e as follows:

17

2.5 / FREQUENCY DISTRIBUTIONS

Variable

Frequency

V

/

9 8 7 6 5 4

I 1 4 3 1 1

T h e a b o v e is a n example of a quantitative frequency distribution, since Y is clearly a m e a s u r e m e n t variable. However, a r r a y s a n d frequency distributions need not be limited to such variables. W e can m a k e frequency distributions of attributes, called qualitative frequency distributions. In these, the various classes are listed in some logical o r a r b i t r a r y order. F o r example, in genetics we might have a qualitative frequency distribution as follows: Phenolype

J

A-

86

an

32

This tells us that there are two classes of individuals, those identifed by the A phenotype, of which 86 were f o u n d , a n d those comprising the h o n i o z y g o t e recessive aa, of which 32 were seen in the sample. An example of a m o r e extensive qualitative frequency distribution is given in Table 2.1, which s h o w s the distribution of m e l a n o m a (a type of skin cancer) over b o d y regions in men a n d w o m e n . This table tells us t h a t the t r u n k a n d limbs are the most frequent sites for m e l a n o m a s and that the buccal cavity, the rest of the gastrointestinal tract, and the genital tract are rarely afflicted by this

ΤΛΒΙ Κ 2.1

Two qualitative frequency distributions. N u m b e r of cases of skin c a n c e r ( m e l a n o m a l d i s t r i b u t e d over b o d y regions of 4599 men a n d 47X6 w o m e n .

Anatomic

site

Mead a n d neck T r u n k and limbs Buccal cavity Rest of g a s t r o i n t e s t i n a l tract Genital tract F.ye Total cases Snune.

Oiilii from I cc

OhseiVi ϊ<1 frequency Men Women J ( 949

645

3243 8 5

3645 11

12

93

21

382

371

4599

4786

CHAPTER 2 / DATA IN BIOSTATISTICS

18 TABI.E 2.2 A

meristic

frequency

distribution.

N u m b e r of p l a n t s of the sedge flacca

Carex

f o u n d in 500 q u a d r a t s .

No. of plants per quadrat Y 0 1 2 3 4

Observed frequency f 181 118 97 54

5

32 9

6 7

5 3

8

1

Total

500

Source: Data from Archibald (1950).

type of cancer. We often e n c o u n t e r o t h e r examples of qualitative frequency d i s t r i b u t i o n s in ecology in the form of tables, o r species lists, of the i n h a b i t a n t s of a sampled ecological area. Such tables c a t a l o g the i n h a b i t a n t s by species o r at a higher t a x o n o m i c level a n d record the n u m b e r of specimens observed for each. T h e a r r a n g e m e n t of such tables is usually alphabetical, o r it m a y follow a special c o n v e n t i o n , as in some botanical species lists. A q u a n t i t a t i v e frequency distribution based on meristic variates is s h o w n in T a b l e 2.2. This is an example f r o m plant ecology: the n u m b e r of p l a n t s per q u a d r a t sampled is listed at the left in the variable c o l u m n ; the observed frequency is shown at the right. Q u a n t i t a t i v e frequency d i s t r i b u t i o n s based on a c o n t i n u o u s variable are the most c o m m o n l y e m p l o y e d frequency distributions; you should b e c o m e t h o r o u g h l y familiar with them. An e x a m p l e is s h o w n in Box 2.1. It is based on 25 femur lengths m e a s u r e d in an aphid p o p u l a t i o n . T h e 25 readings a r e s h o w n at the t o p of Box 2.1 in the o r d e r in which they were o b t a i n e d as m e a s u r e m e n t s . (They could have been arrayed a c c o r d i n g to their magnitude.) T h e d a t a arc next set up in a frequency distribution. T h e variates increase in m a g n i t u d e by unit steps of 0.1. T h e frequency distribution is prepared by entering each variate in turn on the scale a n d indicating a c o u n t by a conventional tally m a r k . W h e n all of (lie items have been tallied in the c o r r e s p o n d i n g class, the tallies are c o n verted into n u m e r a l s indicating frequencies in the next c o l u m n . Their sum is indicated by Σ / . W h a t have we achieved in s u m m a r i z i n g o u r d a t a ? T h e original 25 variates are now represented by only 15 classes. We find that variates 3.6, 3.8, and 4.3 have the highest frequencies. However, wc also n o t e that there arc several classes, such as 3.4 or 3.7, that are not represented by a single aphid. This gives the

2 . 5 / FREQUENCY DISTRIBUTIONS

19

entire frequency distribution a d r a w n - o u t a n d scattered a p p e a r a n c e . T h e reason for this is that we have only 25 aphids, t o o few to put into a frequency distribution with 15 classes. T o o b t a i n a m o r e cohesive and s m o o t h - l o o k i n g distribution, we have to c o n d e n s e our d a t a into fewer classes. This process is k n o w n as grouping of classes of frequency distributions; it is illustrated in Box 2.1 a n d described in the following p a r a g r a p h s . We should realize t h a t g r o u p i n g individual variates into classes of wider range is only an extension of the same process t h a t t o o k place w h e n we o b t a i n e d the initial m e a s u r e m e n t . T h u s , as we have seen in Section 2.3, w h e n we m e a s u r e an aphid and record its femur length as 3.3 units, we imply thereby that the true m e a s u r e m e n t lies between 3.25 a n d 3.35 units, but that we were u n a b l e t o measure to the second decimal place. In recording the m e a s u r e m e n t initially as 3.3 units, we estimated t h a t it fell within this range. H a d we estimated that it exceeded the value of 3.35, for example, we would have given it the next higher score, 3.4. Therefore, all the m e a s u r e m e n t s between 3.25 a n d 3.35 were in fact g r o u p e d into the class identified by the class mark 3.3. O u r class interval was 0.1 units. If we now wish to m a k e wider class intervals, we are d o i n g n o t h i n g but extending the r a n g e within which m e a s u r e m e n t s are placed into one class. Reference to Box 2.1 will m a k e this process clear. We g r o u p the d a t a twice in order to impress u p o n the reader the flexibility of the process. In the first example of grouping, the class interval has been doubled in width; that is, it has been m a d e to equal 0.2 units. If we start at the lower end, the implied class limits will now be f r o m 3.25 to 3.45, the limits for the next class from 3.45 to 3.65, a n d so forth. O u r next task is to find the class marks. This was quite simple in the frequency distribution s h o w n at the left side of Box 2.1, in which the original measurements were used as class marks. However, now we are using a class interval twice as wide as before, and the class m a r k s arc calculated by t a k i n g the midpoint of the new class intervals. T h u s , to find the class mark of the first class, we lake the midpoint between 3.25 and 3.45. which turns out to be 3.35. We note that the class m a r k has one m o r e decimal place than the original measurements. We should not now be led to believe that we have suddenly achieved greater precision. Whenever we designate a class interval whose last significant digit is even (0.2 in this case), the class mark will carry one m o r e decimal place than the original m e a s u r e m e n t s . O n the right side of the table in Box 2.1 the d a t a are grouped once again, using a class interval of 0.3. Because of the o d d last significant digit, the class mark now shows as m a n y decimal places as the original variates, the m i d p o i n t between 3.25 and 3.55 being 3.4. O n c e the implied class limits and the class mark for the first class have been correctly found, the others can be written d o w n bv inspection without any special c o m p u t a t i o n . Simply a d d the class interval repeatedly to each of the values. Thus, starting with the lower limit 3.25, by a d d i n g 0.2 wc obtain 3.45. 3.65, 3.X5, a n d so forth; similarly, for the class marks, we o b t a i n 3.35, 3.55, 3.75, and so forth. It should be o b v i o u s that the wider the class intervals, the m o r e c o m p a c t the d a t a become but also the less precise. However, looking at

BOX 2.1 Preparation of frequency distribution and grouping into fewer classes with wider class intervals. Twenty-five femur lengths of the aphid Pemphigus. Measurements are in m m χ 10~ \

Original

3.8 3.3 3.9 4.1 4.4

3.6 4.3 4.4 4.4 4.1

Original

measurements

4.3 3.9 3.8 4.5 3.6

frequency

Implied limits

3.5 4.3 4.7 3.6 4.2

4.3 3.8 3.6 3.8 3.9

Grouping into 8 classes of interval 0.2

distribution

Tally

Implied limits

Class mark

3.25-3.45

3.35

3.45-3.65

3.55

M

5

3.65-3.85

3.75

mi

4

3.85-4.05

3.95

(Κ

3

4.05-4.25

4.15

III

3

marks

3.25-3.35

3.3

1

3.35-3.45

3.4

0

3.45-3.55

3.5

1

3.55-3.65

3.6

4

3.65-3.75

3.7

0

3.75-3.85

3.8

4

3.85-3.95

3.9

3

3.95-4.05

4.0

0

4.05-4.15

4.1

2

4.15-4.25

4.2

1

425-4.35

4.3

4

4.35-4.45

4.4

3

4.45-4.55

4.5

1

4.55-4.65

4.6

0

4.65-4.75

4.7

If

1

Grouping into S classes of intend 0.3

Implied limits

Class mark

3.25-3.55

3.4

3.55-3.85

3.7

Jff|||

8

3.85-4.15

4.0

j^ff

5

4.15-4.45

4.3

J|ff|||

8

4.45-4.75

4.6

Tally marks

|

1

Tally marks

f

7

4.25-4.45

4.35

4.45-4.65

4.55

|

1

4.65-4.85

4.75

|

_1

25

25

25

Source: Data from R, R. Sokal.

Histogram of the original frequency distribution shown above and of the grouped distribution with 5 classes. Line below abscissa shows class marks for the grouped frequency distribution. Shaded bars represent original frequency distribution; hollow bars represent grouped distribution.

10 r

_] 3.4

I 1 1 i— 3.7 4.0 4.3 4.6 Y (femur length, in units of 0.1 mm)

For a detailed account of the process of grouping, see Section 2.5. Ji

CHAPTER 2 / DATA IN BIOSTATISTICS

22

the frequency d i s t r i b u t i o n of a p h i d f e m u r lengths in Box 2.1, we notice that the initial r a t h e r chaotic s t r u c t u r e is being simplified by grouping. W h e n we g r o u p the frequency distribution into five classes with a class interval of 0.3 units, it b e c o m e s n o t a b l y b i m o d a l (that is, it possesses t w o peaks of frequencies). In setting up frequency distributions, f r o m 12 to 20 classes should be established. This rule need not be slavishly a d h e r e d to, but it should be e m p l o y e d with some of the c o m m o n sense that comes f r o m experience in h a n d l i n g statistical d a t a . T h e n u m b e r of classes d e p e n d s largely on the size of the s a m p l e studied. Samples of less t h a n 40 o r 50 should rarely be given as m a n y as 12 classes, since that w o u l d provide t o o few frequencies per class. O n the o t h e r h a n d , samples of several t h o u s a n d m a y profitably be g r o u p e d into m o r e t h a n 20 classes. If the a p h i d d a t a of Box 2.1 need to be g r o u p e d , they should p r o b a b l y not be g r o u p e d into m o r e t h a n 6 classes. If the original d a t a p r o v i d e us with fewer classes than we think we should have, then n o t h i n g can be d o n e if the variable is meristic, since this is the n a t u r e of the d a t a in question. H o w e v e r , with a c o n t i n u o u s variable a scarcity of classes w o u l d indicate that we p r o b a b l y had not m a d e our m e a s u r e m e n t s with sufficient precision. If we h a d followed the rules on n u m b e r of significant digits for m e a s u r e m e n t s stated in Section 2.3, this could not have h a p p e n e d . W h e n e v e r we c o m e u p with m o r e t h a n the desired n u m b e r of classes, g r o u p ing should be u n d e r t a k e n . W h e n the d a t a are meristic, the implied limits of c o n t i n u o u s variables are meaningless. Yet with m a n y meristic variables, such as a bristle n u m b e r varying f r o m a low of 13 to a high of 81, it would p r o b a b l y be wise to g r o u p the variates into classes, each c o n t a i n i n g several counts. This can best be d o n e by using an o d d n u m b e r as a class interval so that the class m a r k representing the d a t a will be a whole rather than a fractional n u m b e r . T h u s , if we were to g r o u p the bristle n u m b e r s 13, 14, 15, a n d 16 into o n e class, the class m a r k would have to be 14.5, a meaningless value in terms of bristle n u m b e r . It would therefore be better to use a class ranging over 3 bristles or 5 bristles, giving the integral value 14 or 15 as a class m a r k . G r o u p i n g data into frequency d i s t r i b u t i o n s was necessary when c o m p u tations were d o n e by pencil a n d paper. N o w a d a y s even t h o u s a n d s of variatcs can be processed efficiently by c o m p u t e r without prior grouping. However, frequency d i s t r i b u t i o n s are still extremely useful as a tool for d a t a analysis. This is especially true in an age in which it is all l o o easy for a researcher to o b t a i n a numerical result f r o m a c o m p u t e r p r o g r a m without ever really e x a m i n i n g the d a t a for outliers or for o t h e r ways in which the sample m a y not c o n f o r m to the a s s u m p t i o n s of the statistical m e t h o d s . Rather t h a n using tally m a r k s to set u p a frequency distribution, as was d o n e in Box 2.1, we can e m p l o y T u k e y ' s stem-and-leaf display. This t e c h n i q u e is an i m p r o v e m e n t , since it not only results in a frequency distribution of the variates of a sample but also permits easy checking of the variates a n d o r d e r i n g them into an array (neither of which is possible with tally marks). This technique will therefore be useful in c o m p u t i n g the m e d i a n of a sample (see Section 3.3) a n d in c o m p u t i n g various tests that require ordered a r r a y s of the sample variates c . . , . ι Ι Ο Ί

..ηΛ

t Ί

23

2.5 / FREQUENCY DISTRIBUTIONS

T o learn how to construct a stem-and-leaf display, let us look a h e a d to Table 3.1 in the next c h a p t e r , which lists 15 b l o o d n e u t r o p h i l c o u n t s . T h e unordered m e a s u r e m e n t s are as follows: 4.9, 4.6, 5.5, 9.1, 16.3, 12.7, 6.4, 7.1, 2.3, 3.6, 18.0, 3.7, 7.3, 4.4, a n d 9.8. T o p r e p a r e a stem-and-leaf display, we scan the variates in the s a m p l e to discover the lowest a n d highest leading digit or digits. Next, we write d o w n the entire range of leading digits in unit increments to the left of a vertical line (the "stem"), as s h o w n in the a c c o m p a n y i n g illustration. We then put the next digit of the first variate (a "leaf") at that level of the stem c o r r e s p o n d i n g to its leading digit(s). T h e first o b s e r v a t i o n in o u r s a m p l e is 4.9. W e therefore place a 9 next to the 4. T h e next variate is 4.6. It is entered by finding the stem level for the leading digit 4 a n d recording a 6 next to the 9 that is already there. Similarly, for the third variate, 5.5, we record a 5 next to the leading digit 5. W e c o n t i n u e in this way until all 15 variates have been entered (as "leaves") in sequence a l o n g the a p p r o p r i a t e leading digits of the stem. The completed a r r a y is the equivalent of a frequency distribution a n d has the a p p e a r a n c e of a histogram or bar d i a g r a m (see the illustration). M o r e o v e r , it permits the efficient o r d e r i n g of the variates. T h u s , f r o m the c o m p l e t e d array it becomes o b v i o u s that the a p p r o p r i a t e o r d e r i n g of the 15 variates is 2.3, 3.6, 3.7, 4.4, 4.6, 4.9, 5.5, 6.4, 7.1, 7.3, 9.1, 9.8, 12.7, 16.3, 18.0. T h e m e d i a n can easily be read off the stem-and-leaf display. It is clearly 6.4. F o r very large samples, stem-and-leaf displays m a y b e c o m e a w k w a r d . In such cases a conventional frequency distribution as in Box 2.1 w o u l d be preferable.

Step I

Step 2

Ί

Ί

3 4 5 6 7 X 9

3 9

4 5 6 7 X 9

...

Step 7

.. .

Completed (Step

7 96

3 4 5 6 7 X 9 10

96 5 4

1

3 4 5 6 7 X 9

3 67 964 5 4

13 IX

10 11

11

10 11

12

12

12

13

13

13

13

14

14

16

15 16

14 15

14

15

16

17

17

17

17

IX

IX

18

IX

10

array )

11 7

12

7

15 3

16

3 0

W h e n the shape of a frequency distribution is of particular interest, wish to present the distribution in graphic form when discussing the This is generally d o n e by m e a n s of frequency d i a g r a m s , of which there c o m m o n types. F o r a distribution of meristic d a t a we e m p l o y a bar

wc may results. are two diagram.

24

CHAPTER 2 / DATA IN BIOSTATISTICS

FIGURE 2 . 3

F r e q u e n c y polygon. Birth weights of 9465 males infants. C h i n e s e third-class p a t i e n t s in S i n g a p o r e , 1950 a n d 1951. D a t a f r o m Millis a n d Seng (1954).

the variable (in o u r case, the n u m b e r of p l a n t s per q u a d r a t ) , a n d the o r d i n a t e represents the frequencies. T h e i m p o r t a n t point a b o u t such a d i a g r a m is t h a t the bars d o not t o u c h each other, which indicates that the variable is not c o n tinuous. By contrast, c o n t i n u o u s variables, such as the frequency distribution of the femur lengths of a p h i d stem m o t h e r s , are g r a p h e d as a histogram. In a h i s t o g r a m the width of each bar a l o n g the abscissa represents a class interval of the frequency distribution a n d the bars t o u c h each other to s h o w that the actual limits of the classes a r e contiguous. T h e m i d p o i n t of the bar c o r r e s p o n d s to the class mark. At the b o t t o m of Box 2.! are shown h i s t o g r a m s of the frequency distribution of the aphid data, u n g r o u p e d a n d g r o u p e d . T h e height of each bar represents the frequency of the c o r r e s p o n d i n g class. T o illustrate that h i s t o g r a m s are a p p r o p r i a t e a p p r o x i m a t i o n s to the c o n t i n u o u s distributions f o u n d in nature, we may take a histogram a n d m a k e the class intervals m o r e n a r r o w , p r o d u c i n g m o r e classes. T h e h i s t o g r a m would then clearly have a closer fit to a c o n t i n u o u s distribution. We can c o n t i n u e this p r o cess until the class intervals b e c o m e infinitesimal in width. At this point the h i s t o g r a m becomes the c o n t i n u o u s distribution of the variable. Occasionally the class intervals of a g r o u p e d c o n t i n u o u s frequency distribution are unequal. For instance, in a frequency distribution of ages we might have m o r e detail on the different ages of y o u n g individuals and less a c c u r a t e identification of the ages of old individuals. In such cases, the class intervals lor the older age g r o u p s would be wider, those lor the y o u n g e r age groups, narrower. In representing such d a t a , the bars of the histogram are d r a w n with different widths. f igure 2.3 shows a n o t h e r graphical m o d e of representation of a frequency distribution of a c o n t i n u o u s variable (in this case, birth weight in infants). As we shall see later the shapes of distributions seen in such frequency polygons can reveal much a b o u t the biological situations affecting the given variable.

2.6 The handling of data D a t a must be handled skillfully a n d expeditiously so that statistics can be practiced successfully. Readers should therefore a c q u a i n t themselves with (he var-

2.6 / THE HANDLING OF DATA

25

In this b o o k we ignore " p e n c i l - a n d - p a p e r " short-cut m e t h o d s for c o m p u t a tions, f o u n d in earlier t e x t b o o k s of statistics, since we a s s u m e that t h e s t u d e n t has access to a calculator or a c o m p u t e r . S o m e statistical m e t h o d s are very easy to use because special tables exist that provide answers for s t a n d a r d statistical problems; thus, almost no c o m p u t a t i o n is involved. An example is Finney's table, a 2-by-2 contingency table c o n t a i n i n g small frequencies that is used for the test of independence ( P e a r s o n a n d Hartley, 1958, T a b l e 38). F o r small problems, Finney's table can be used in place of Fisher's m e t h o d of finding exact probabilities, which is very tedious. O t h e r statistical techniques are so easy to carry out that no mechanical aids are needed. S o m e are inherently simple, such as the sign test (Section 10.3). O t h e r m e t h o d s are only a p p r o x i m a t e but can often serve the p u r p o s e adequately; for example, we m a y sometimes substitute an casy-to-evaluate m e d i a n (defined in Section 3.3) for the m e a n (described in Sections 3.1 a n d 3.2) which requires c o m p u t a t i o n . We can use m a n y new types of e q u i p m e n t to p e r f o r m statistical c o m p u t a t i o n s — m a n y m o r e than we could have when Introduction to Biostalistics was first published. T h e o n c e - s t a n d a r d electrically driven mechanical desk calculator has completely d i s a p p e a r e d . M a n y new electronic devices, f r o m small pocket calculators to larger d e s k - t o p c o m p u t e r s , have replaced it. Such dcvices are so diverse that we will not try to survey the field here. Even if we did, the rate of a d v a n c e in (his area would be so rapid that w h a t e v e r we might say would soon become obsolete. We c a n n o t really d r a w the line between the m o r e sophisticated electronic calculators, on the o n e h a n d , a n d digital c o m p u t e r s . T h e r e is no a b r u p t increase in capabilities between the more versatile p r o g r a m m a b l e calculators a n d the simpler m i c r o c o m p u t e r s , just as there is n o n e as we progress f r o m m i c r o c o m puters to m i n i c o m p u t e r s a n d so on up to the large c o m p u t e r s that o n e associates with the central c o m p u t a t i o n center of a large university or research l a b o r a t o r y . All can perform c o m p u t a t i o n s automatically a n d be controlled by a set of detailed instructions p r e p a r e d by the user. Most of these devices, including prog r a m m a b l e small calculators, are a d e q u a t e for all of the c o m p u t a t i o n s described in this book, even for large sets of d a t a . T h e m a t e r i a l in this b o o k c o n s i s t s of r e l a t i v e l y s t a n d a r d statistical c o m p u t a t i o n s that arc a v a i l a b l e in m a n y statistical p r o g r a m s . BI()Mstat : , : is a statistical s o f t w a r e p a c k a g e that i n c l u d e s most of the statistical m e t h o d s c o v e r e d in this b o o k . T h e ti.se of m o d e r n d a t a processing procedures has o n e inherent danger. O n e can all too easily cither feed in e r r o n e o u s d a t a or choose an i n a p p r o p r i a t e p r o g r a m . Users must select p r o g r a m s carefully to ensure that those p r o g r a m s perform the desired c o m p u t a t i o n s , give numerically reliable results, and are as free from e r r o r as possible. When using a p r o g r a m for the first time, one should test it using d a t a f r o m t e x t b o o k s with which o n e is familiar. S o m e p r o g r a m s * Ι·'οι· information or l<> order, contact Hxcter S o f t w a r e . Websile:hUp://www.cxelcrM>ilwaa\com. li-mail: siilcs(«'cxctcrsoflwai'e.com. ' t h e s e programs are compatible with Windows XI' and Vista.

CHAPTER 2 / DATA IN BIOSTATISTICS

26

a r e n o t o r i o u s because the p r o g r a m m e r has failed to g u a r d against excessive r o u n d i n g e r r o r s or o t h e r p r o b l e m s . Users of a p r o g r a m should carefully check the d a t a being analyzed so t h a t typing e r r o r s are not present. In a d d i t i o n , p r o g r a m s should help users identify a n d remove b a d d a t a values a n d should p r o v i d e them with t r a n s f o r m a t i o n s so that they can m a k e sure that their d a t a satisfy the a s s u m p t i o n s of various analyses.

Exercises 2.1

R o u n d t h e f o l l o w i n g n u m b e r s t o t h r e e s i g n i f i c a n t figures: 1 0 6 . 5 5 , 0 . 0 6 8 1 9 , 3 . 0 4 9 5 , 7815.01, 2.9149, a n d 20.1500. W h a t a r e t h e implied limits b e f o r e a n d after r o u n d ing? R o u n d these s a m e n u m b e r s t o o n e decimal place.

2.2

D i f f e r e n t i a t e b e t w e e n t h e f o l l o w i n g p a i r s of t e r m s a n d g i v e a n e x a m p l e o f e a c h , (a) S t a t i s t i c a l a n d b i o l o g i c a l p o p u l a t i o n s , ( b ) V a n a l e a n d i n d i v i d u a l , (c) A c c u r a c y a n d p r e c i s i o n ( r e p e a t a b i l i t y ) , ( d ) C l a s s i n t e r v a l a n d c l a s s m a r k , (e) B a r d i a g r a m a n d h i s t o g r a m , (f) A b s c i s s a a n d o r d i n a t e .

2.3

G i v e n 2 0 0 m e a s u r e m e n t s r a n g i n g f r o m 1.32 t o 2 . 9 5 m m , h o w w o u l d y o u g r o u p t h e m i n t o a f r e q u e n c y d i s t r i b u t i o n ? G i v e class limits a s well a s c l a s s m a r k s .

2.4

G r o u p t h e f o l l o w i n g 4 0 m e a s u r e m e n t s of i n t e r o r b i t a l w i d t h of a s a m p l e o f d o m e s t i c p i g e o n s i n t o a f r e q u e n c y d i s t r i b u t i o n a n d d r a w its h i s t o g r a m ( d a t a f r o m O l s o n a n d M i l l e r , 1958). M e a s u r e m e n t s a r e in m i l l i m e t e r s .

A N S . F o r t h e first v a l u e : 107; 1 0 6 . 5 4 5

2.5

2.6

2.7

12.2

12.9

11.8

11.9

10.7 12.1 10.8

1 1.5

1 1.3 10.4 10.4

11.2 10.7 10.7

11.9 11.6

106.555; 1 0 6 . 5 - 1 0 7 . 5 ; 106.6

11.6 1 1.6

11.1 11.9

10.8 12.0

11.0 12.4

12.3

12.2

11.8

13.3 11.9 11.7

11.2 10.2 11.8

10.5 10.9 1 1.3

11.8 11.1 11.6 11.1

H o w p r e c i s e l y s h o u l d y o u m e a s u r e t h e w i n g l e n g t h of a s p e c i e s of m o s q u i t o e s in a s t u d y of g e o g r a p h i c v a r i a t i o n if t h e s m a l l e s t s p c c i m c n h a s a l e n g t h of a b o u t 2.8 m m a n d t h e l a r g e s t a l e n g t h of a b o u t 3.5 mm'. 1 T r a n s f o r m t h e 4 0 m e a s u r e m e n t s in E x e r c i s e 2.4 i n l o c o m m o n l o g a r i t h m s ( u s e a t a b i c o r c a l c u l a t o r ) a n d m a k e a f r e q u e n c y d i s t r i b u t i o n of t h e s e t r a n s f o r m e d v a r i a t e s . C o m m e n t o n t h e r e s u l t i n g c h a n g e in t h e p a t t e r n of t h e f r e q u e n c y d i s tribution from that found before f o r t h e d a t a of T a h l e s 2.1 a n d 2.2 i d e n t i f y t h e i n d i v i d u a l o b s e r v a t i o n s , s a m p l e s , populations, and variables.

2.8

M a k e a s t e m - a n d - l c a f d i s p l a y of t h e d a t a g i v e n in E x c r c i s c 2.4.

2.9

T h e d i s t r i b u t i o n o f a g e s of s t r i p e d b a s s c a p t u r e d by h o o k a n d l i n e f r o m t h e E a s t R i v e r a n d t h e H u d s o n R i v e r d u r i n g 1 9 8 0 w e r e r e p o r t e d a s f o l l o w s ( Y o u n g , 1981):

A
I

1

13

2

49

3 4

96 28

5

16

6

X

S h o w t h i s d i s t r i b u t i o n in t h e f o r m of a b a r d i a g r a m .

CHAPTER

Descriptive

Statistics

An early a n d f u n d a m e n t a l stage in any seienec is the descriptive stage. Until p h e n o m e n a c a n be accurately described, a n analysis of their causes is p r e m a t u r e . T h e question " W h a t ? " comes before " H o w ? " Unless we k n o w s o m e t h i n g a b o u t the usual distribution of the sugar c o n t e n t of blood in a p o p u l a t i o n of guinea pigs, as well as its fluctuations f r o m day to d a y a n d within days, we shall be unable to ascertain the effect of a given dose of a d r u g u p o n this variable. In a sizable s a m p l e it w o u l d be tedious to o b t a i n o u r knowledge of the material by c o n t e m p l a t i n g each individual o b s e r v a t i o n . W e need s o m e f o r m of s u m m a r y to permit us to deal with the d a t a in m a n a g e a b l e form, as well as to be able to share o u r findings with o t h e r s in scientific talks a n d publications. A hist o g r a m or bar d i a g r a m of the frequency distribution would be o n e type of s u m m a r y . However, for most purposes, a numerical s u m m a r y is needed to describe concisely, yet accurately, t h e properties of the o b s e r v e d frequency distribution. Q u a n t i t i e s p r o v i d i n g such a s u m m a r y are called descriptive statistics. This c h a p t e r will i n t r o d u c e you to some of them a n d s h o w how they arc c o m p u t e d . T w o kinds of descriptive statistics will be discussed in this c h a p t e r : statistics of location and statistics of dispersion. T h e statistics of location (also k n o w n as

28

CHAPTER 3 /' DESCRIPTIVE STATISTICS

measures of central tendency) describe the position of a sample along a given dimension representing a variable. F o r example, after we m e a s u r e t h e length of the a n i m a l s within a sample, we will then w a n t to k n o w w h e t h e r the a n i m a l s a r e closer, say, to 2 cm o r to 20 cm. T o express a representative value for t h e s a m p l e of o b s e r v a t i o n s — f o r the length of the a n i m a l s — w e use a statistic of location. But statistics of location will n o t describe the s h a p e of a frequency distribution. T h e s h a p e m a y be long or very n a r r o w , m a y be h u m p e d or Us h a p e d , m a y c o n t a i n t w o h u m p s , or m a y be m a r k e d l y asymmetrical. Q u a n t i tative m e a s u r e s of such aspects of frequency distributions a r e required. T o this e n d we need to define a n d study t h e statistics of dispersion. T h e a r i t h m e t i c m e a n , described in Section 3.1, is u n d o u b t e d l y the most i m p o r t a n t single statistic of location, but o t h e r s (the geometric m e a n , the h a r m o n i c mean, the m e d i a n , a n d the m o d e ) are briefly m e n t i o n e d in Sections 3.2, 3.3, a n d 3.4. A simple statistic of dispersion (the range) is briefly n o t e d in Section 3.5, a n d the s t a n d a r d deviation, the most c o m m o n statistic for describing dispersion, is explained in Section 3.6. O u r first e n c o u n t e r with c o n t r a s t s between s a m p l e statistics a n d p o p u l a t i o n p a r a m e t e r s occurs in Section 3.7, in c o n n e c t i o n with statistics of location a n d dispersion. In Section 3.8 there is a description of practical m e t h o d s for c o m p u t i n g the m e a n a n d s t a n d a r d deviation. T h e coefficient of variation (a statistic that permits us to c o m p a r e the relative a m o u n t of dispersion in different samples) is explained in the last section (Section 3.9). T h e techniques that will be at y o u r disposal after you have mastered this c h a p t e r will not be very powerful in solving biological problems, but they will be indispensable tools for any further w o r k in biostatistics. O t h e r descriptive statistics, of b o t h location and dispersion, will be taken up in later chapters. An important note: We shall first e n c o u n t e r the use of l o g a r i t h m s in this c h a p t e r . T o avoid c o n f u s i o n , c o m m o n logarithms have been consistently abbreviated as log, a n d n a t u r a l l o g a r i t h m s as In. T h u s , log \ m e a n s l o g , 0 χ a n d In v m e a n s log,, x.

3.1 The arithmetic mean T h e most c o m m o n statistic of location is familiar to everyone. It is the arithmetic mean, c o m m o n l y called the mean or average. T h e m e a n is calculated by s u m m i n g all the individual o b s e r v a t i o n s or items of a s a m p l e and dividing this s u m by the n u m b e r of items in the sample. F o r instance, as the result of a gas analysis in a respirometer an investigator o b t a i n s the following four readings of oxygen percentages a n d s u m s them: 14.9 10.8

12.3 23.3 Sum

=- 6 1 7 3

3.1 / THE ARITHMETIC MEAN

29

T h e investigator calculates the m e a n oxygen percentage as the s u m of the four items divided by the n u m b e r of items. T h u s the average oxygen p e r c e n t a g e is Mean =

15.325%

Calculating a m e a n presents us with the o p p o r t u n i t y for learning statistical symbolism. W e have already seen (Section 2.2) t h a t a n individual o b s e r v a t i o n is symbolized by Y|·, which s t a n d s for t h e ith o b s e r v a t i o n in t h e sample. F o u r observations could be written symbolically as follows: Υ»

v2,

Y3,

^

W e shall define n, t h e sample size, as the n u m b e r of items in a sample. In this particular instance, the sample size η is 4. T h u s , in a large sample, we c a n symbolize the a r r a y f r o m the first to the nth item as follows: Yl,

Υ2,.·.,Υη

W h e n we wish to s u m items, we use the following n o t a t i o n : Σ"

Yi

= y> +

y

2 + ·· ·+

η

i= 1

T h e capital Greek sigma, Σ, simply m e a n s the sum of the items indicated. T h e i = 1 m e a n s that the items should be s u m m e d , starting with the first o n e a n d ending with the nth one, as indicated by the i = η a b o v e the Σ. T h e subscript a n d superscript are necessary to indicate how m a n y items s h o u l d be s u m m e d . T h e "/ = " in the superscript is usually o m i t t e d as superfluous. F o r instance, if we h a d wished t o s u m only the first three items, we would have written Σ?=, Y{. O n the o t h e r h a n d , h a d we wished to sum all of them except the first one, we would have written Σ " = 2 ν ; . W i t h some exceptions (which will a p p e a r in later chapters), it is desirable to omit subscripts a n d superscripts, which generally add to the a p p a r e n t complexity of the f o r m u l a and, when they are unnecessary, distract the s t u d e n t ' s a t t e n t i o n f r o m the i m p o r t a n t relations expressed by the formula. Below are seen increasing simplifications of the c o m p l e t e s u m m a t i o n n o t a t i o n shown at the extreme left:

Σ 1 Yi = ιΣ 1 γί

!

= Σγ< ;

= Σ

γ

=

Σ

γ

T h e third symbol might be interpreted as meaning, " S u m the Y t 's over all available values of /." This is a frequently used n o t a t i o n , a l t h o u g h we shall not employ it in this b o o k . T h e next, with η as a superscript, tells us to sum η items of V; note (hat the i subscript of the Y has been d r o p p e d as unnecessary. Finally, the simplest n o t a t i o n is s h o w n at the right. It merely says sum the Vs. This will be the form we shall use most frequently: if a s u m m a t i o n sign precedes a variable, the s u m m a t i o n will be u n d e r s t o o d to be over η items (all the items in the sample) unless subscripts or superscripts specifically tell us otherwise.

CHAPTER 3 /' DESCRIPTIVE STATISTICS

30

W e shall use the s y m b o l Y for the a r i t h m e t i c m e a n of the variable Y. Its f o r m u l a is ^written as follows: y y L Y = = ~YY η

(3.1)

η""

This f o r m u l a tells us, " S u m all the («) items a n d divide the s u m by n." T h e mean of a sample is the center of gravity of the obsen'ations in the sample. If you were to d r a w a h i s t o g r a m of an observed frequency d i s t r i b u t i o n o n a sheet of c a r d b o a r d a n d then cut out the h i s t o g r a m a n d lay it flat against a b l a c k b o a r d , s u p p o r t i n g it with a pencil b e n e a t h , chances a r e t h a t it would be out of balance, t o p p l i n g to either the left o r the right. If you m o v e d the s u p p o r t i n g pencil p o i n t to a position a b o u t which the h i s t o g r a m w o u l d exactly balance, this point of b a l a n c e would c o r r e s p o n d to the a r i t h m e t i c m e a n . W e often m u s t c o m p u t e averages of m e a n s or of o t h e r statistics that m a y differ in their reliabilities because they are based on different sample sizes. At o t h e r times we m a y wish the individual items to be averaged to have different weights or a m o u n t s of influence. In all such cases we c o m p u t e a weighted average. A general f o r m u l a for calculating the weighted average of a set of values Yt is as follows:

t

=

(3.2)

Σ »·.w h e r e η variates, each weighted by a factor w„ are being averaged. T h e values of Yi in such cases are unlikely to represent variates. They are m o r e likely to be s a m p l e m e a n s Yt or s o m e o t h e r statistics of different reliabilities. T h e simplest case in which this arises is when the V, are not individual variates but are means. T h u s , if the following three m e a n s are based on differing s a m p l e sizes, as shown,

>;

n,

3.85 5.21 4.70

12 25 Η

their weighted average will be -

=

(12)(3.85) + (25)(5.2I| + (8X4.70) 12T25 1 S

=

214.05 45

N o t e that in this example, c o m p u t a t i o n of Ihc weighted mean is exactly equivalent to a d d i n g up all the original m e a s u r e m e n t s a n d dividing the sum by the total n u m b e r of the m e a s u r e m e n t s . Thus, the s a m p l e with 25 observations, having the highest m e a n , will influence the weighted average in p r o p o r t i o n to ils size.

31

3.2 / OTHER MEANS

3.2 Other means W e shall see in C h a p t e r s 10 a n d 11 t h a t variables are s o m e t i m e s t r a n s f o r m e d into their l o g a r i t h m s or reciprocals. If we calculate the m e a n s of such transformed variables a n d then c h a n g e the m e a n s back into the original scale, these m e a n s will not be the s a m e as if we h a d c o m p u t e d the arithmetic m e a n s of t h e original variables. T h e resulting m e a n s have received special n a m e s in statistics. T h e b a c k - t r a n s f o r m e d m e a n of the logarithmically t r a n s f o r m e d variables is called the geometric mean. It is c o m p u t e d as follows: GMv = antilog - Υ log Y

(3.3)

η

which indicates that the geometric m e a n GMr is the a n t i l o g a r i t h m of the m e a n of the l o g a r i t h m s of variable Y. Since a d d i t i o n of logarithms is equivalent t o multiplication of their antilogarithms, there is a n o t h e r way of representing this quantity; it is GMY = ^Y^YiT77Yn

(3.4)

T h e geometric m e a n p e r m i t s us to b e c o m e familiar with a n o t h e r o p e r a t o r symbol: capital pi, Π , which m a y be read as " p r o d u c t . " Just as Σ symbolizes s u m m a t i o n of the items that follow it, so Π symbolizes the multiplication of the items that follow it. T h e subscripts a n d superscripts have exactly the same m e a n i n g as in the s u m m a t i o n case. T h u s , Expression (3.4) for the geometric m e a n can be rewritten m o r e c o m p a c t l y as follows: GMr=nY\Yi

(3.4a) I

T h e c o m p u t a t i o n of the geometric m e a n by Expression (3.4a) is quite In practice, the geometric m e a n has to be c o m p u t e d by t r a n s f o r m i n g the into logarithms. The reciprocal of the arithmetic m e a n of reciprocals is called the mean. If we symbolize it by HY, the f o r m u l a for the h a r m o n i c m e a n written in concise form (without subscripts a n d superscripts) as 1

1 „

tedious. variates harmonic can be

1

You may wish to convince yourself that the geometric mean a n d the h a r m o n i c m e a n of the four oxygen percentages are 14.65% a n d 14.09%, respectively. U n less the individual items d o not vary, the geometric m e a n is always less than the arithmetic m e a n , and the h a r m o n i c m e a n is always less t h a n the geometric mean. S o m e beginners in statistics have difficulty in accepting the fact that measures of location or central tendency o t h e r t h a n the arithmetic m e a n are permissible or even desirable. T h e y feel that the arithmetic m e a n is the "logical"

32

CHAPTER 3 /' DESCRIPTIVE STATISTICS

average, a n d that any o t h e r m e a n would be a distortion. This whole p r o b l e m relates t o the p r o p e r scale of m e a s u r e m e n t for representing d a t a ; this scale is not always the linear scale familiar to everyone, but is sometimes by preference a logarithmic or reciprocal scale. If you have d o u b t s a b o u t this question, we shall try to allay t h e m in C h a p t e r 10, where we discuss the reasons for t r a n s f o r m i n g variables.

3.3 The median T h e median Μ is a statistic of location occasionally useful in biological research. It is defined as that value of the variable (in an o r d e r e d array) that has an equal number of items on either side of it. Thus, the m e d i a n divides a frequency distribution into two halves. In the following sample of five m e a s u r e m e n t s , 14, 15, 16, 19, 23 Μ ~ 16, since the third o b s e r v a t i o n has an equal n u m b e r of o b s e r v a t i o n s on b o t h sides of it. We can visualize the m e d i a n easily if we think of an a r r a y f r o m largest t o s m a l l e s t — f o r example, a row of m e n lined u p by their heights. T h e m e d i a n individual will then be that m a n having an equal n u m b e r of m e n on his right a n d left sides. His height will be the median height of the s a m ple considered. This quantity is easily evaluated f r o m a sample a r r a y with an o d d n u m b e r of individuals. W h e n the n u m b e r in the s a m p l e is even, the m e d i a n is conventionally calculated as the m i d p o i n t between the (n/2)th a n d the [(«/2) + 1 j t h variate. T h u s , for the s a m p l e of four m e a s u r e m e n t s 14, 15, 16, 19 the median would be the m i d p o i n t between the second and third items, or 15.5. Whenever any o n e value of a variatc occurs m o r e than once, p r o b l e m s may develop in locating the m e d i a n . C o m p u t a t i o n of the median item b e c o m e s m o r e involved because all the m e m b e r s of a given class in which the m e d i a n item is located will have the s a m e class m a r k . T h e median then is the {n/2)lh variate in the frequency distribution. It is usually c o m p u t e d as that point between the class limits of the m e d i a n class where the median individual would be located (assuming the individuals in the class were evenly distributed). T h e median is just o n e of a family of statistics dividing a frequency distribution into equal areas. It divides the distribution into two halves. T h e three quartiles cut the d i s t r i b u t i o n at the 25, 50, and 75% p o i n t s — t h a t is, at points dividing the distribution into first, second, third, and f o u r t h q u a r t e r s by area (and frequencies). T h e second quarlile is, of course, the median. (There are also quintiles, deciles, a n d percentiles, dividing the distribution into 5. 10, a n d 100 equal portions, respectively.) M e d i a n s arc most often used for d i s t r i b u t i o n s that d o not c o n f o r m to the s t a n d a r d probability models, so that n o n p a r a m e t r i c m e t h o d s (sec C h a p t e r 10) must be used. Sometimes (he median is a m o r e representative m e a s u r e of location than the a r i t h m e t i c m e a n . Such instances almost always involve a s y m m e t r i c

3.4 / THE MODE

33

distributions. An often q u o t e d example f r o m economics w o u l d be a suitable m e a s u r e of location for the "typical" salary of a n employee of a c o r p o r a t i o n . T h e very high salaries of the few senior executives would shift the arithmetic m e a n , the center of gravity, t o w a r d a completely unrepresentative value. T h e m e d i a n , on the o t h e r h a n d , would be little affected by a few high salaries; it w o u l d give the p a r t i c u l a r point o n the salary scale a b o v e which lie 50% of the salaries in the c o r p o r a t i o n , the o t h e r half being lower t h a n this figure. In biology an example of the preferred application of a m e d i a n over the arithmetic m e a n m a y be in p o p u l a t i o n s showing skewed distribution, such as weights. T h u s a m e d i a n weight of American males 50 years old m a y be a more meaningful statistic than the average weight. T h e m e d i a n is also of i m p o r t a n c e in cases where it m a y be difficult or impossible to o b t a i n a n d m e a s u r e all the items of a sample. F o r example, s u p p o s e an animal behaviorist is studying the time it takes for a s a m p l e of a n i m a l s to perform a certain behavioral step. T h e variable he is m e a s u r i n g is the time from the beginning of the experiment until each individual has performed. W h a t he w a n t s to o b t a i n is an average time of p e r f o r m a n c e . Such an average time, however, can be calculated only after records have been o b t a i n e d on all the individuals. It m a y t a k e a long lime for the slowest a n i m a l s to complete their p e r f o r m a n c e , longer t h a n the observer wishes to spend. (Some of them may never respond a p p r o p r i a t e l y , m a k i n g the c o m p u t a t i o n of a m e a n impossible.) Therefore, a convenient statistic of location to describe these a n i m a l s may be the median time of p e r f o r m a n c e . Thus, so long as the observer k n o w s what the total sample size is, he need not have m e a s u r e m e n t s for the right-hand tail of his distribution. Similar e x a m p l e s would be the responses to a d r u g or poison in a g r o u p of individuals (the median lethal or effective dose. LD 5 ( I or F.D S 0 ) or the median time for a m u t a t i o n to a p p e a r in a n u m b e r of lines of a species.

3.4 The mode T h e mode

r e f e r s t o the

value

represented

by

the

greatest

number

of

individuals.

When seen on a frequency distribution, the m o d e is the value of the variable at which the curve peaks. In grouped frequency distributions the m o d e as a point has little meaning. It usually sulliccs It) identify the m o d a l class. In biology, the m o d e does not have m a n y applications. Distributions having two peaks (equal or unequal in height) are called bimodal; those with m o r e than two peaks are multimodal. In those rare distributions that are U-shaped, we refer to the low point at the middle of the distribution as an

antimode.

In evaluating the relative merits of the arithmetic mean, the median, a n d the mode, a n u m b e r of c o n s i d e r a t i o n s have to be kept in mind. T h e m e a n is generally preferred in statistics, since it has a smaller s t a n d a r d e r r o r than o t h e r statistics of location (see Section 6.2), it is easier to work with mathematically, and it has an a d d i t i o n a l desirablc p r o p e r t y (explained in Section 6.1): it will tend to be normally distributed even if the original data are not. T h e mean is

CHAPTER 3 /' DESCRIPTIVE STATISTICS

34

20

η = 120

18

uh

Hi

14

12 c" ct-

10

U.

8

:i. ι

:i.(i

;is

i.o

lVl"!'!']

1.2

1,1

!.(>

4.8

5,0

bul I r r f a t

HGURi·: 3.1 An a s y m m e t r i c a l f r e q u e n c y d i s t r i b u t i o n ( s k e w e d t o the right) s h o w i n g l o c a t i o n of t h e m e a n , m e d i a n , a n d m o d e . P e r c e n t b u t t e r f a t in 120 s a m p l e s of milk ( f r o m a C a n a d i a n c a t t l e b r e e d e r s ' r e c o r d b o o k ) .

m a r k e d l y affected by outlying observations; the m e d i a n and m o d e are not. T h e mean is generally m o r e sensitive to c h a n g e s in the s h a p e of a frequency distribution, a n d if it is desired to have a statistic reflecting such changes, the m e a n may be preferred. In symmetrical, u n i m o d a l d i s t r i b u t i o n s the mean, the median, a n d the m o d e are all identical. A prime example of this is the well-known n o r m a l distribution of C h a p t e r 5. In a typical asymmetrical d i s t r i b u t i o n , such as the o n e s h o w n in Figure 3.1, the relative positions of the mode, median, and mean are generally these: the mean is closest to the d r a w n - o u t tail of the distribution, the m o d e is farthest, and the m e d i a n is between these. An easy way to r e m e m b e r this seq u e n c e is to recall that they occur in alphabetical o r d e r from the longer tail of t h e distribution.

3.5 The ran}>e We now turn to measures of dispersion, f igure 3.2 d e m o n s t r a t e s that radically different-looking distributions may possess the identical arithmetic mean. It is

35

3.5 / THE RANGE

10 8

6 4 2

0 10 8

£α; 6 αϊ 4

Uh

2

0 10

8 1

(i

I

0 FIGURE 3 . 2

T h r e e frequency d i s t r i b u t i o n s h a v i n g identical m e a n s a n d s a m p l e si/.es but differing in dispersion pattern.

O n e simple m e a s u r e of dispersion is the range, which is defined as the difference

between

the

largest

and

the

smallest

items

in a sample.

Thus, the

range

of the four oxygen percentages listed earlier (Section 3.1) is R a n g e = 23.3 - 10.8 = 12.5";, a n d the range of the a p h i d femur lengths (Box 2.1) is Range = 4.7 - 3.3 = 1.4 units of 0.1 m m Since the range is a m e a s u r e of the s p a n of the variates a l o n g the scale of the variable, it is in the same units as the original m e a s u r e m e n t s . T h e range is clearly affected by even a single outlying value a n d for this reason is only a rnuoh estimate of the dtsriersion of all the items in the samtnle.

36

CHAPTER 3 /' DESCRIPTIVE STATISTICS

3.6 The standard deviation W e desire t h a t a m e a s u r e of dispersion t a k e all items of a d i s t r i b u t i o n i n t o c o n s i d e r a t i o n , weighting e a c h item by its distance f r o m the center of the distrib u t i o n . W e shall n o w try t o c o n s t r u c t such a statistic. In T a b l e 3.1 we s h o w a s a m p l e of 15 b l o o d n e u t r o p h i l c o u n t s f r o m p a t i e n t s with t u m o r s . C o l u m n (1) s h o w s the variates in t h e o r d e r in which they were reported. T h e c o m p u t a t i o n of t h e m e a n is s h o w n below the table. T h e m e a n n e u t r o p h i l c o u n t t u r n s o u t to be 7.713. T h e distance of e a c h variate f r o m t h e m e a n is c o m p u t e d as t h e following deviation: y = Y - Y E a c h individual deviation, or deviate, is by c o n v e n t i o n c o m p u t e d as the individual o b s e r v a t i o n m i n u s t h e m e a n , Υ — Ϋ, r a t h e r t h a n the reverse, Ϋ — Y. D e v i a t e s are symbolized by lowercase letters c o r r e s p o n d i n g to the capital letters of t h e variables. C o l u m n (2) in T a b l e 3.1 gives the deviates c o m p u t e d in this manner. W e n o w wish to calculate a n average d e v i a t i o n t h a t will s u m all t h e deviates and divide t h e m by the n u m b e r of deviates in the sample. But n o t e that when TABLE

3.1

The standard deviation. L o n g m e t h o d , not r e c o m m e n d e d for h a n d or c a l c u l a t o r c o m p u t a t i o n s but s h o w n here to illust r a t e t h e m e a n i n g of t h e s t a n d a r d deviation. T h e d a t a a r e b l o o d n e u t r o p h i l c o u n t s (divided by 1000) per microliter, in 15 p a t i e n t s with n o n h e m a t o l o g i c a l t u m o r s . (/) Y

y

(2) Υ - Y

(i)

y2

4.9

-2.81

7.9148

4.6 5.5

-3.11 -2.21

9.6928 4.8988

9.1 16.3 12.7 6.4

1.39 8.59 4.99

1.9228 73.7308

-1.31

24.8668 1.7248

-0.61 -5.41

0.3762 29.3042

-4.11 10.29

105.8155

-4.01 -0.41

16.1068 0.1708

-3.31 2.09

10.9782

9.8 15.7

0.05

308.7770

7.1 2.3 3.6 18.0 3.7 7.3 4.4

Total

=

Mean

Y

ΣΥ

16.9195

4.3542

I Is.7 -

7.713

37

3.7 / SAMPLE STATISTICS AND PARAMETERS

we s u m o u r deviates, negative a n d positive deviates cancel out, as is s h o w n by the s u m at the b o t t o m of c o l u m n (2); this sum a p p e a r s to be u n e q u a l to zero only because of a r o u n d i n g error. D e v i a t i o n s f r o m the a r i t h m e t i c m e a n always s u m to zero because the m e a n is the center of gravity. C o n s e q u e n t l y , an average based o n the s u m of deviations w o u l d also always e q u a l zero. Y o u are urged to study A p p e n d i x A l . l , which d e m o n s t r a t e s that the s u m of deviations a r o u n d the m e a n of a s a m p l e is equal t o zero. S q u a r i n g t h e deviates gives us c o l u m n (3) of Table 3.1 a n d e n a b l e s us to reach a result o t h e r t h a n zero. (Squaring the deviates also h o l d s o t h e r m a t h e matical a d v a n t a g e s , which we shall t a k e u p in Sections 7.5 a n d 11.3.) T h e sum of the s q u a r e d deviates (in this case, 308.7770) is a very i m p o r t a n t q u a n t i t y in statistics. It is called t h e sum of squares a n d is identified symbolically as Σγ2. A n o t h e r c o m m o n symbol for the s u m of s q u a r e s is SS. T h e next step is t o o b t a i n the average of the η s q u a r e d deviations. T h e resulting q u a n t i t y is k n o w n as the variance, or the mean square'. Variance =

X>· 2 __ 308.7770 15

= 20.5851

T h e variance is a m e a s u r e of f u n d a m e n t a l i m p o r t a n c e in statistics, a n d we shall employ it t h r o u g h o u t this b o o k . At the m o m e n t , we need only r e m e m b e r that because of the s q u a r i n g of the deviations, the variance is expressed in squared units. T o u n d o the effect of the squaring, we now take the positive s q u a r e r o o t of the variance a n d o b t a i n the standard deviation:

Thus, s t a n d a r d deviation is again expressed in the original units of measurement, since it is a s q u a r e r o o t of the squared units of the variance. An important note: T h e technique just learned a n d illustrated in T a b l e 3.1 is not the simplest for direct c o m p u t a t i o n of a variance a n d s t a n d a r d deviation. However, it is often used in c o m p u t e r p r o g r a m s , where accuracy of c o m p u t a tions is an i m p o r t a n t consideration. Alternative a n d simpler c o m p u t a t i o n a l m e t h o d s are given in Section 3.8. T h e o b s e r v a n t reader m a y have noticed that we have avoided assigning any symbol to either the variance o r the s t a n d a r d deviation. We shall explain why in the next section. 3.7 Sample statistics and parameters U p to now we have calculated statistics f r o m samples without giving t o o m u c h t h o u g h t to what these statistics represent. W h e n correctly calculated, a m e a n and s t a n d a r d deviation will always be absolutely true measures of location a n d dispersion for the samples on which they are based. T h u s , the true m e a n of the four oxygen percentage readings in Section 3.1 is 15.325".",. T h e s t a n d a r d deviation of the 15 n e u t r o p h i l c o u n t s is 4.537. However, only rarely in biology (or f ι,Λ,-,ιί^,Λ .,,-,,ι ,ι;..

38

CHAPTER 3 /' DESCRIPTIVE STATISTICS

only as descriptive s u m m a r i e s of the samples we have studied. Almost always we are interested in the populations f r o m which t h e samples h a v e been t a k e n . W h a t we w a n t to k n o w is not the m e a n of the particular four oxygen precentages, but r a t h e r the t r u e oxgyen percentage of the universe of readings f r o m which the f o u r readings have been sampled. Similarly, we would like t o k n o w the true m e a n neutrophil c o u n t of the p o p u l a t i o n of patients with n o n h e m a t o l o g i c a l t u m o r s , n o t merely the m e a n of the 15 individuals m e a s u r e d . W h e n s t u d y i n g dispersion we generally wish to learn the true s t a n d a r d deviations of t h e p o p u lations a n d not those of t h e samples. These p o p u l a t i o n statistics, however, are u n k n o w n a n d (generally speaking) are u n k n o w a b l e . W h o would be able t o collect all the patients with this p a r t i c u l a r disease a n d m e a s u r e their n e u t r o p h i l c o u n t s ? T h u s we need to use sample statistics as e s t i m a t o r s of population statistics or parameters. It is c o n v e n t i o n a l in statistics to use G r e e k letters for p o p u l a t i o n p a r a m e t e r s a n d R o m a n letters for s a m p l e statistics. T h u s , the sample m e a n Ϋ estimates μ, the p a r a m e t r i c m e a n of the p o p u l a t i o n . Similarly, a sample variance, symbolized by s 2 , estimates a p a r a m e t r i c variance, symbolized by a 2 . Such e s t i m a t o r s should be unbiased. By this we m e a n that samples (regardless of the sample size) t a k e n f r o m a p o p u l a t i o n with a k n o w n p a r a m e t e r should give sample statistics that, when averaged, will give the p a r a m e t r i c value. An estimator that d o e s not d o so is called biased. T h e s a m p l e m e a n Ϋ is an unbiased e s t i m a t o r of the p a r a m e t r i c m e a n μ. H o w e v e r , the s a m p l e variance as c o m p u t e d in Section 3.6 is not unbiased. O n the average, it will u n d e r e s t i m a t e the m a g n i t u d e of the p o p u l a t i o n variance a 1 . T o o v e r c o m e this bias, m a t h e m a t i c a l statisticians have shoWn t h a t w h e n s u m s of squares are divided by π — 1 rather than by η the resulting s a m p l e variances will be unbiased estimators of the p o p u l a t i o n variance. F o r this reason, it is c u s t o m a r y to c o m p u t e variances by dividing the sum of squares by η — 1. T h e f o r m u l a for the s t a n d a r d deviation is therefore customarily given as follows: (3.6) In the n e u t r o p h i l - c o u n t d a t a the s t a n d a r d deviation would thus be c o m p u t e d as

We note that this value is slightly larger than o u r previous estimate of 4.537. Of course, the greater the s a m p l e size, the less difference there will be between division by η a n d by n I. However, regardless of sample size, it is good practice to divide a sum of s q u a r e s by η — 1 when c o m p u t i n g a variance or s t a n d a r d deviation. It m a y be assumed that when the symbol s2 is e n c o u n t e r e d , it refers to a variance o b t a i n e d by division of the sum of squares by the degrees of freedom, as the q u a n t i t y η — 1 is generally referred to. Division of the s u m of s q u a r e s by η is a p p r o p r i a t e only when the interest of the investigator is limited to the s a m p l e at h a n d a n d to its variance a n d

3.8 / PRACTICAL METHODS FOR COMPUTING MEAN AND STANDARD DEVIATION

39

s t a n d a r d deviation as descriptive statistics of the sample. This w o u l d be in c o n t r a s t t o using these as estimates of the p o p u l a t i o n p a r a m e t e r s . T h e r e are also the rare cases in which the investigator possesses d a t a on the entire p o p u lation; in such cases division by η is perfectly justified, because then the investigator is not e s t i m a t i n g a p a r a m e t e r but is in fact e v a l u a t i n g it. T h u s the variance of the wing lengths of all a d u l t w h o o p i n g cranes w o u l d b e a p a r a m e t r i c value; similarly, if the heights of all winners of the N o b e l Prize in physics h a d been m e a s u r e d , their variance w o u l d be a p a r a m e t e r since it w o u l d be based on the entire p o p u l a t i o n . 3.8 Practical methods for computing mean and standard deviation T h r e e steps are necessary for c o m p u t i n g the s t a n d a r d deviation: (1) find Σ>>2, the s u m of squares; (2) divide by η — 1 to give the variance; a n d (3) take the s q u a r e r o o t of the variance to o b t a i n the s t a n d a r d deviation. T h e p r o c e d u r e used t o c o m p u t e the s u m of squares in Section 3.6 can be expressed by the following f o r m u l a : £y2 = X
2

(3.7)

This f o r m u l a t i o n explains most clearly the m e a n i n g of the sum of squares, alt h o u g h it m a y be inconvenient for c o m p u t a t i o n by h a n d or calculator, since one must first c o m p u t e the m e a n before one can s q u a r e a n d sum the deviations. A quicker c o m p u t a t i o n a l f o r m u l a for this q u a n t i t y is v

r

V>"

(3.8) 11

Let us see exactly w h a t this f o r m u l a represents. T h e first term o n the right side of the e q u a t i o n , Σ Υ 2 , is the sum of all individual Y's, each s q u a r e d , as follows: £y

2

-

Y 2 + >1 + >1 + · • • + Y2„

W h e n referred to by name, Σ Υ 2 should be called the "sum of Y s q u a r e d " and should be carefully distinguished f r o m Σ>>2, "the sum of squares of Y." These names are u n f o r t u n a t e , but they are t o o well established to think of a m e n d i n g them. T h e o t h e r q u a n t i t y in Expression (3.8) is (ΣΥ) 2 />ι. It is often called the correction term (CT). T h e n u m e r a t o r of this term is the s q u a r e of the sum of the Y's; t h a t is, all t h e Y's are first s u m m e d , and this s u m is then s q u a r e d . In general, this q u a n t i t y is different f r o m Σ Υ 2 , which first squares the y ' s a n d then sums them. These two terms a r c identical only if all the Y's arc equal. If you arc not certain a b o u t this, you can convince yourself of this fact by calculating these two quantities for a few n u m b e r s . T h e d i s a d v a n t a g e of Expression (3.8) is that the quantities Σ Y2 a n d (Σ Y)2hi may b o t h be quite large, so that accuracy may be lost in c o m p u t i n g their difference unless one takes the precaution of c a r r y i n g sufficient significant figures. W h y is Expression (3.8) identical with Expression (3.7)? T h e proof of this identity is very simple a n d is given in Appendix A 1.2. You are urged to work

40

CHAPTER 3 /' DESCRIPTIVE STATISTICS

t h r o u g h it t o build u p y o u r confidence in h a n d l i n g statistical s y m b o l s a n d formulas. It is s o m e t i m e s possible t o simplify c o m p u t a t i o n s by recoding variates into simpler f o r m . W e shall use the term additive coding for the a d d i t i o n or s u b t r a c t i o n of a c o n s t a n t (since s u b t r a c t i o n is only a d d i t i o n of a negative n u m b e r ) . W e shall similarly use multiplicative coding to refer to the multiplication or division by a c o n s t a n t (since division is multiplication by the reciprocal of the divisor). W e shall use the t e r m combination coding to m e a n the a p p l i c a t i o n of b o t h additive a n d multiplicative c o d i n g t o the same set of d a t a . In A p p e n d i x A 1.3 we e x a m i n e the c o n s e q u e n c e s of the three types of coding in the c o m p u t a t i o n of means, variances, a n d s t a n d a r d deviations. F o r the case of means, the f o r m u l a for c o m b i n a t i o n coding a n d d e c o d i n g is the most generally applicable one. If the c o d e d variable is Yc = D(Y + C), then

where C is an additive c o d e a n d D is a multiplicative code. O n considering the effects of c o d i n g variates on the values of variances and standard deviations, we find that additive codes have no effect o n the s u m s of squares, variances, or s t a n d a r d deviations. T h e m a t h e m a t i c a l proof is given in A p p e n d i x A 1.3, but we can see this intuitively, because an additive code has n o effect on the distance of an item f r o m its m e a n . T h e distance f r o m an item of 15 to its m e a n of 10 would be 5. If we were to code the variates by subtracting a c o n s t a n t of 10, the item would now be 5 a n d the m e a n zero. T h e difference between t h e m would still be 5. T h u s , if only additive c o d i n g is employed, the only statistic in need of d e c o d i n g is the mean. But multiplicative coding does have an effect on s u m s of squares, variances, a n d s t a n d a r d deviations. T h e s t a n d a r d deviations have to be divided by the multiplicative code, just as had to be d o n e for the m e a n . However, the s u m s of squares or variances have to be divided by the multiplicative codes s q u a r e d , because they are s q u a r e d terms, and the multiplicative factor becomcs s q u a r e d d u r i n g the o p e r a t i o n s . In c o m b i n a t i o n coding the additive code can be ignored. W h e n the d a t a are u n o r d e r e d , the c o m p u t a t i o n of the m e a n and s t a n d a r d deviation proceeds as in Box 3.1, which is based on the u n o r d e r e d n e u t r o p h i l c o u n t d a t a s h o w n in T a b l e 3.1. W e chose not to apply coding to these d a t a , since it would not have simplified the c o m p u t a t i o n s appreciably. W h e n the d a t a are a r r a y e d in a frequency distribution, the c o m p u t a t i o n s can be m a d e m u c h simpler. W h e n c o m p u t i n g the statistics, you can often avoid the need for m a n u a l entry of large n u m b e r s of individual variatcs if you first set u p a frequency distribution. Sometimes the d a t a will c o m e to you already in the form of a frequency distribution, having been g r o u p e d by the researcher. T h e c o m p u t a t i o n of Ϋ a n d s f r o m a frequency distribution is illustrated in Box 3.2. T h e d a t a are the birth weights of male Chinese children, first e n c o u n t e r e d in Figure 2.3. T h e calculation is simplified by coding to remove the a w k w a r d class m a r k s . This is d i m e bv s u b t r a c t i n g 59.5. the lowest class m a r k of the arrav.

3.8 / PRACTICAL METHODS FOR COMPUTING MEAN AND STANDARD DEVIATION

41

BOX 3.1 Calculation of Ϋ and s from unordered data. Neutrophil counts, unordered as shown in Table 3.1. Computation

n = 15 £ 7 = 115.7 y = - T y = 7.713 η

ΣΥ

2

= 1201.21

= 308.7773

S

Σ / , 308.7773 ~ η- 1~ 14 = 22.056 s = V22.056 = 4.696

T h e resulting class m a r k s are values such as 0, 8, 16, 24, 32, a n d so on. T h e y are then divided by 8, which c h a n g e s them to 0, 1, 2, 3, 4, and so on, which is the desired f o r m a t . T h e details of the c o m p u t a t i o n can be learned f r o m the box. W h e n checking the results of calculations, it is frequently useful to have an a p p r o x i m a t e m e t h o d for e s t i m a t i n g statistics so that gross e r r o r s in c o m p u tation can be detected. A simple m e t h o d for e s t i m a t i n g the m e a n is to average the largest a n d smallest o b s e r v a t i o n to obtain the so-called miJrunye. F o r the neutrophil c o u n t s of T a b l e 3.1, this value is (2.3 + 18.0J/2 = 10.15 (not a very good estimate). S t a n d a r d deviations can be estimated f r o m ranges by a p p r o priate division of the range, as follows:

/•'or samples of 10 30 100 500 1000

Divide the range by 3 4 5

6 6

42

CHAPTER 3 /' DESCRIPTIVE STATISTICS

BOX 3.2 Calculation of F, s, and Κ from a frequency distribution. Birth weights of male Chinese in ounces.

(/) Class

y

(2)

mark

Coifei

c/iJM mark

/

59.5 67.5 75.5 83.5 91.5 99.5 107.5 115.5 123.5 131.5 139.5 147.5 155.5 163.5 171.5

2 6 39 385 888 1729 2240 2007 1233 641 201 74 14 5 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

9465 = η Source: Millis and Seng (1954).

Computation

Coding

v

Σ JX = 59,629

Code: Yc =

Ϋ, = 6.300

To decode

2

^ /Y f = 402,987 CT =

η

and

decoding

Y — 59 5 y = 8fr+59.5 = 50-4 + 59.5 = 109.9 oz

= 375,659.550

Z / > ' c = Z / n 2 - C T = 27,327.450 s? =

η— 1

sc = 1.6991

= 2.888 To decode sf: s = 8sc = 13.593 oz V = i X 100 - ^ ^ X 100 = 12.369% Y 109.9

3.9 / THE COEFFICIENT OF VARIATION

43

T h e range of the neutrophil c o u n t s is 15.7. W h e n this value is divided by 4, we get a n estimate for the s t a n d a r d deviation of 3.925, which c o m p a r e s with the calculated value of 4.696 in Box 3.1. H o w e v e r , w h e n we estimate m e a n a n d s t a n d a r d deviation of the a p h i d f e m u r lengths of Box 2.1 in this m a n n e r , we o b t a i n 4.0 a n d 0.35, respectively. These are g o o d estimates of the a c t u a l values of 4.004 a n d 0.3656, the s a m p l e m e a n a n d s t a n d a r d deviation. 3.9 The coefficient of variation H a v i n g o b t a i n e d the s t a n d a r d deviation as a m e a s u r e of t h e a m o u n t of v a r i a t i o n in the d a t a , y o u m a y be led to ask, " N o w w h a t ? " At this stage in o u r c o m prehension of statistical theory, n o t h i n g really useful comes of the c o m p u t a t i o n s we have carried out. H o w e v e r , the skills j u s t learned are basic to all later statistical w o r k . So far, the only use t h a t we might have for the s t a n d a r d deviation is as an estimate of the a m o u n t of variation in a p o p u l a t i o n . T h u s , we may wish to c o m p a r e the m a g n i t u d e s of the s t a n d a r d deviations of similar p o p u l a tions a n d see w h e t h e r p o p u l a t i o n A is m o r e or less variable than p o p u l a t i o n B. W h e n p o p u l a t i o n s differ appreciably in their means, the direct c o m p a r i s o n of their variances o r s t a n d a r d deviations is less useful, since larger o r g a n i s m s usually vary m o r e t h a n smaller one. F o r instance, the s t a n d a r d deviation of the tail lengths of e l e p h a n t s is obviously m u c h greater than the entire tail length of a mouse. T o c o m p a r e the relative a m o u n t s of variation in p o p u l a t i o n s having different means, the coefficient of variation, symbolized by V (or occasionally CV), has been developed. This is simply the s t a n d a r d deviation expressed as a percentage of the m e a n . Its f o r m u l a is

F o r example, the coefficient of variation of the birth weights in Box 3.2 is 12.37%, as s h o w n at the b o t t o m of that box. T h e coefficient of variation is independent of the unit of m e a s u r e m e n t a n d is expressed as a percentage. Coefficients of variation are used when one wishes t o c o m p a r e the variation of two p o p u l a t i o n s without considering the m a g n i t u d e of their means. (It is p r o b a b l y of little interest to discover whether the birth weights of the Chinese children are m o r e or less variable t h a n the femur lengths of the aphid stem mothers. However, we can calculate V for the latter as (0.3656 χ Ι00)/4.004 = 9.13%, which would suggest that the birth weights arc m o r e variable.) O f t e n , we shall wish to test whether a given biological sample is m o r e variable for o n e character than for a n o t h e r . T h u s , for a s a m p l e of rats, is b o d y weight m o r e variable than b l o o d sugar content? A second, frequent type of c o m p a r i s o n , especially in systcmatics, is a m o n g different p o p u l a t i o n s for the same c h a r a c t e r . Thus, we m a y have m e a s u r e d wing length in samples of birds f r o m several localities. We wish t o k n o w w h e t h e r any o n e of these p o p u l a t i o n s is m o r e variable than the others. An a n s w e r to this question can be o b t a i n e d by examining the coefficients of variation of wing length in these samples.

44

CHAPTER 3 /' DESCRIPTIVE STATISTICS

Exercises 3.1

F i n d f , s, V, a n d t h e m e d i a n f o r t h e f o l l o w i n g d a t a ( m g o f g l y c i n e p e r m g o f c r e a t i n i n e in t h e u r i n e o f 3 7 c h i m p a n z e e s ; f r o m G a r t l e r , F i r s c h e i n , a n d D o b z h a n s k y , 1956). A N S . Y = 0 . 1 1 5 , s = 0 . 1 0 4 0 4 . .008

.018

.056

.055

.135

.052

.077

.026

.025

.043

.080

.110 .110

.100

.070

.100 .050

.120

.011

.036 .060

.110

.350 .120

.100

.155

.370

.019

.100

.100

.116

.300

.440 .100

.300 .100

.133

3.2

F i n d t h e m e a n , s t a n d a r d d e v i a t i o n , a n d c o e f f i c i e n t of v a r i a t i o n f o r t h e p i g e o n d a t a g i v e n i n E x e r c i s e 2.4. G r o u p t h e d a t a i n t o t e n c l a s s e s , r e c o m p u t e Ϋ a n d s, a n d c o m p a r e t h e m with the results o b t a i n e d from u n g r o u p e d data. C o m p u t e the m e d i a n for the g r o u p e d data.

3.3

T h e f o l l o w i n g a r e p e r c e n t a g e s of b u t t e r f a t f r o m 120 r e g i s t e r e d t h r e e - y e a r - o l d A y r s h i r e c o w s selected at r a n d o m f r o m a C a n a d i a n stock r e c o r d b o o k . (a) C a l c u l a t e Y, s, a n d V d i r e c t l y f r o m t h e d a t a . (b) G r o u p t h e d a t a i n a f r e q u e n c y d i s t r i b u t i o n a n d a g a i n c a l c u l a t e Y, s, a n d V. C o m p a r e t h e r e s u l t s w i t h t h o s e o f (a). H o w m u c h p r e c i s i o n h a s b e e n l o s t b y grouping? Also calculate the median. 4.32

4.24

4.29

4.00

3.96 3.74

4.48

3.89 4.20

4.02

4.10

4.00

4.33 4.23

4.16 4.67

4.33 3.88 3.74

3.81 4.81 4.25

4.28 4.15 4.49 4.67 4.60 4.00 4.71

4.03 4.29 4.05 4.11 4.38 4.46 3.96

4.42 4.27 3.97 4.24 3.72 4.82

4.09 4.38 4.32

4.38 4.06 3.97 4.31

4.16 4.08 3.97 3.70

4.30

4.17

3.97

4.20

4.51 4.24

3.86

4.36

4.18

4.05

4.05

3.56

3.94

3.89

4.17 4.06 3.93

3.82 3.89 4.20

4.58 3.70 4.07

3.99 4.33 3.58

3.89

4.38

4.14 3.47

4.66 3.92

4.60 3.97 4.91

4.38 3.91

4.12

4.52

4.10

4.09

4.34 3.98

4.09 3.86

4.88 4.58

4.22 3.95 4.35 4.09 4.28 3.4

4.42

3.66 3.77 3.66 4.20 3.83

3.87

5.00 3.99 3.91 4.10 4.40 4.70 4.41 4.24

W h a t clfect w o u l d a d d i n g a c o n s t a n t

5.2 t o all o b s e r v a t i o n s

have upon

n u m e r i c a l v a l u e s o f t h e f o l l o w i n g s t a t i s t i c s : Υ, .s, V, a v e r a g e d e v i a t i o n ,

the

median.

EXERCISES

3.5

3.6

45

mode, range? What would be the effect of adding 5.2 and then multiplying the sums by 8.0? Would it make any difference in the above statistics if we multiplied by 8.0 first and then added 5.2? Estimate μ and σ using the midrange and the range (see Section 3.8) for the data in Exercises 3.1, _3.2, and 3.3. How well do these estimates agree with the estimates given by Y and s? ANS. Estimates of μ and σ for Exercise 3.2 are 0.224 and 0.1014. Show that the equation for the variance can also be written as ,

s2 = ^

ΤΥ2-ηΫ2 η — 1

3.7

3.8

Using the striped _bass age distribution given in Exercise 2.9, compute the following statistics: Y, s2, s, V, median, and mode. ANS. 7 = 3.043, s2 = 1.2661, s = 1.125, V = 36.98%, median = 2.948, mode = 3. Use a calculator and compare the results of using Equations 3.7 and 3.8 to compute s 2 for the following artificial data sets: (a) 1 , 2 , 3 , 4 , 5 (b) 9001, 9002, 9003, 9004, 9005 (c) 90001, 90002, 90003, 90004, 90005 (d) 900001, 900002, 900003, 900004, 900005 Compare your results with those of one or more computer programs. What is the correct answer? Explain your results.

CHAPTER

Introduction Distributions: Poisson

to

Probability

The Binomial and

Distributions

In Section 2.5 we first e n c o u n t e r e d frequency distributions. F o r example, T a b l e 2.2 s h o w s a distribution for a meristic, or discrete (discontinuous), variable, the n u m b e r of sedge p l a n t s per q u a d r a t . Examples of distributions for c o n t i n u o u s variables are the f e m u r lengths of a p h i d s in Box 2.1 and the h u m a n birth weights in Box 3.2. Each of these d i s t r i b u t i o n s i n f o r m s us a b o u t the a b s o l u t e f r e q u e n c y of a n y given class a n d permits us to c o m p u t a t e the relative frequencies of a n y class of variable. T h u s , m o s t of the q u a d r a t s c o n t a i n e d either n o sedges or o n e or t w o plants. In the 139.5-oz class of birth weights, we find only 201 out of the total of 9465 babies recorded; that is, a p p r o x i m a t e l y only 2.1% of the infants are in t h a t birth weight class. W e realize, of course, that these frequency d i s t r i b u t i o n s are only samples f r o m given p o p u l a t i o n s . T h e birth weights, for example, represent a p o p u l a t i o n of male Chinese infants f r o m a given geographical area. But if we k n e w o u r s a m p l e to be representative of that p o p u l a t i o n , we could m a k e all sorts of predictions based u p o n the s a m p l e frequency distribution. F o r instance, we could say t h a t a p p r o x i m a t e l y 2.1% of male Chinese babies b o r n in this p o p u l a t i o n should weigh between 135.5 a n d 143.5 oz at birth. Similarly, we might say that

47CHAPTER4/INTRODUCTION TO PROBABILITY DISTRIBUTIONS

the p r o b a b i l i t y t h a t the weight at birth of any o n e b a b y in this p o p u l a t i o n will be in t h e 139.5-oz b i r t h class is quite low. If all of the 9465 weights were mixed up in a h a t a n d a single o n e pulled out, t h e probability t h a t we w o u l d pull out one of the 201 in the 139.5-oz class w o u l d be very low i n d e e d — o n l y 2.1%. It would be m u c h m o r e p r o b a b l e t h a t we w o u l d sample a n infant of 107.5 or 115.5 oz, since the infants in these classes are represented by frequencies 2240 a n d 2007, respectively. Finally, if we were t o s a m p l e f r o m a n u n k n o w n p o p u l a tion of babies a n d find t h a t the very first individual sampled h a d a b i r t h weight of 170 oz, we w o u l d p r o b a b l y reject a n y hypothesis t h a t the u n k n o w n p o p u l a t i o n was the same as t h a t sampled in Box 3.2. W e w o u l d arrive at this conclusion because in the distribution in Box 3.2 only o n e out of a l m o s t 10,000 infants h a d a birth weight t h a t high. T h o u g h it is possible t h a t we could have sampled f r o m the p o p u l a t i o n of male Chinese babies a n d o b t a i n e d a birth weight of 170 oz, the probability t h a t t h e first individual s a m p l e d would have such a value is very low indeed. It seems m u c h m o r e r e a s o n a b l e t o s u p p o s e t h a t the u n k n o w n p o p u l a t i o n f r o m which we are s a m p l i n g has a larger m e a n t h a t the o n e sampled in Box 3.2. W e have used this empirical frequency distribution to m a k e certain predictions (with w h a t frequency a given event will occur) or to m a k e j u d g m e n t s a n d decisions (is it likely t h a t an infant of a given birth weight belongs to this population?). In m a n y cases in biology, however, we shall m a k e such predictions not f r o m empirical distributions, b u t on the basis of theoretical c o n s i d e r a t i o n s that in o u r j u d g m e n t are pertinent. W e m a y feel t h a t the d a t a should be distributed in a certain way because of basic a s s u m p t i o n s a b o u t the n a t u r e of the forces acting o n the e x a m p l e at h a n d . If o u r actually observed d a t a d o not c o n f o r m sufficiently to the values expected on the basis of these a s s u m p t i o n s , we shall have serious d o u b t s a b o u t o u r a s s u m p t i o n s . This is a c o m m o n use of frequency distributions in biology. T h e a s s u m p t i o n s being tested generally lead to a theoretical frequency distribution k n o w n also as a probability distribution. This m a y be a simple two-valued distribution, such as the 3:1 ratio in a Mendelian cross; or it m a y be a m o r e complicated function, as it would be if we were trying to predict the n u m b e r of plants in a q u a d r a t . If we find that the observed d a t a d o not fit the expectations on the basis of theory, we are often led to the discovery of s o m e biological m e c h a n i s m causing this deviation f r o m expectation. T h e p h e n o m e n a of linkage in genetics, of preferential m a t i n g between different p h e n o t y p e s in animal behavior, of c o n g r e g a t i o n of a n i m a l s at certain favored places or, conversely, their territorial dispersion are cases in point. We shall thus m a k e use of probability theory to test o u r a s s u m p t i o n s a b o u t the laws of occurrence of certain biological p h e n o m e n a . Wc should point out to the reader, however, t h a t probability theory underlies the entire s t r u c t u r e of statistics, since, owing to the n o n m a t h e m a t i c a l o r i e n t a t i o n of this b o o k , this m a y not be entirely obvious. In this c h a p t e r we shall first discuss probability, in Section 4.1, but only to the extent necessary for c o m p r e h e n s i o n of the sections that follow at the intended level of m a t h e m a t i c a l sophistication. Next, in Section 4.2, we shall take up the

48

CHAPTER 4 / INTRODUCTION TO PROBABILITY DISTRIBUTIONS

b i n o m i a l frequency distribution, which is not only i m p o r t a n t in certain types of studies, such as genetics, but also f u n d a m e n t a l to an u n d e r s t a n d i n g of t h e various k i n d s of p r o b a b i l i t y d i s t r i b u t i o n s t o be discussed in this b o o k . T h e Poisson d i s t r i b u t i o n , which follows in Section 4.3, is of wide applicability in biology, especially for tests of r a n d o m n e s s of occurrence of certain events. B o t h the b i n o m i a l a n d P o i s s o n d i s t r i b u t i o n s are discrete p r o b a b i l i t y distributions. T h e m o s t c o m m o n c o n t i n u o u s p r o b a b i l i t y distribution is the n o r m a l frequency d i s t r i b u t i o n , discussed in C h a p t e r 5.

4.1 Probability, random sampling, and hypothesis testing W e shall start this discussion with an e x a m p l e t h a t is n o t biometrical o r biological in the strict sense. W e have often f o u n d it pedagogically effective t o i n t r o d u c e new c o n c e p t s t h r o u g h situations t h o r o u g h l y familiar to the s t u d e n t , even if the e x a m p l e is n o t relevant to the general subject m a t t e r of biostatistics. Let us b e t a k e ourselves to M a t c h l e s s University, a state institution s o m e w h e r e between the A p p a l a c h i a n s a n d the Rockies. L o o k i n g at its e n r o l l m e n t figures, we notice the following b r e a k d o w n of the student body: 70% of the s t u d e n t s a r e American u n d e r g r a d u a t e s (AU) a n d 26% are American g r a d u a t e s t u d e n t s (AG); the r e m a i n i n g 4% are f r o m a b r o a d . Of these, 1% are foreign u n d e r g r a d u a t e s ( F U ) a n d 3% are foreign g r a d u a t e s t u d e n t s (FG). In m u c h of o u r w o r k we shall use p r o p o r t i o n s r a t h e r t h a n percentages as a useful c o n v e n t i o n . T h u s the enrollment consists of 0.70 AU's, 0.26 AG's, 0.01 F U ' s , a n d 0.03 F G ' s . T h e total student b o d y , c o r r e s p o n d i n g to 100%, is therefore represented by the figure 1.0. If we were to assemble all the s t u d e n t s a n d s a m p l e 100 of t h e m at r a n d o m , we would intuitively expect that, on the average, 3 would be foreign g r a d u a t e students. T h e actual o u t c o m e might vary. T h e r e might not be a single F G s t u d e n t a m o n g the 100 sampled, or there might be quite a few m o r e t h a n 3. T h e ratio of the n u m b e r of foreign g r a d u a t e s t u d e n t s sampled divided by the total n u m b e r of s t u d e n t s sampled might therefore vary f r o m zero to c o n s i d e r a b l y greater than 0.03. If we increased o u r s a m p l e size to 500 or 1000, it is less likely t h a t t h e ratio would fluctuate widely a r o u n d 0.03. T h e greater the s a m p l e taken, the closer the r a t i o of F G s t u d e n t s sampled t o the total s t u d e n t s s a m p l e d will a p p r o a c h 0.03. In fact, the probability of s a m p l i n g a foreign s t u d e n t can be defined as the limit as s a m p l e size keeps increasing of the ratio of foreign s t u d e n t s to the total n u m b e r of s t u d e n t s sampled. T h u s , we may formally s u m m a r i z e the situation by stating that the probability that a student at Matchless University will be a foreign g r a d u a t e student is P [ F G ] = 0.03. Similarly, the probability of s a m p l i n g a foreign u n d e r g r a d u a t e is P [ F U ] = 0 . 0 1 , that of s a m p l i n g an American u n d e r g r a d u a t e is /-"[AUJ = 0.70, and that for American g r a d u a t e students, P [ A G ] = 0.26. N o w let us imagine the following experiment: We try to sample a student at r a n d o m f r o m a m o n g the student body at Matchless University. This is not as easy a task as might be imagined. If we w a n t e d to d o this o p e r a t i o n physically,

4 . 1 / PROBABILITY, RANDOM SAMPLING, AND HYPOTHESIS TESTING

49

we w o u l d h a v e t o set u p a collection o r t r a p p i n g s t a t i o n s o m e w h e r e o n c a m p u s . A n d t o m a k e certain t h a t the s a m p l e was truly r a n d o m with respect t o t h e entire s t u d e n t p o p u l a t i o n , we w o u l d have t o k n o w t h e ecology of s t u d e n t s o n c a m p u s very t h o r o u g h l y . W e should try to locate o u r t r a p a t s o m e s t a t i o n where e a c h s t u d e n t h a d a n e q u a l probability of passing. F e w , if a n y , such places can be f o u n d in a university. T h e s t u d e n t u n i o n facilities a r e likely t o be frequented m o r e by i n d e p e n d e n t a n d foreign students, less by t h o s e living in organized houses a n d d o r m i t o r i e s . F e w e r foreign a n d g r a d u a t e s t u d e n t s m i g h t be f o u n d a l o n g fraternity row. Clearly, we w o u l d n o t wish t o place o u r t r a p near the I n t e r n a t i o n a l C l u b o r H o u s e , because o u r p r o b a b i l i t y of s a m p l i n g a foreign s t u d e n t w o u l d be greatly e n h a n c e d . In f r o n t of the b u r s a r ' s w i n d o w we might s a m p l e s t u d e n t s p a y i n g tuition. But those o n scholarships m i g h t n o t be found there. W e d o n o t k n o w w h e t h e r the p r o p o r t i o n of scholarships a m o n g foreign o r g r a d u a t e s t u d e n t s is t h e s a m e as o r different f r o m t h a t a m o n g t h e American or u n d e r g r a d u a t e students. Athletic events, political rallies, dances, and the like w o u l d all d r a w a differential s p e c t r u m of the s t u d e n t body; indeed, n o easy solution seems in sight. T h e time of s a m p l i n g is equally i m p o r t a n t , in the seasonal as well as the d i u r n a l cycle. T h o s e a m o n g t h e r e a d e r s w h o are interested in s a m p l i n g o r g a n i s m s f r o m n a t u r e will already h a v e perceived parallel p r o b l e m s in their w o r k . If we were to s a m p l e only s t u d e n t s wearing t u r b a n s or saris, their p r o b a b i l i t y of being foreign s t u d e n t s w o u l d b e a l m o s t 1. W e could n o longer speak of a r a n d o m sample. In the familiar ecosystem of t h e university these violations of p r o p e r sampling p r o c e d u r e a r e o b v i o u s t o all of us, b u t they are not nearly so o b v i o u s in real biological instances where we a r e unfamiliar with the true n a t u r e of the environment. H o w s h o u l d we proceed t o o b t a i n a r a n d o m s a m p l e of leaves f r o m a tree, of insects f r o m a field, o r of m u t a t i o n s in a culture? In s a m p l i n g at r a n d o m , we are a t t e m p t i n g t o permit the frequencies of v a r i o u s events occurring in n a t u r e t o be r e p r o d u c e d unalteredly in o u r records; t h a t is, we h o p e t h a t o n the average the frequencies of these events in o u r s a m p l e will be the same as they a r e in the n a t u r a l situation. A n o t h e r way of saying this is that in a r a n d o m s a m p l e every individual in the p o p u l a t i o n being s a m p l e d has a n equal probability of being included in the sample. We might go a b o u t o b t a i n i n g a r a n d o m s a m p l e by using records representing the student b o d y , such as the student directory, selecting a page f r o m it at r a n d o m a n d a n a m e at r a n d o m f r o m the page. O r we could assign an an a r b i t r a r y n u m b e r t o each s t u d e n t , write each o n a chip or disk, put these in a large c o n t a i n e r , stir well, a n d then pull out a n u m b e r . I m a g i n e n o w t h a t we s a m p l e a single s t u d e n t physically by the t r a p p i n g m e t h o d , after carefully p l a n n i n g t h e placement of the t r a p in such a way as to m a k e s a m p l i n g r a n d o m . W h a t a r e the possible o u t c o m e s ? Clearly, the student could be either a n A U , A G , F U or F G . T h e set of these four possible o u t c o m e s exhausts the possibilities of this experiment. This set, which we c a n represent as {AU, A G , F U , F G } is called the sample space. Any single trial of the experiment described a b o v e w o u l d result in only o n e of the f o u r possible o u t c o m e s (elements)

CHAPTER 4 / INTRODUCTION TO PROBABILITY DISTRIBUTIONS

50

in t h e set. A single element in a s a m p l e space is called a simple event. It is distinguished f r o m an event, which is a n y subset of the sample-space. T h u s , in the s a m p l e space defined a b o v e {AU}, {AG}, {FU}, a n d { F G } a r e e a c h simple events. T h e following s a m p l i n g results a r e some of the possible events: {AU, A G , F U } , {AU, A G , F G } , {AG, F G } , {AU, F G } , . . . By t h e definition of "event," simple events as well as t h e entire s a m p l e space a r e also events. T h e m e a n i n g of these events s h o u l d be clarified. T h u s {AU, A G , F U } implies being either a n A m e r i c a n o r a n u n d e r g r a d u a t e , or b o t h . Given the s a m p l i n g space described above, the event A = {AU, A G } enc o m p a s s e s all possible o u t c o m e s in the space yielding a n A m e r i c a n student. Similarly, the event Β = {AG, F G } s u m m a r i z e s the possibilities for o b t a i n i n g a g r a d u a t e student. T h e intersection of events A a n d B, written Α η Β, describes only those events t h a t a r e shared by A a n d B. Clearly only A G qualifies, as can be seen below: A = {AU, A G } Β =

{AG, F G }

T h u s , Α η Β is that event in the s a m p l e space giving rise to the s a m p l i n g of a n A m e r i c a n g r a d u a t e s t u d e n t . W h e n the intersection of t w o events is e m p t y , as in Β η C, where C = {AU, F U } , events Β a n d C are m u t u a l l y exclusive. T h u s there is n o c o m m o n element in these t w o events in the s a m p l i n g space. W e m a y also define events t h a t are unions of t w o o t h e r events in the s i m p l e space. T h u s Α υ Β indicates t h a t A or Β or b o t h A a n d Β occur. As defined above, A u Β would describe all s t u d e n t s w h o are either American students, g r a d u a t e students, o r A m e r i c a n g r a d u a t e students. W h y a r e we c o n c e r n e d with defining s a m p l e spaces a n d events? Because these concepts lead us to useful definitions a n d o p e r a t i o n s r e g a r d i n g the p r o b a b i l i t y of various o u t c o m e s . If we can assign a n u m b e r p, where 0 < ρ < 1, t o each simple event in a s a m p l e space such t h a t the sum of these p's over all simple events in the space e q u a l s unity, then the space b e c o m e s a (finite) probability space. In o u r e x a m p l e above, the following n u m b e r s were associated with the a p p r o p r i a t e simple events in the s a m p l e space: {AU, AG, F U ,

FG}

{0.70,0.26, 0.01,0.03} G i v e n this p r o b a b i l i t y space, we a r e n o w able to m a k e s t a t e m e n t s r e g a r d i n g the probability of given events. F o r example, w h a t is the p r o b a b i l i t y that a s t u d e n t sampled at r a n d o m will be an A m e r i c a n g r a d u a t e s t u d e n t ? Clearly, it is P [ { A G } ] = 0.26. W h a t is the p r o b a b i l i t y that a student is either American o r a g r a d u a t e s t u d e n t ? In terms of the events defined earlier, this is PLAuBj

= P[{AU,AG}]

+ P[{AG,

= 0.96 + 0.29

0.26

= 0.99

FG]]

-

P[{AG]]

4.1 /

PROBABILITY, R A N D O M SAMPLING, A N D HYPOTHESIS TESTING

51

W e s u b t r a c t P [ { A G } ] f r o m the s u m on the right-hand side of t h e e q u a t i o n because if we did n o t d o so it w o u l d be included twice, once in P [ A ] a n d once in P [ B ] , a n d w o u l d lead to the a b s u r d result of a p r o b a b i l i t y greater t h a n 1. N o w let us a s s u m e t h a t we have sampled o u r single s t u d e n t f r o m the s t u d e n t b o d y of Matchless University a n d t h a t s t u d e n t t u r n s o u t to be a foreign g r a d u a t e student. W h a t c a n we c o n c l u d e f r o m this? By c h a n c e alone, this result w o u l d h a p p e n 0.03, or 3%, of the t i m e — n o t very frequently. T h e a s s u m p t i o n t h a t we have sampled at r a n d o m should p r o b a b l y be rejected, since if we accept the hypothesis of r a n d o m sampling, the o u t c o m e of the experiment is i m p r o b a b l e . Please n o t e that we said improbable, n o t impossible. It is o b v i o u s t h a t we could have chanced u p o n a n F G as the very first one t o be s a m p l e d . H o w e v e r , it is not very likely. T h e p r o b a b i l i t y is 0.97 t h a t a single s t u d e n t s a m p l e d will be a n o n - F G . If we could be certain t h a t o u r s a m p l i n g m e t h o d was r a n d o m (as when d r a w i n g s t u d e n t n u m b e r s o u t of a container), we w o u l d have t o decide that an i m p r o b a b l e event h a s occurred. T h e decisions of this p a r a g r a p h are all based on o u r definite k n o w l e d g e t h a t the p r o p o r t i o n of s t u d e n t s at Matchless University is indeed as specified by t h e p r o b a b i l i t y space. If we were uncertain a b o u t this, we w o u l d be led to a s s u m e a higher p r o p o r t i o n of foreign g r a d u a t e students as a c o n s e q u e n c e of the o u t c o m e of o u r sampling experiment. W e shall n o w extend o u r experiment a n d s a m p l e two s t u d e n t s r a t h e r t h a n just one. W h a t a r e the possible o u t c o m e s of this s a m p l i n g e x p e r i m e n t ? T h e new sampling space can best be depicted by a d i a g r a m (Figure 4.1) t h a t shows the set of t h e 16 possible simple events as p o i n t s in a lattice. T h e simple events are the following possible c o m b i n a t i o n s . I g n o r i n g which student was sampled first, they are (AU, AU), (AU, AG), (AU, FU), (AU, FG), (AG, AG), (AG, FU), (AG, FG), ( F U , FU), ( F U , FG), a n d ( F G , FG).

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0.0078

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(1.0070

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Kirs! s t u d e n t n c a κι 4.1 S a m p l e space lor s a m p l i n g Iwo students from Matchless University.

52

CHAPTER 4 / INTRODUCTION TO PROBABILITY DISTRIBUTIONS

W h a t are t h e expected probabilities of these o u t c o m e s ? W e k n o w t h e expected o u t c o m e s for s a m p l i n g o n e s t u d e n t f r o m the f o r m e r p r o b a b i l i t y space, b u t w h a t will be the p r o b a b i l i t y space c o r r e s p o n d i n g to t h e new s a m p l i n g space of 16 elements? N o w t h e n a t u r e of the s a m p l i n g p r o c e d u r e b e c o m e s q u i t e imp o r t a n t . W e m a y s a m p l e with or w i t h o u t replacement: we m a y r e t u r n t h e first s t u d e n t s a m p l e d to the p o p u l a t i o n (that is, replace the first student), o r we m a y k e e p him or her o u t of the p o o l of t h e individuals to be s a m p l e d . If we d o n o t replace the first individual s a m p l e d , the p r o b a b i l i t y of s a m p l i n g a foreign g r a d u a t e s t u d e n t will n o longer be exactly 0.03. This is easily seen. Let us a s s u m e t h a t M a t c h l e s s University h a s 10,000 students. T h e n , since 3% are foreign g r a d u a t e students, there m u s t be 300 F G s t u d e n t s at the university. After s a m p l i n g a foreign g r a d u a t e s t u d e n t first, this n u m b e r is reduced t o 299 o u t of 9999 students. C o n s e q u e n t l y , the p r o b a b i l i t y of s a m p l i n g a n F G s t u d e n t n o w b e c o m e s 299/9999 = 0.0299, a slightly lower probability t h a n the value of 0.03 for s a m p l i n g the first F G student. If, o n the o t h e r h a n d , we r e t u r n the original foreign student t o the s t u d e n t p o p u l a t i o n a n d m a k e certain that the p o p u l a t i o n is t h o r o u g h l y r a n d o m i z e d before being sampled again (that is, give the student a c h a n c e to lose him- o r herself a m o n g the c a m p u s crowd or, in d r a w i n g student n u m b e r s out of a container, mix u p the disks with the n u m b e r s on them), the probability of s a m p l i n g a second F G student is the s a m e as before—0.03. In fact, if we keep on replacing the sampled individuals in the original p o p u l a t i o n , we can s a m p l e from it as t h o u g h it were a n infinite-sized population. Biological p o p u l a t i o n s are, of course, finite, but they are frequently so large t h a t for p u r p o s e s of s a m p l i n g e x p e r i m e n t s we can consider t h e m effectively infinite whether we replace s a m p l e d individuals or not. After all, even in this relatively small p o p u l a t i o n of 10,000 students, the probability of s a m p l i n g a second foreign g r a d u a t e student (without replacement) is only minutely different f r o m 0.03. F o r the rest of this section we shall consider s a m p l i n g to be with replacement, so that the p r o b a b i l i t y level of o b t a i n i n g a foreign s t u d e n t d o e s not change. T h e r e is a second potential source of difficulty in this design. W e h a v e to a s s u m e not only that the p r o b a b i l i t y of s a m p l i n g a second foreign s t u d e n t is equal to that of the first, but also that it is independent of it. By independence of events we m e a n that the probability that one event will occur is not affected by whether or not another event has or has not occurred. In the case of the students, if we have s a m p l e d o n e foreign s t u d e n t , is it m o r e o r less likely t h a t a second s t u d e n t sampled in the same m a n n e r will also be a foreign s t u d e n t ? Independence of the events may d e p e n d on where we sample the s t u d e n t s or o n the m e t h o d of sampling. If we have sampled s t u d e n t s on c a m p u s , it is quite likely that the events are not i n d e p e n d e n t ; that is, if o n e foreign student has been s a m p l e d , the probability that the second student will be foreign is increased, since foreign s t u d e n t s tend to congregate. T h u s , at M a t c h l e s s University the probability that a student walking with a foreign g r a d u a t e s t u d e n t is also an F G will be greater than 0.03.

4 . 1 / PROBABILITY, RANDOM SAMPLING, AND HYPOTHESIS TESTING

53

Events D a n d Ε in a s a m p l e space will be defined as i n d e p e n d e n t whenever P [ D η E ] = P [ D ] P [ E ] , T h e p r o b a b i l i t y values assigned to the sixteen p o i n t s in the sample-space lattice of F i g u r e 4.1 have been c o m p u t e d t o satisfy the a b o v e c o n d i t i o n . T h u s , letting P [ D ] e q u a l the p r o b a b i l i t y t h a t t h e first s t u d e n t will be an A U , t h a t is, P ^ A ^ A U ^ A U X A G 2 , A U ^ U , , A U ^ G , } ] , a n d letting P [ E ] equal the p r o b a b i l i t y t h a t the second s t u d e n t will be a n F G , t h a t is, P [ { A U ! F G 2 , A G j F G j , F U 1 F G 2 , F G 1 F G 2 } ] , we n o t e t h a t t h e intersection D n E is { A U , F G 2 } · T h i s h a s a value of 0.0210 in the p r o b a b i l i t y space of Figure 4.1. W e find t h a t this value is t h e p r o d u c t P [ { A U } ] P [ { F G } ] = 0.70 χ 0.03 = 0.0210. These m u t u a l l y i n d e p e n d e n t relations have been deliberately imposed u p o n all p o i n t s in the p r o b a b i l i t y space. Therefore, if the s a m p l i n g probabilities for t h e second s t u d e n t a r e i n d e p e n d e n t of t h e type of s t u d e n t sampled first, we c a n c o m p u t e t h e probabilities of the o u t c o m e s simply as the p r o d u c t of the i n d e p e n d e n t probabilities. T h u s the probability of o b t a i n i n g t w o F G students is P [ { F G } ] P [ { F G } ] = 0.03 χ 0.03 = 0.0009. T h e p r o b a b i l i t y of o b t a i n i n g o n e A U a n d o n e F G student in t h e s a m p l e should be the p r o d u c t 0.70 χ 0.03. However, it is in fact twice that p r o b a bility. It is easy t o see why. T h e r e is only o n e way of o b t a i n i n g t w o F G students, namely, by s a m p l i n g first o n e F G a n d then again a n o t h e r F G . Similarly, there is only o n e w a y to s a m p l e t w o A U students. H o w e v e r , s a m p l i n g one of each type of s t u d e n t can be d o n e by s a m p l i n g first an A U a n d then an F G or by s a m p l i n g first a n F G a n d then an AU. T h u s the probability is 2 P [ { A U } ] P [ { F G } ] = 2 χ 0.70 χ 0.03 = 0.0420. If we c o n d u c t e d such a n experiment a n d o b t a i n a sample of two F G students, we would be led to the following conclusions. O n l y 0.0009 of the samples of 1% o r 9 out of 10,000 cases) w o u l d be expected to consist of t w o foreign g r a d u a t e students. It is quite i m p r o b a b l e to o b t a i n such a result by c h a n c e alone. Given P [ { F G } ] = 0.03 as a fact, we would therefore suspect that s a m p l i n g was not r a n d o m or that the events were not i n d e p e n d e n t (or t h a t b o t h assumptions r a n d o m s a m p l i n g a n d independence of e v e n t s — w e r e incorrect). R a n d o m s a m p l i n g is sometimes confused with r a n d o m n e s s in nature. T h e former is the faithful r e p r e s e n t a t i o n in the s a m p l e of the distribution of the events in nature; the latter is the i n d e p e n d e n c e of the events in n a t u r e . T h e first of these generally is or should be u n d e r the control of the e x p e r i m e n t e r a n d is related to the strategy of g o o d sampling. T h e second generally describes an i n n a t e p r o p e r t y of the objects being sampled a n d thus is of greater biological interest. T h e confusion between r a n d o m s a m p l i n g a n d i n d e p e n d e n c e of events arises because lack of either can yield observed frequencies of events differing f r o m expectation. We have already seen how lack of i n d e p e n d e n c e in samples of foreign s t u d e n t s can be interpreted f r o m both points of view in o u r illustrative example f r o m M a t c h l e s s University. T h e a b o v e a c c o u n t of probability is a d e q u a t e for o u r present p u r p o s e s but far t o o sketchy to convey a n u n d e r s t a n d i n g of the field. R e a d e r s interested in extending their k n o w l e d g e of the subject are referred to M o s i m a n n (1968) for a simple i n t r o d u c t i o n .

54

CHAPTER 4 / INTRODUCTION TO PROBABILITY DISTRIBUTIONS

4.2 The binomial distribution F o r p u r p o s e s of the discussion to follow we shall simplify o u r s a m p l e space to consist of only t w o elements, foreign a n d American students, a n d i g n o r e w h e t h e r the s t u d e n t s are u n d e r g r a d u a t e s or g r a d u a t e s ; we shall represent t h e s a m p l e space by the set {F, A}. Let us symbolize the probability space by {p, q}, where ρ — .P[F], the p r o b a b i l i t y t h a t the s t u d e n t is foreign, a n d q = P [ A ] , t h e p r o b ability t h a t the s t u d e n t is American. As before, we c a n c o m p u t e the p r o b a b i l i t y space of samples of t w o s t u d e n t s as follows: { F F , FA, AA} { P2, 2pq, q2 } If we were to s a m p l e three s t u d e n t s independently, the probability space of samples of three s t u d e n t s w o u l d be as follows: { F F F , F F A , FAA, AAA} { p \

3p2q,

3pq2,

q3

}

S a m p l e s of three foreign o r three A m e r i c a n s t u d e n t s can again be o b t a i n e d in only o n e way, a n d their probabilities a r e p3 a n d q3, respectively. H o w e v e r , in s a m p l e s of three there are three ways of o b t a i n i n g t w o s t u d e n t s of o n e k i n d a n d o n e s t u d e n t of the other. As before, if A s t a n d s for American a n d F s t a n d s for foreign, then the s a m p l i n g sequence c a n be A F F , F A F , F F A for t w o foreign s t u d e n t s a n d o n e American. T h u s the p r o b a b i l i t y of this o u t c o m e will be 3 p 2 q . Similarly, the p r o b a b i l i t y for t w o A m e r i c a n s a n d o n e foreign s t u d e n t is 3 p q 2 . A convenient way to s u m m a r i z e these results is by m e a n s of the b i n o m i a l e x p a n s i o n , which is applicable to samples of a n y size f r o m p o p u l a t i o n s in which objects occur i n d e p e n d e n t l y in only t w o c l a s s e s — s t u d e n t s w h o m a y be foreign or American, o r individuals w h o m a y be d e a d or alive, male o r female, black o r white, r o u g h or s m o o t h , a n d so forth. This is accomplished by e x p a n d i n g the b i n o m i a l term (p + q f , where k e q u a l s s a m p l e size, ρ e q u a l s the p r o b a b i l i t y of occurrence of the first class, a n d q e q u a l s the probability of o c c u r r e n c e of the second class. By definition, ρ + q = 1; hence q is a f u n c t i o n of p: q = 1 — p. W e shall e x p a n d the expression for samples of k f r o m 1 to 3:

F o r samples of 1, (p + q)1 = ρ + q F o r samples of 2, (p + q)2 = p2 + 2pq + q2 F o r s a m p l e s of 3, (p + q)3 = p3 + 3 p 2 q + 3pq 2 + q3 It will be seen t h a t these expressions yield the s a m e probability spaces discussed previously. T h e coefficients (the n u m b e r s before the p o w e r s of ρ a n d q) express the n u m b e r of ways a p a r t i c u l a r o u t c o m e is o b t a i n e d . An easy m e t h o d for e v a l u a t i n g the coefficients of the e x p a n d e d t e r m s of the binomial expression

55

4 . 2 / THE BINOMIAL DISTRIBUTION

is t h r o u g h t h e use of Pascal's triangle: k 1 2 3 4 5

1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Pascal's triangle p r o v i d e s the coefficients of the b i n o m i a l e x p r e s s i o n — t h a t is, the n u m b e r of possible o u t c o m e s of the v a r i o u s c o m b i n a t i o n s of events. F o r k = 1 t h e coefficients a r e 1 a n d 1. F o r the second line (k = 2), write 1 at the left-hand m a r g i n of the line. T h e 2 in the middle of this line is the s u m of the values t o the left a n d right of it in the line above. T h e line is c o n c l u d e d with a 1. Similarly, the values at the beginning a n d e n d of the third line a r e 1, a n d the o t h e r n u m b e r s a r e s u m s of the values t o their left a n d right in t h e line above; t h u s 3 is the s u m of 1 a n d 2. This principle c o n t i n u e s for every line. Y o u can w o r k out the coefficients for any size s a m p l e in this m a n n e r . T h e line for fc = 6 w o u l d consist of the following coefficients: 1, 6, 15, 20, 15, 6, 1. T h e ρ a n d q values bear p o w e r s in a consistent p a t t e r n , which should be easy to imitate for any value of k. W e give it here for k = 4: pV

+ pV

+ p V

+ pY

+ p°q4

T h e p o w e r of ρ decreases f r o m 4 t o 0 (k to 0 in t h e general case) as t h e p o w e r of q increases f r o m 0 t o 4 (0 to k in the general case). Since a n y value t o t h e power 0 is 1 a n d a n y term to the p o w e r 1 is simply itself, we c a n simplify this expression as s h o w n below a n d at the s a m e time provide it with t h e coefficients f r o m Pascal's triangle for t h e case k — 4: p 4 + 4p3<7 + 6 p V + 4pq3 + q4 T h u s we are able t o write d o w n a l m o s t by inspection the e x p a n s i o n of the b i n o m i a l to any r e a s o n a b l e power. Let us n o w practice o u r newly learned ability to e x p a n d the binomial. S u p p o s e we have a p o p u l a t i o n of insects, exactly 40% of which a r e infected with a given virus X. If we t a k e samples of k = 5 insects each a n d e x a m i n e e a c h insect separately for presence of the virus, w h a t distribution of s a m p l e s could we expect if the p r o b a b i l i t y of infection of each insect in a s a m p l e were i n d e p e n d e n t of that of o t h e r insects in the sample? In this case ρ = 0.4, the p r o p o r t i o n infected, a n d q = 0.6, the p r o p o r t i o n not infected. It is a s s u m e d t h a t the p o p u l a t i o n is so large t h a t the question of w h e t h e r s a m p l i n g is with o r w i t h o u t replacement is irrelevant for practical purposes. T h e expected p r o p o r tions would be t h e e x p a n s i o n of t h e binomial: (p + q)k = (0.4 + 0.6) 5

CHAPTER 4 / INTRODUCTION TO PROBABILITY DISTRIBUTIONS

56

With the aid of Pascal's triangle this e x p a n s i o n is {p 5 + 5 p*q + 10p3q2 + 1 0 p V + 5 pq4 + q5} or (0.4) 5 + 5(0.4) 4 (0.6) + 10(0.4) 3 (0.6) 2 + 10(0.4) 2 (0.6) 3 + 5(0.4)(0.6) 4 + (0.6) 5 representing t h e expected p r o p o r t i o n s of samples of five infected insects, f o u r infected a n d o n e noninfected insects, three infected a n d t w o noninfected insects, a n d so on. T h e reader h a s p r o b a b l y realized by n o w t h a t the terms of t h e b i n o m i a l e x p a n s i o n actually yield a type of f r e q u e n c y distribution for these different o u t c o m e s . Associated with e a c h o u t c o m e , such as "five infected insects," there is a p r o b a b i l i t y of o c c u r r e n c e — i n this case (0.4) 5 = 0.01024. This is a theoretical frequency d i s t r i b u t i o n o r probability distribution of events that can o c c u r in t w o classes. It describes the expected distribution of o u t c o m e s in r a n d o m s a m p l e s of five insects f r o m a p o p u l a t i o n in which 40% are infected. T h e p r o b a b i l i t y d i s t r i b u t i o n described here is k n o w n as the binomial distribution, a n d the b i n o mial e x p a n s i o n yields the expected frequencies of the classes of t h e b i n o m i a l distribution. A convenient l a y o u t for p r e s e n t a t i o n a n d c o m p u t a t i o n of a b i n o m i a l d i s t r i b u t i o n is s h o w n in T a b l e 4.1. T h e first c o l u m n lists t h e n u m b e r of infected insects per sample, the second c o l u m n shows decreasing p o w e r s of ρ f r o m p s to p°, a n d the third c o l u m n shows increasing powers of q f r o m q° to q5. T h e b i n o m i a l coefficients f r o m Pascal's triangle are s h o w n in c o l u m n (4). T h e relative

TABLE 4 . 1

Expected frequencies of infected insects in samples of 5 insects sampled from an infinitely large population with an assumed infection rate of 40",,. U) Number of infected insects per sample V

(2) Powers of ρ = 0.4

(3) Powers of q = 0.6

Μ Binomial coefficients

(5) Relative expected frequencies

(6) Absolute expected frequencies

(7) Observed frequencies

L

f

f

1.00000 0.60000

1

0.01024

5

0.07680

3

0.02560 0.06400

0.36000

10

0.23040

2

0.16000

0.21600

10

0.34560

1

0.40000 1.00000

0.12960 0.07776

5 1

5 4

0

0.01024

Σ /

or

Z

y

Mean Standard

197

558.3 837.4

535 817

628.0 188.4

643

0.07776 1.00000 2.00000

2423.0 4846.1

2423

2.00000 1.09545

2.00004

0.25920

£ / ( = « )

deviation

29

24.8 186.1

1.09543

202

4815 1.98721 1.11934

4.2 / THE BINOMIAL DISTRIBUTION

57

expected frequencies, w h i c h a r e t h e p r o b a b i l i t i e s of the v a r i o u s o u t c o m e s , a r e s h o w n in c o l u m n (5). W e label s u c h e x p e c r e d f r e q u e n c i e s / r e l . T h e y a r e simply the p r o d u c t of c o l u m n s (2), (3), a n d (4). T h e i r s u m is e q u a l t o 1.0, since t h e events listed in c o l u m n (1) e x h a u s t the p o s s i b l e o u t c o m e s . W e see f r o m c o l u m n (5) in T a b l e 4.1 t h a t o n l y a b o u t 1% of s a m p l e s a r e e x p e c t e d t o c o n s i s t of 5 infected insects, a n d 25.9% a r e e x p e c t e d t o c o n t a i n 1 infected a n d 4 n o n i n f e c t e d insects. W e shall test w h e t h e r these p r e d i c t i o n s h o l d in a n a c t u a l e x p e r i m e n t . Experiment 4.1. S i m u l a t e t h e s a m p l i n g o f i n f e c t e d i n s e c t s b y u s i n g a t a b l e o f r a n d o m n u m b e r s s u c h a s T a b l e I in A p p e n d i x A l . T h e s e a r e r a n d o m l y c h o s e n o n e - d i g i t n u m b e r s in w h i c h e a c h d i g i t 0 t h r o u g h 9 h a s a n e q u a l p r o b a b i l i t y of a p p e a r i n g . T h e n u m b e r s a r e g r o u p e d in b l o c k s o f 2 5 f o r c o n v e n i e n c e . S u c h n u m b e r s c a n a l s o b e o b t a i n e d f r o m r a n d o m n u m b e r k e y s o n s o m e p o c k e t c a l c u l a t o r s a n d b y m e a n s of p s e u d o r a n d o m n u m b e r - g e n e r a t i n g a l g o r i t h m s in c o m p u t e r p r o g r a m s . (In fact, this e n t i r e e x p e r i m e n t can be p r o g r a m m e d a n d performed a u t o m a t i c a l l y — e v e n on a small computer.) Since t h e r e is a n e q u a l p r o b a b i l i t y f o r a n y o n e d i g i t t o a p p e a r , y o u c a n let a n y f o u r d i g i t s ( s a y , 0, 1, 2, 3) s t a n d f o r t h e i n f e c t e d i n s e c t s a n d t h e r e m a i n i n g d i g i t s (4, 5, 6, 7, 8, 9) s t a n d for the n o n i n f e c t e d insects. T h e p r o b a b i l i t y that a n y o n e digit selected f r o m the t a b l e will r e p r e s e n t a n i n f e c t e d i n s e c t ( t h a t is, will b e a 0, 1, 2, o r 3) is t h e r e f o r e 4 0 % , o r 0.4, s i n c e t h e s e a r e f o u r o f t h e t e n p o s s i b l e d i g i t s . A l s o , s u c c e s s i v e d i g i t s a r e a s s u m e d t o b e i n d e p e n d e n t of t h e v a l u e s o f p r e v i o u s d i g i t s . T h u s t h e a s s u m p t i o n s o f t h e b i n o m i a l d i s t r i b u t i o n s h o u l d b e m e t in t h i s e x p e r i m e n t . E n t e r t h e t a b l e o f r a n d o m n u m b e r s a t a n a r b i t r a r y p o i n t ( n o t a l w a y s at t h e b e g i n n i n g ! ) a n d l o o k at s u c c e s s i v e g r o u p s of five d i g i t s , n o t i n g in e a c h g r o u p h o w m a n y of t h e d i g i t s a r e 0, 1, 2, o r 3. T a k e a s m a n y g r o u p s of five a s y o u c a n f i n d t i m e t o d o , b u t n o f e w e r t h a n 100 g r o u p s .

C o l u m n (7) in T a b l e 4.1 s h o w s the results of o n e such e x p e r i m e n t d u r i n g o n e year by a b i o s t a t i s t i c s class. A total of 2423 s a m p l e s of five n u m b e r s were o b t a i n e d f r o m t h e t a b l e of r a n d o m n u m b e r s ; the d i s t r i b u t i o n of t h e f o u r digits s i m u l a t i n g t h e p e r c e n t a g e of infection is s h o w n in this c o l u m n . T h e o b s e r v e d f r e q u e n c i e s a r c l a b e l e d / . T o c a l c u l a t e the e x p e c t c d f r e q u e n c i e s for this a c t u a l e x a m p l e we m u l t i p l i e d the relative f r e q u e n c i e s / r c , of c o l u m n (5) times η = 2423, the n u m b e r of s a m p l e s t a k e n . T h i s results in absolute expected frequencies, labeled / , s h o w n in c o l u m n (6). W h e n w e c o m p a r e the o b s e r v e d f r e q u e n c i e s in c o l u m n (7) with the e x p e c t e d f r e q u e n c i e s in c o l u m n (6) we n o t e general a g r e e m e n t b e t w e e n t h e t w o c o l u m n s of figures. T h e t w o d i s t r i b u t i o n s a r e a l s o illustrated in F i g u r e 4.2. If the o b s e r v e d f r e q u e n c i e s did not fit expected f r e q u e n c i e s , we might believe t h a t the lack of fit w a s d u e t o c h a n c e a l o n e . O r w e might be led t o reject o n e o r m o r e of the f o l l o w i n g h y p o t h e s e s : (I) t h a t the t r u e p r o p o r t i o n of digits 0, 1 , 2 , a n d 3 is 0.4 (rejection of this h y p o t h e s i s w o u l d n o r m a l l y not be r e a s o n a b l e , f o r w e m a y rely o n the fact t h a t t h e p r o p o r t i o n of digits 0, 1. 2, a n d 3 in a t a b l e of r a n d o m n u m b e r s is 0.4 o r very close to it); (2) t h a t s a m p l i n g w a s at r a n d o m ; a n d (3) t h a t e v e n t s were i n d e p e n d e n t . T h e s e s t a t e m e n t s c a n be r e i n t e r p r e t e d in t e r m s of the o r i g i n a l infection m o d e l with w h i c h we s t a r t e d this d i s c u s s i o n . If, i n s t e a d of a s a m p l i n g e x p e r i m e n t of digits by a biostatistics class, this h a d been a real s a m p l i n g e x p e r i m e n t of insects, we w o u l d c o n c l u d e that the insects h a d indeed been r a n d o m l y s a m p l e d

chapter 4 / introduction t o probability

58

distributions

Observed frequencies

900 Π

H00

Expected frequencies

700 C,'

f

">00

k

400 :ioo '200

100

ο

0

1

2

I

5

N u m b e r of i n f e c t e d insects per s a m p l e FIGURE 4 . 2

B a r d i a g r a m of o b s e r v e d a n d expected frequencies given in T a b l e 4.1.

a n d t h a t we h a d n o evidence to reject the hypothesis t h a t the p r o p o r t i o n of infected insects was 40%. If the observed frequencies h a d not fitted the expected frequencies, the lack of fit m i g h t be a t t r i b u t e d t o chance, o r t o t h e conclusion t h a t the true p r o p o r t i o n of infection was not 0.4; o r we w o u l d h a v e h a d to reject o n e o r b o t h of the following a s s u m p t i o n s : (1) t h a t s a m p l i n g was at r a n d o m a n d (2) t h a t the o c c u r r e n c e of infected insects in these samples w a s i n d e p e n d e n t . E x p e r i m e n t 4.1 w a s designed to yield r a n d o m samples a n d i n d e p e n d e n t events. H o w could we simulate a s a m p l i n g p r o c e d u r e in which the occurrences of the digits 0, 1, 2, a n d 3 were not i n d e p e n d e n t ? We could, for example, instruct the sampler t o s a m p l e as indicated previously, but, every time a 3 w a s f o u n d a m o n g t h e first four digits of a sample, t o replace t h e following digit with a n o t h e r o n e of the four digits s t a n d i n g for infected individuals. T h u s , once a 3 w a s f o u n d , the p r o b a b i l i t y w o u l d be 1.0 t h a t a n o t h e r o n e of the indicated digits w o u l d be included in the sample. After repeated samples, this would result in higher frequencies of s a m p l e s of t w o o r m o r e indicated digits a n d in lower frequencies t h a n expected (on the basis of the b i n o m i a l distribution) of s a m p l e s of o n e such digit. A variety of such different s a m p l i n g schemes could be devised. It s h o u l d b e q u i t e clear t o t h e reader that t h e probability of the second event's o c c u r r i n g w o u l d be different f r o m that of the first a n d d e p e n d e n t on it. H o w w o u l d we interpret a large d e p a r t u r e of the observed frequencies f r o m expectation? W e have not as yet learned techniques for testing w h e t h e r observed frequencies differ f r o m those expected by m o r e t h a n c a n be a t t r i b u t e d to c h a n c e alone. T h i s will be taken u p in C h a p t e r 13. A s s u m e t h a t such a test h a s been carried o u t a n d t h a t it has s h o w n us that o u r observed frequencies a r e significantly different f r o m expectation. T w o m a i n types of d e p a r t u r e f r o m exp e c t a t i o n can be characterized: (1) clumping a n d (2) repulsion, s h o w n in fictitious

4 . 2 / THE BINOMIAL DISTRIBUTION

59

TABLE 4 . 2

Artificial distributions to illustrate clumping and repulsion. E x p e c t e d frequencies f r o m T a b l e 4.1.

(/) Number of infected insects per sample Y

5 4 3 2 1 0 Σ/or

Σγ

η

Mean Standard deviation


(2) Absolute expected frequencies

Clumped (contagious) frequencies

f

f

(4) Deviation from expectation

+

(5) Repulsed frequencies f

14 157 548 943 618 143

24.8 186.1 558.3 837.4 628.0 188.4

47 227 558 663 703 225

2423.0 4846.1

2423 4846

2423.0 4846

2.00004 1.09543

2.00000 1.20074

2.00000 1.01435

-ΙΟ —

+ +

«5) Deviation from expectation

_ — -

+ — -

examples in T a b l e 4.2. In actual examples we w o u l d have n o a priori n o t i o n s a b o u t the m a g n i t u d e of p, the probability of o n e of the t w o possible o u t c o m e s . In such cases it is c u s t o m a r y to o b t a i n ρ f r o m the observed s a m p l e a n d to calculate the expected frequencies, using the s a m p l e p. This w o u l d m e a n that the hypothesis t h a t ρ is a given value c a n n o t be tested, since by design the expected frequencies will have t h e same ρ value as the observed frequencies. Therefore, the hypotheses tested a r e w h e t h e r the samples a r e r a n d o m a n d the events independent. T h e c l u m p e d frequencies in T a b l e 4.2 have an excess of o b s e r v a t i o n s at the tails of the frequency distribution a n d consequently a s h o r t a g e of o b s e r v a t i o n s at the center. Such a distribution is also said to be contagious. ( R e m e m b e r that the total n u m b e r of items m u s t be the s a m e in b o t h observed a n d expected frequencies in o r d e r t o m a k e t h e m c o m p a r a b l e . ) In the repulsed frequency distrib u t i o n there are m o r e o b s e r v a t i o n s t h a n expected at the center of the distribution a n d fewer at the tails. These discrepancies are most easily seen in c o l u m n s (4) and (6) of T a b l e 4.2, where t h e deviations of observed f r o m expected frequencies are s h o w n as plus o r m i n u s signs. W h a t d o these p h e n o m e n a imply? In the c l u m p e d frequencies, m o r e samples were entirely infected (or largely infected), a n d similarly, m o r e samples were entirely noninfected (or largely noninfected) t h a n you would expect if p r o b a bilities of infection were independent. This could be d u e t o p o o r s a m p l i n g design. If, for example, the investigator in collecting samples of five insects always tended t o pick out like o n e s — t h a t is, infected o n e s or noninfected o n e s — t h e n such a result would likely a p p e a r . But if the s a m p l i n g design is s o u n d , the results b e c o m e m o r e interesting. C l u m p i n g would then m e a n t h a t the samples of five a r e in some way related, so that if o n e insect is infected, o t h e r s in the

60

CHAPTER 4 / INTRODUCTION TO PROBABILITY DISTRIBUTIONS

s a m e s a m p l e a r e m o r e likely to be infected. This could be t r u e if they c o m e f r o m a d j a c e n t l o c a t i o n s in a s i t u a t i o n in which n e i g h b o r s are easily infected. O r they could be siblings j o i n t l y exposed to a source of infection. O r possibly the infection m i g h t s p r e a d a m o n g m e m b e r s of a sample between t h e time that the insects are s a m p l e d a n d the time they a r e examined. T h e o p p o s i t e p h e n o m e n o n , repulsion, is m o r e difficult t o interpret b i o logically. T h e r e are fewer h o m o g e n e o u s g r o u p s a n d m o r e mixed g r o u p s in such a distribution. T h i s involves the idea of a c o m p e n s a t o r y p h e n o m e n o n : if s o m e of the insects in a s a m p l e a r e infected, the o t h e r s in the s a m p l e are less likely to be. If the infected insects in the s a m p l e could in some w a y t r a n s m i t imm u n i t y to their associates in t h e sample, such a situation could arise logically, b u t it is biologically i m p r o b a b l e . A m o r e r e a s o n a b l e i n t e r p r e t a t i o n of such a finding is t h a t for e a c h s a m p l i n g unit, there were only a limited n u m b e r of p a t h o g e n s available; then once several of the insects have b e c o m e infected, t h e o t h e r s go free of infection, simply because there is n o m o r e infectious agent. This is an unlikely situation in microbial infections, but in situations in which a limited n u m b e r of p a r a s i t e s enter t h e b o d y of the host, repulsion m a y b e m o r e reasonable. F r o m the expected a n d observed frequencies in T a b l e 4.1, we m a y calculate the m e a n a n d s t a n d a r d deviation of the n u m b e r of infected insects per sample. These values are given at the b o t t o m of c o l u m n s (5), (6), a n d (7) in T a b l e 4.1. W e n o t e that the m e a n s a n d s t a n d a r d deviations in c o l u m n s (5) a n d (6) a r e a l m o s t identical a n d differ only trivially because of r o u n d i n g errors. C o l u m n (7), being a sample f r o m a p o p u l a t i o n whose p a r a m e t e r s are the s a m e as those of the expected frequency distribution in c o l u m n (5) or (6), differs s o m e w h a t . T h e m e a n is slightly smaller a n d the s t a n d a r d deviation is slightly greater t h a n in the expected frequencies. If we wish to k n o w the m e a n a n d s t a n d a r d d e v i a t i o n of expected binomial frequency distributions, we need not go t h r o u g h the c o m p u t a t i o n s s h o w n in T a b l e 4.1. T h e m e a n a n d s t a n d a r d deviation of a b i n o m i a l frequency distribution are, respectively,

μ = kp

σ = \Jkpq

S u b s t i t u t i n g the values k = 5, ρ = 0.4, a n d q = 0.6 of the a b o v e example, we o b t a i n μ = 2.0 a n d σ = 1.095,45, which are identical to the values c o m p u t e d f r o m c o l u m n (5) in T a b l e 4.1. N o t e that we use the G r e e k p a r a m e t r i c n o t a t i o n here because μ a n d a are p a r a m e t e r s of an expected frequency d i s t r i b u t i o n , not s a m p l e statistics, as a r e the m e a n a n d s t a n d a r d deviation in c o l u m n (7). T h e p r o p o r t i o n s ρ a n d q a r e p a r a m e t r i c values also, and strictly speaking, they should be distinguished from s a m p l e p r o p o r t i o n s . In fact, in later c h a p t e r s we resort to ρ a n d q for p a r a m e t r i c p r o p o r t i o n s (rather t h a n π, which c o n v e n t i o n ally is used as the ratio of the circumfcrence to the d i a m e t e r of a circle). Here, however, we prefer to keep o u r n o t a t i o n simple. If we wish to express o u r variable as a p r o p o r t i o n r a t h e r than as a c o u n t — t h a t is, to indicate m e a n incidence of infection in the insccts as 0.4, r a t h e r t h a n as 2 per sample of 5 we can use o t h e r f o r m u l a s for the m e a n a n d s t a n d a r d deviation in a binomial

61

4.2 / THE BINOMIAL DISTRIBUTION

distribution: μ = ρ

σ

=

It is interesting t o look at the s t a n d a r d deviations of the c l u m p e d a n d replused frequency distributions of T a b l e 4.2. W e n o t e that the c l u m p e d distrib u t i o n h a s a s t a n d a r d deviation greater t h a n expected, a n d t h a t of the repulsed one is less t h a n expected. C o m p a r i s o n of sample s t a n d a r d deviations with their expected values is a useful m e a s u r e of dispersion in such instances. W e shall n o w e m p l o y the b i n o m i a l distribution t o solve a biological p r o b lem. O n the basis of o u r k n o w l e d g e of the cytology a n d biology of species A, we expect the sex r a t i o a m o n g its offspring to be 1:1. T h e study of a litter in n a t u r e reveals t h a t of 17 offspring 14 were females a n d 3 were males. W h a t conclusions can we d r a w f r o m this evidence? A s s u m i n g t h a t (the probability of being a female offspring) = 0.5 a n d t h a t this probability is i n d e p e n d e n t a m o n g the m e m b e r s of t h e sample, the pertinent p r o b a b i l i t y distribution is the b i n o m i a l for s a m p l e size k = 17. E x p a n d i n g the b i n o m i a l t o the p o w e r 17 is a f o r m i d a b l e task, which, as we shall see, f o r t u n a t e l y need n o t be d o n e in its entirety. H o w ever, we m u s t have the b i n o m i a l coefficients, which c a n be o b t a i n e d either from an e x p a n s i o n of Pascal's triangle (fairly tedious unless once o b t a i n e d a n d stored for f u t u r e use) or by w o r k i n g o u t t h e expected frequencies for any given class of Y f r o m the general f o r m u l a for any term of the b i n o m i a l distribution C(k,

Y)prqk

Y

(4.1)

T h e expression C(/c, y ) s t a n d s for the n u m b e r of c o m b i n a t i o n s t h a t can be formed f r o m k items t a k e n Y at a time. This can be evaluated as kl/[ Y!(k — V)!], where ! m e a n s "factorial." In m a t h e m a t i c s k factorial is the p r o d u c t of all the integers from 1 u p t o a n d including k. T h u s , 5! = 1 χ 2 χ 3 χ 4 χ 5 = 120. By convention, 0! = 1. In w o r k i n g out fractions c o n t a i n i n g factorials, n o t e that any factorial will always cancel against a higher factorial. T h u s 5!/3! = (5 χ 4 χ 3!)/ 3! = 5 χ 4. F o r example, the binomial coefficient for the expected frequency of samples of 5 items c o n t a i n i n g 2 infected insects is C(5, 2) = 5!/2!3! = (5 χ 4)/2 = 10. T h e s e t u p of t h e e x a m p l e is s h o w n in T a b l e 4.3. Decreasing p o w e r s of ρ f r o m p,17 d o w n a n d increasing powers of q . a r e c o m p u t e d (from p o w e r 0 to p o w e r 4). Since we require the probability of 14 females, we n o t e that for the p u r p o s e s of o u r p r o b l e m , we need not proceed b e y o n d the term for 13 females a n d 4 males. C a l c u l a t i n g the relative expected frequencies in c o l u m n (6), we note that the probability of 14 females a n d 3 males is 0.005,188,40, a very small value. If we a d d t o this value all "worse" o u t c o m e s — t h a t is, all o u t c o m e s that are even m o r e unlikely t h a n 14 females a n d 3 males on the a s s u m p t i o n of a 1:1 h y p o t h e s i s — w e o b t a i n a probability of 0.006,363,42, still a very small value. (In statistics, we often need to calculate the p r o b a b i l i t y of observing a deviation as large as or larger than a given value.)

chapter 4 / introduction to probability

62

distributions

TABLE 4 . 3

S o m e expected frequencies of males and females for samples of 17 offspring on the assumption that the sex ratio is 1:1 [ p v = 0.5, q. = 0.5; ( p . + q-)k = (0.5 + 0 . 5 ) ' 7 ] .

(/)

(2)

$$ 17 16 15 14 13

1 2 3 4

(3)

(4)

Pi

is

0.000,007,63 0.000,015,26 0.000,030,52 0.000,061,04 0.000,122,07

1 0.5 0.25 0.125 0.0625

(5) Binomial coefficients

1 17 136 680 2380

(6) Relative expected frequencies

L 0.000,007,631 0.000,129,711 0.001,037,681 υ · υ υ 6 ' · 3 Μ ' 4 2 0.005,188,40j 0.018,157,91

O n the basis of these findings o n e o r m o r e of the following a s s u m p t i o n s is unlikely: (1) t h a t the t r u e sex r a t i o in species A is 1:1, (2) t h a t we have s a m p l e d at r a n d o m in t h e sense of o b t a i n i n g an u n b i a s e d sample, or (3) t h a t the sexes of t h e offspring are i n d e p e n d e n t of o n e a n o t h e r . Lack of i n d e p e n d e n c e of events m a y m e a n t h a t a l t h o u g h the average sex r a t i o is 1:1, the individual sibships, o r litters, are largely unisexual, so that the offspring f r o m a given m a t i n g w o u l d tend to be all (or largely) females or all (or largely) males. T o c o n f i r m this hypothesis, we w o u l d need t o have m o r e samples a n d then examine the distrib u t i o n of samples for clumping, which w o u l d indicate a tendency for unisexual sibships. W e m u s t be very precise a b o u t the q u e s t i o n s we ask of o u r d a t a . T h e r e are really t w o questions we could ask a b o u t the sex ratio. First, are the sexes u n e q u a l in frequency so t h a t females will a p p e a r m o r e often t h a n males? Second, a r e the sexes u n e q u a l in frequency? It m a y be t h a t we k n o w from past experience t h a t in this particular g r o u p of o r g a n i s m s the males are never m o r e f r e q u e n t t h a n females; in that case, we need be c o n c e r n e d only with the first of these t w o questions, and the r e a s o n i n g followed a b o v e is a p p r o p r i a t e . H o w e v e r , if we k n o w very little a b o u t this g r o u p of organisms, a n d if o u r q u e s t i o n is simply whether the sexes a m o n g the offspring are unequal in frequency, then we have to consider b o t h tails of the binomial f r e q u e n c y distribution; d e p a r t u r e s f r o m the 1:1 ratio could occur in either direction. We should then consider not only the p r o b a b i l i t y of samples with 14 females a n d 3 males (and all worse cases) but also the probability of samples of 14 males a n d 3 females (and all worse cases in t h a t direction). Since this probability distribution is symmetrical (because p, = q . = 0.5), we simply d o u b l e the c u m u l a t i v e probability of 0.006,363,42 o b tained previously, which results in 0.012,726,84. This new value is still very small, m a k i n g it quite unlikely that the true sex ratio is 1:1. This is y o u r first experience with o n e of the m o s t i m p o r t a n t a p p l i c a t i o n s of statistics —hypothesis testing. A f o r m a l i n t r o d u c t i o n t o this field will be deferred

4.2

/ t h e BINoMiAL d i s t r i b u t i o n

63

until Section 6.8. W e m a y simply p o i n t o u t here t h a t the t w o a p p r o a c h e s followed a b o v e are k n o w n a p p r o p r i a t e l y as one-tailed, tests a n d two-tailed tests, respectively. S t u d e n t s sometimes have difficulty k n o w i n g which of the t w o tests to apply. In f u t u r e examples we shall try to p o i n t out in each case w h y a onetailed or a two-tailed test is being used. W e have said t h a t a tendency for unisexual sibships w o u l d result in a clumped distribution of observed frequencies. An actual case of this n a t u r e is a classic in t h e literature, the sex r a t i o d a t a o b t a i n e d by Geissler (1889) f r o m hospital records in Saxony. T a b l e 4.4 r e p r o d u c e s sex ratios of 6115 sibships of 12 children each f r o m t h e m o r e extensive study by Geissler. All c o l u m n s of the table should by n o w be familiar. T h e expected frequencies were not calculated on the basis of a 1:1 hypothesis, since it is k n o w n t h a t in h u m a n p o p u l a t i o n s the sex ratio at b i r t h is n o t 1:1. As the sex r a t i o varies in different h u m a n populations, the best estimate of it for the p o p u l a t i o n in S a x o n y was simply o b t a i n e d using the m e a n p r o p o r t i o n of males in these d a t a . This can be o b t a i n e d by calculating the average n u m b e r of males per sibship ( F = 6.230,58) for the 6115 sibships a n d c o n v e r t i n g this i n t o a p r o p o r t i o n . This value t u r n s out to be 0.519,215. C o n s e q u e n t l y , t h e p r o p o r t i o n of females = 0.480,785. In the deviations of the observed frequencies f r o m the a b s o l u t e expected frequencies s h o w n in c o l u m n (9) of T a b l e 4.4, we notice considerable clumping. T h e r e are m a n y m o r e instances of families with all male or all female children (or nearly so) than i n d e p e n d e n t probabilities would indicate. T h e genetic basis for this is not clear, b u t it is evident t h a t there a r e some families which " r u n t o girls" a n d similarly those which " r u n t o boys." Evidence of c l u m p i n g can also be seen f r o m the fact t h a t s2 is m u c h larger t h a n we would expect on the basis of t h e b i n o m i a l distribution (σ 2 = kpq = 12(0.519,215)0.480,785 = 2.995,57). T h e r e is a distinct c o n t r a s t between the d a t a in T a b l e 4.1 a n d those in Table 4.4. In the insect infection d a t a of T a b l e 4.1 we h a d a hypothetical p r o p o r tion of infection based on outside knowledge. In the sex r a t i o d a t a of T a b l e 4.4 we h a d n o such knowledge; we used an empirical value of ρ obtained from the data, r a t h e r t h a n a hypothetical value external to the data. This is a distinction whose i m p o r t a n c e will b e c o m e a p p a r e n t later. In the sex r a t i o d a t a of T a b l e 4.3, as in m u c h w o r k in M e n d e l i a n genetics, a hypothetical value of ρ is used.

4.3 The Poisson distribution In the typical a p p l i c a t i o n of the binomial we h a d relatively small s a m p l e s (2 students, 5 insects, 17 offspring, 12 siblings) in which t w o alternative states occurred at varying frequencies (American a n d foreign, infected a n d n o n i n fected, male a n d female). Q u i t e frequently, however, we study cases in which sample size k is very large a n d o n e of the events (represented by p r o b a b i l i t y q) is very m u c h m o r e f r e q u e n t t h a n the o t h e r (represented by probability p). W e have seen t h a t the e x p a n s i o n of the binomial (p + qf is quite tiresome w h e n k is large. S u p p o s e y o u h a d to e x p a n d t h e expression (0.001 + 0.999) 1 0 0 0 . In such cases we are generally interested in o n e tail of the distribution only. This is the

ο — '5 οε

+ + + +

i + + + +

+

^ O O h N ^ ' t n h M O N

iu

C o'

3 ξ I"-, -oCO a . 3 χ β" δ .5! 5/ϊ

a» r—ν©

·~ qj

•I a

Μ

m —c oo Ο <*ϊ NO 00 < ON ο >/"i 00 oo fN Ο 1Λ
in oo n o fS rn CI s γ-'
|

00 Ο Ο γ λ ο ο ρt— o n V) ο οο τ γ ρ·^ ίΝ r- σν crC n o <Ν r i fS Ο m — <οN<Ν Ο Ο Ο ο ο ο ο ο ο ο

s s§ g

5 -Si

ν t

o\

Tt

=0 ο

ΜΙΛΓ'κηΟΟ'Λ'Ί'^Ι— r-- —η ^ \o m o\ oo ci O ·—• —' rn c-f lr'> r i •—' τ»·<Ν—'OOOOOO O O O O O O O O O O O

IS JS

Τ 'C * (X

cn ρ
§

Ο Η

4.2

65

/ t h e BINoMiAL d i s t r i b u t i o n

tail represented by the terms

p°q\ C(k,

1 ) / > Y ~ \ C(k, 2 ) p V " 2 , C(k,

3)p3qk~\ . -.

T h e first term represents n o r a r e events a n d k frequent events in a s a m p l e of k events. T h e second t e r m represents o n e r a r e event a n d k — 1 f r e q u e n t events. T h e third term represents t w o r a r e events a n d k — 2 f r e q u e n t events, a n d so forth. T h e expressions of the f o r m C(k, i) are the b i n o m i a l coefficients, represented by the c o m b i n a t o r i a l terms discussed in the previous section. A l t h o u g h the desired tail of the curve could be c o m p u t e d by this expression, as long as sufficient decimal accuracy is m a i n t a i n e d , it is c u s t o m a r y in such cases to c o m p u t e a n o t h e r distribution, the Poisson distribution, which closely a p p r o x i m a t e s the desired results. As a rule of t h u m b , we m a y use the P o i s s o n distribution to a p p r o x i m a t e the b i n o m i a l distribution w h e n the p r o b a b i l i t y of the rare event ρ is less t h a n 0.1 a n d the p r o d u c t kp (sample size χ probability) is less t h a n 5. T h e Poisson distribution is also a discrete frequency distribution of the n u m b e r of times a r a r e event occurs. But, in c o n t r a s t to the binomial distribution, t h e Poisson distribution applies t o cases where the n u m b e r of times that a n event does n o t occur is infinitely large. F o r p u r p o s e s of o u r t r e a t m e n t here, a P o i s s o n variable will be studied in samples t a k e n over space o r time. An example of the first would be the n u m b e r of m o s s p l a n t s in a s a m p l i n g q u a d r a t on a hillside or the n u m b e r of parasites on a n individual host; a n e x a m p l e of a t e m p o r a l sample is the n u m b e r of m u t a t i o n s occurring in a genetic strain in the time interval of o n e m o n t h or the r e p o r t e d cases of influenza in o n e t o w n d u r i n g one week. T h e Poisson variable Y will be the n u m b e r of events per sample. It can a s s u m e discrete values f r o m 0 o n up. T o be d i s t r i b u t e d in Poisson fashion the variable m u s t have t w o properties: (I) Its m e a n m u s t be small relative t o the m a x i m u m possible n u m b e r of events per sampling unit. T h u s the event should be "rare." But this m e a n s t h a t o u r s a m p l i n g unit of space o r time must be large e n o u g h to a c c o m m o d a t e a potentially substantial n u m b e r of events. F o r example, a q u a d r a t in which m o s s plants are c o u n t e d m u s t be large e n o u g h t h a t a substantial n u m b e r of m o s s plants could occur there physically if the biological c o n d i t i o n s were such as to favor the d e v e l o p m e n t of n u m e r o u s moss plants in the q u a d r a t . A q u a d r a t consisting of a 1-cm s q u a r e w o u l d be far t o o small for mosses to be distributed in Poisson fashion. Similarly, a time s p a n of 1 m i n u t e would be unrealistic for r e p o r t i n g new influenza cases in a t o w n , b u t within 1 week a great m a n y such cases could occur. (2) An occurrence of the event m u s t be i n d e p e n d e n t of prior occurrences within the s a m p l i n g unit. T h u s , the presence of o n e m o s s plant in a q u a d r a t m u s t not e n h a n c e or diminish the probability t h a t o t h e r m o s s plants a r e developing in the q u a d r a t . Similarly, the fact t h a t o n e influenza case has been r e p o r t e d m u s t not affect the probability of r e p o r t i n g s u b s e q u e n t influenza cases. Events t h a t meet these c o n d i t i o n s (rare and random events) should be d i s t r i b u t e d in Poisson fashion. T h e p u r p o s e of fitting a Poisson distribution to n u m b e r s of rare events in n a t u r e is to test w h e t h e r the events occur independently with respect to each

66

chapter 4 / introduction to probability

distributions

other. If they do, they will follow the P o i s s o n distribution. If the o c c u r r e n c e of o n e event e n h a n c e s the p r o b a b i l i t y of a second such event, we o b t a i n a c l u m p e d , or c o n t a g i o u s , distribution. If the occurrence of one event impedes t h a t of a second such event in t h e s a m p l i n g unit, we o b t a i n a repulsed, o r spatially o r temporally u n i f o r m , distribution. T h e Poisson can be used as a test for r a n d o m ness or i n d e p e n d e n c e of d i s t r i b u t i o n not only spatially but also in time, as s o m e examples below will show. T h e Poisson distribution is n a m e d after the F r e n c h m a t h e m a t i c i a n P o i s s o n , w h o described it in 1837. It is an infinite series w h o s e terms a d d t o 1 (as m u s t be t r u e for a n y probability distribution). T h e series can be represented as ^ ·

1!έ>"'

2!ί>"'

3!<>μ'

(4.2)

4!?"

w h e r e the terms are the relative expected frequencies c o r r e s p o n d i n g t o the following c o u n t s of the rare event Y: 0,

1,

2,

3,

4,

T h u s , the first of these t e r m s represents the relative expected f r e q u e n c y of samples c o n t a i n i n g n o rare event; the second term, o n e rare event; t h e third term, t w o rare events; a n d so on. T h e d e n o m i n a t o r of each term c o n t a i n s e where e is t h e base of the n a t u r a l , or N a p i e r i a n , logarithms, a c o n s t a n t w h o s e value, a c c u r a t e to 5 decimal places, is 2.718,28. W e recognize μ as the p a r a m e t r i c m e a n of the distribution; it is a c o n s t a n t for a n y given problem. T h e e x c l a m a t i o n m a r k after the coefficient in the d e n o m i n a t o r m e a n s "factorial," as explained in the previous section. O n e way to learn m o r e a b o u t the Poisson distribution is to apply it to a n actual case. At the t o p of Box 4.1 is a well-known result f r o m the early statistical literature based on the d i s t r i b u t i o n of yeast cells in 400 squares of a h e m a c y t o meter, a c o u n t i n g c h a m b e r such as is used in m a k i n g c o u n t s of b l o o d cells a n d o t h e r microscopic objects suspended in liquid. C o l u m n (1) lists the n u m b e r of yeast cells observed in each h e m a c y t o m e t e r square, and c o l u m n (2) gives the observed f r e q u e n c y — t h e n u m b e r of squares c o n t a i n i n g a given n u m b e r of yeast cells. We n o t e that 75 s q u a r e s c o n t a i n e d n o yeast cells, but that m o s t s q u a r e s held either 1 or 2 cells. O n l y 17 s q u a r e s c o n t a i n e d 5 or m o r e yeast cells. W h y would we expect this frequency distribution to be d i s t r i b u t e d in Poisson fashion? W e have here a relatively rare event, the frequency of yeast cells per h e m a c y t o m e t e r s q u a r e , the m e a n of which has been calculated a n d f o u n d to be 1.8. T h a t is, on the average there are 1.8 cells per square. Relative to the a m o u n t of space provided in each s q u a r e and the n u m b e r of cells t h a t could have c o m e to rest in a n y o n e square, the actual n u m b e r f o u n d is low indeed. We might also expect that the occurrence of individual yeast cells in a s q u a r e is independent of the occurrence of o t h e r yeast cells. This is a c o m m o n l y e n c o u n t e r e d class of application of the Poisson distribution. T h e m e a n of the rare event is the only q u a n t i t y that we need to k n o w to calculate the relative expected frequencies of a Poisson distribution. Since we d o

4.2 / t h e BINoMiAL d i s t r i b u t i o n

67

BOX 4 1

Calculation of expected Poisson frequencies. Yeast cells in 400 squares of a hemacytometer: f = 1.8 cells per square; η - 400 squares sampled.

(/)

(3)

Μ

Absolute expected frequencies

Deviation from expectation /--/

(2)

Observed frequencies

Number of cells per square Y

/

/

66.1 119.0 107.1 64.3 28.9 10.41 3.1 0.8 •14.5 0.2

75 103 121 54 30 13Ί 2 1 •17 0 lj 40)

0 1 2 3 4 5 6 7 8 9

0.0.

+ —

+ —

+

4—

+

•

+

—

+

399.9

Source: "Student" (1907).

Computational

steps

Flow of computation based on Expression (4.3) multiplied by n, since we wish to obtain absolute expected frequencies,/. 1. Find e f in a table of exponentials or compute it using an exponential key: J _ e* „1.8 6.0496 η 400 66.12 2. f 0 6.0496 66.12(1.8)

3 . / W o ? 4. Λ

/ 2J t ^

5./3=/

2

Y

3 =

=119.02

= 107.11 '(f)- 64.27 107.1

119.02

t )

6 . / W 3

64.27

28.92

7.

28.92

10.41

Λ - A y

; Y 8·/6= Λ

"(τ) -

3.12

68

CHAPTER 4 / i n t r o d u c t i o n t o p r o b a b i l i t y

distributions

BOX 4.1 Continued 3 . 1 2 ^

-

0.80

10. Λ - Λ I = 0.8θ(~^ = 0.18 Total

39935

A

f

9

and beyond

0.05

At step 3 enter Ϋ as a constant multiplier. Then multiply it by n/er (quantity 2). At each subsequent step multiply the result of the previous step by ? and then divide by the appropriate integer.

•

n o t k n o w the p a r a m e t r i c m e a n of t h e yeast cells in this p r o b l e m , we e m p l o y a n e s t i m a t e (the sample m e a n ) a n d calculate expected frequencies of a Poisson d i s t r i b u t i o n with μ equal to the m e a n of the observed frequency d i s t r i b u t i o n of Box 4.1. It is c o n v e n i e n t for t h e p u r p o s e of c o m p u t a t i o n t o rewrite Expression (4.2) as a recursion f o r m u l a as follows: h = L

,(y)

for i = 1, 2, . . . ,

where / 0 =

(4.3)

N o t e first of all that the p a r a m e t r i c m e a n μ h a s been replaced by the s a m p l e m e a n Ϋ. Each term developed by this recursion f o r m u l a is m a t h e m a t i c a l l y exactly the same as its c o r r e s p o n d i n g term in Expression (4.2). It is i m p o r t a n t to m a k e n o c o m p u t a t i o n a l error, since in such a chain multiplication the correctness of each term d e p e n d s o n the accuracy of the term before it. Expression (4.3) yields relative expected frequencies. If, as is m o r e usual, a b s o l u t e expected frequencies are desired, simply set the first term / 0 to n/ey, where η is the n u m b e r of samples, a n d then proceed with the c o m p u t a t i o n a l steps as before. T h e a c t u a l c o m p u t a t i o n is illustrated in Box 4.1, a n d the expected frequencies so o b t a i n e d a r e listed in c o l u m n (3) of t h e frequency d i s t r i b u t i o n . W h a t have we learned f r o m this c o m p u t a t i o n ? W h e n we c o m p a r e the observed with the expected frequencies, we notice quite a good fit of o u r o b served frequencies t o a Poisson d i s t r i b u t i o n of m e a n 1.8, a l t h o u g h we have not as yet learned a statistical test for g o o d n e s s of fit (this will be covered in C h a p ter 13). N o clear p a t t e r n of deviations f r o m expectation is s h o w n . We c a n n o t test a hypothesis a b o u t the m e a n , because the m e a n of the expected distribution was t a k e n f r o m the s a m p l e m e a n of the observed variates. As in the binomial distribution, c l u m p i n g o r a g g r e g a t i o n w o u l d indicate that the probability that a second yeast cell will be f o u n d in a s q u a r e is not i n d e p e n d e n t of the prcs-

4.2

/ t h e BINoMiAL d i s t r i b u t i o n

69

ence of t h e first one, b u t is higher t h a n the p r o b a b i l i t y for the first cell. This would result in a c l u m p i n g of the items in the classes at the tails of the distrib u t i o n so t h a t there w o u l d be s o m e s q u a r e s with larger n u m b e r s of cells t h a n expected, o t h e r s with fewer n u m b e r s . T h e biological i n t e r p r e t a t i o n of the dispersion p a t t e r n varies with the p r o b l e m . T h e yeast cells seem to be r a n d o m l y distributed in t h e c o u n t i n g c h a m b e r , indicating t h o r o u g h mixing of the suspension. Red b l o o d cells, o n the o t h e r h a n d , will often stick t o g e t h e r because of a n electrical c h a r g e unless the p r o p e r suspension fluid is used. T h i s so-called r o u l e a u x effect w o u l d be indicated by c l u m p i n g of t h e observed frequencies. N o t e t h a t in Box 4.1, as in the s u b s e q u e n t tables giving examples of the application of the P o i s s o n distribution, we g r o u p the low frequencies at o n e tail of the curve, uniting t h e m by m e a n s of a bracket. This t e n d s t o simplify the p a t t e r n s of d i s t r i b u t i o n s o m e w h a t . However, the m a i n r e a s o n for this g r o u p ing is related t o the G test for g o o d n e s s of fit (of observed t o expected f r e q u e n cies), which is discussed in Section 13.2. F o r p u r p o s e s of this test, n o expected frequency / should be less t h a n 5. Before we t u r n t o o t h e r examples, we need to learn a few m o r e facts a b o u t the P o i s s o n distribution. Y o u p r o b a b l y noticed t h a t in c o m p u t i n g expected frequencies, we needed t o k n o w only o n e p a r a m e t e r — t h e m e a n of the distribution. By c o m p a r i s o n , in the b i n o m i a l distribution we needed t w o parameters, ρ and k. T h u s , the m e a n completely defines the s h a p e of a given Poisson distribution. F r o m this it follows that the variance is some f u n c t i o n of the m e a n . In a P o i s s o n distribution, we have a very simple relationship between the two: μ = σ 2 , t h e variance being equal to the m e a n . T h e variance of the n u m b e r of yeast cells per s q u a r e based o n the observed frequencies in Box 4.1 e q u a l s 1.965, not m u c h larger t h a n t h e m e a n of 1.8, indicating again that the yeast cells are distributed in Poisson fashion, hence r a n d o m l y . This r e l a t i o n s h i p between variance a n d m e a n suggests a rapid test of w h e t h e r an observed frequency distribution is distributed in Poisson fashion even w i t h o u t fitting expected frequencies to the d a t a . We simply c o m p u t e a coefficient of dispersion

This value will be near 1 in distributions that are essentially Poisson distributions, will be > 1 in c l u m p e d samples, a n d will be < 1 in cases of repulsion. In the yeast cell example, CD = 1.092. T h e shapes of five Poisson d i s t r i b u t i o n s of different m e a n s are s h o w n in Figure 4.3 as frequency polygons (a frequency polygon is formed by the line connecting successive m i d p o i n t s in a bar diagram). We notice that for the low value of μ = 0.1 the frequency polygon is extremely L-shapcd, but with an increase in the value of μ the d i s t r i b u t i o n s b e c o m e h u m p e d a n d eventually nearly symmetrical. We c o n c l u d e o u r study of the Poisson distribution with a c o n s i d e r a t i o n of two examples. T h e first e x a m p l e (Table 4.5) s h o w s the d i s t r i b u t i o n of a n u m b e r

chapter 4 / introduction t o probability

70

distributions

1.0

L^i 2

0

I^^t-^gc^—. ι 4 6 8

I

I I 10

ι ι ι ι — I - ' 12 14 16

' 18

N u m b e r of r a r e e v e n t s p e r s a m p l e figure

4.3

F r e q u e n c y p o l y g o n s of t h e P o i s s o n d i s t r i b u t i o n for v a r i o u s values of t h e m e a n .

of accidents per w o m a n f r o m an accident record of 647 w o m e n w o r k i n g in a m u n i t i o n s factory d u r i n g a five-week period. T h e s a m p l i n g unit is o n e w o m a n d u r i n g this period. T h e rare event is the n u m b e r of accidents t h a t h a p p e n e d t o a w o m a n in this period. T h e coefficient of dispersion is 1.488, a n d this is clearly reflected in the observed frequencies, which are greater t h a n expected in the tails a n d less t h a n expected in the center. T h i s relationship is easily seen in the deviations in the last c o l u m n (observed m i n u s expected frequencies) a n d shows a characteristic c l u m p e d p a t t e r n . T h e m o d e l assumes, of course, t h a t the accidents a r e n o t fata! o r very serious a n d thus d o not remove the individual f r o m f u r t h e r exposure. T h e noticeable c l u m p i n g in these d a t a p r o b a b l y arises

t a b l e

4.5

Accidents in 5 weeks to 647 women working on high-explosive shells.

Poisson expected frequencies f

Observed frequencies f

0 2 3 4 5+ Total

447 132 42

406.3 189.0 44.0

647

647.0

7 = 0.4652

= 0.692

Source: Greenwood and Yule (1920).

+

CD = 1.488

71

exercises

t a b l e

4.6

Azuki bean weevils (Callosobruchus chinensis) 112 Azuki beans (Phaseolus radiatus). U) Number of weevils emerging per bean Y

0 1 2 3 4

(2) Observed frequencies

f

f

61 50 η oil

Total

(3) Poisson expected frequencies

112

? = 0.4643

emerging from

(4) Deviation from expectation f - f

70.4 32.7 7.6] 1.2}· 8.9 0.1 J

•

-1 -J

112.0 x2 = 0.269

CD = 0.579

Source: Utida (1943).

either because some w o m e n are accident-prone or because some w o m e n have m o r e d a n g e r o u s j o b s t h a n others. Using only information on the distributions of accidents, one c a n n o t distinguish between the two alternatives, which suggest very different changes that should be m a d e to reduce the n u m b e r s of accidents. The second example (Table 4.6) is extracted f r o m an experimental study of the effects of different densities of the Azuki bean weevil. Larvae of these weevils enter the beans, feed and p u p a t e inside them, and then emerge through an emergence hole. T h u s the n u m b e r of holes per bean is a good measure of the n u m b e r of adults that have emerged. T h e rare event in this case is the presence of the weevil in the bean. W e note that the distribution is strongly repulsed. There are m a n y m o r e beans containing one weevil than the Poisson distribution would predict. A statistical finding of this sort leads us to investigate the biology of the p h e n o m e n o n . In this case it was found that the adult female weevils tended to deposit their eggs evenly rather than r a n d o m l y over the available beans. This prevented the placing of too m a n y eggs on any one bean and precluded heavy competition a m o n g the developing larvae on any one bean. A contributing factor was competition a m o n g remaining larvae feeding on the same bean, in which generally all but one were killed or driven out. Thus, it is easily understood h o w the above biological p h e n o m e n a would give rise to a repulsed distribution. Exercises 4.1

The two columns below give fertility of eggs of the CP strain of Drosophila melanogaster raised in 100 vials of 10 eggs each (data from R. R. Sokal). Find the expected frequencies on the assumption of independence of mortality for

72

chapter 4 / introduction t o probability

distributions

each egg in a vial. Use the observed mean. Calculate the expected variance and compare it with the observed variance. Interpret results, knowing that the eggs of each vial are siblings and that the different vials contain descendants from different parent pairs. ANS. σ 2 = 2.417, s 2 = 6.636. There is evidence that mortality rates are different for different vials.

Number of eggs hatched Y

0 1 2 3 4 5 6 7 8 9 10

4.2

43

4.4 4.5

4.6

Number

of vials f

1 3 8 10 6 15 14 12 13 9 9

In human beings the sex ratio of newborn infants is about 100?V': 105 J J . Were we to take 10,000 random samples of 6 newborn infants from the total population of such infants for one year, what would be the expected frequency of groups of 6 males, 5 males, 4 males, and so on? The Army Medical Corps is concerned over the intestinal disease X. From previous experience it knows that soldiers suffering from the disease invariably harbor the pathogenic organism in their feces and that to all practical purposes every stool specimen from a diseased person contains the organism. However, the organisms are never abundant, and thus only 20% of all slides prepared by the standard procedure will contain some. (We assume that if an organism is present on a slide it will be seen.) How many slides should laboratory technicians be directed to prepare and examine per stool specimen, so that in case a specimen is positive, it will be erroneously diagnosed negative in fewer than 1 % of the cases (on the average)? On the basis of your answer, would you recommend that the Corps attempt to improve its diagnostic methods? ANS. 21 slides. Calculate Poisson expected frequencies for the frequency distribution given in Table 2.2 (number of plants of the sedge Carex flacca found in 500 quadrats). A cross is made in a genetic experiment in Drosophila in which it is expected that { of the progeny will have white eyes and 5 will have the trait called "singed bristles." Assume that the two gene loci segregate independently, (a) What proportion of the progeny should exhibit both traits simultaneously? (b) If four flies are sampled at random, what is the probability that they will all be white-eyed? (c) What is the probability that none of the four flies will have either white eyes or "singed bristles?" (d) If two flies are sampled, what is the probability that at least one of the flies will have either white eyes or "singed bristles" or both traits? ANS. (a) (b) (i) 4 ; (c) [(1 - i)(l - i)] 4 ; (d) 1 - [(1 - i)(l Those readers who have had a semester or two of calculus may wish to try to prove that Expression (4.1) tends to Expression (4.2) as k becomes indefinitely

exercises

73

large (and ρ becomes infinitesimal, so that μ = kp remains constant). HINT: /

4.7

*Y 1 -» e x as η oo V "/ If the frequency of the gene A is ρ and the frequency of the gene a is q, what are the expected frequencies of the zygotes A A, Aa, and aa (assuming a diploid zygote represents a random sample of size 2)? What would the expected frequency be for an autotetraploid (for a locus close to the centromere a zygote can be thought of as a random sample of size 4)? ANS. P{AA} = p2, P{Aa} = 2pq, P{aa} = q2, f o r a d i p l o i d ; a n d P{AAAA} = p4, P{AAAa} 6 p 2 q 2 , P{Aaaa} = 4 p q 3 , P{aaaa} = q4, f o r a t e t r a p l o i d .

4.8 4.9

=

Summarize and compare the assumptions and parameters on which the binomial and Poisson distributions are based. A population consists of three types of individuals, A„ A2, and A3, with relative frequencies of 0.5,0.2, and 0.3, respectively, (a) What is the probability of obtaining only individuals of type Αλ in samples of size 1, 2, 3 , . . . , n? (b) What would be the probabilities of obtaining only individuals that were not of type Α γ or A 2 in a sample of size n? (c) What is the probability of obtaining a sample containing at least one representation of each type in samples of size 1, 2, 3, 4, 5 , . . . , n? ANS. (a) I i , I , . . . , 1/2". (b) (0.3)". (c) 0, 0, 0.18, 0.36, 0.507, for n:

4.10

= 4 p 3 q , P{AAaa}

"f

"£ '

"'.

.·=ι

i= ι

Aj\(n-i-j)\

|0.5|'(0.2Κ(0.3)" • '

If the average number of weed seeds found in a j o u n c e sample of grass seed is 1.1429, what would you expect the frequency distribution of weed seeds lo be in ninety-eight 4-ounce samples? (Assume there is random distribution of the weed seeds.)

CHAPTER

The

Normal

Probability

Distribution

T h e theoretical frequency d i s t r i b u t i o n s in C h a p t e r 4 were discrete. T h e i r variables a s s u m e d values that c h a n g e d in integral steps (that is, they were meristic variables). T h u s , the n u m b e r of infected insects per sample could be 0 or 1 o r 2 but never an i n t e r m e d i a t e value between these. Similarly, the n u m b e r of yeast cells per h e m a c y t o m e t e r s q u a r e is a meristic variable a n d requires a discrete probability f u n c t i o n to describe it. However, most variables e n c o u n t e r e d in biology either are c o n t i n u o u s (such as the aphid femur lengths or the infant birth weights used as e x a m p l e s in C h a p t e r s 2 a n d 3) or can be treated as cont i n u o u s variables for m o s t practical purposes, even t h o u g h they a r e inherently meristic (such as the n e u t r o p h i l c o u n t s e n c o u n t e r e d in the same chapters). C h a p t e r 5 will deal m o r e extensively with the distributions of c o n t i n u o u s variables. Section 5.1 introduces frequency d i s t r i b u t i o n s of c o n t i n u o u s variables. In Section 5.2 we show o n e way of deriving the m o s t c o m m o n such distribution, the n o r m a l probability distribution. T h e n we e x a m i n e its properties in Section 5.3. A few a p p l i c a t i o n s of the n o r m a l d i s t r i b u t i o n are illustrated in Section 5.4. A g r a p h i c technique for pointing out d e p a r t u r e s f r o m normality and for cstimat-

5.1 / f r e q u e n c y " d i s t r i b u t i o n s o f c o n t i n u o u s

variables

75

ing m e a n a n d s t a n d a r d deviation in a p p r o x i m a t e l y n o r m a l d i s t r i b u t i o n s is given in Section 5.5, as are s o m e of the reasons for d e p a r t u r e f r o m n o r m a l i t y in observed frequency distributions.

5.1 Frequency distributions of continuous variables F o r c o n t i n u o u s variables, t h e theoretical p r o b a b i l i t y distribution, or probability density function, can be represented by a c o n t i n u o u s curve, as s h o w n in F i g u r e 5.1. T h e o r d i n a t e of the curve gives the density for a given value of the variable s h o w n a l o n g the abscissa. By density we m e a n the relative c o n c e n t r a t i o n of variates a l o n g t h e Y axis (as indicated in F i g u r e 2.1). In o r d e r to c o m p a r e the theoretical with the observed frequency distribution, it is necessary to divide the t w o into c o r r e s p o n d i n g classes, as s h o w n by the vertical lines in F i g u r e 5.1. Probability density f u n c t i o n s are defined so t h a t the expected frequency of observations between two class limits (vertical lines) is given by the area between these limits u n d e r t h e curve. T h e total area u n d e r the curve is therefore equal t o the s u m of the expected frequencies (1.0 or n, d e p e n d i n g on w h e t h e r relative or absolute expected frequencies have been calculated). W h e n you f o r m a frequency distribution of o b s e r v a t i o n s of a c o n t i n u o u s variable, y o u r choice of class limits is arbitrary, because all values of a variable are theoretically possible. In a c o n t i n u o u s distribution, one c a n n o t evaluate the probability t h a t the variable will be exactly equal to a given value such as 3 or 3.5. O n e can only estimate the frequency of o b s e r v a t i o n s falling between two limits. This is so because the area of the curve c o r r e s p o n d i n g to a n y point a l o n g the curve is an infinitesimal. T h u s , to calculate expected frequencies for a c o n t i n u o u s distribution, we have t o calculate the area u n d e r the curve between the class limits. In Sections 5.3 and 5.4, we shall see how this is d o n e for the n o r m a l frequency distribution. C o n t i n u o u s frequency distributions m a y start and t e r m i n a t e at finite points a l o n g the Y axis, as s h o w n in F i g u r e 5.1, or o n e o r both e n d s of the curve may extend indefinitely, as will be seen later in Figures 5.3 and 6.11. T h e idea of an area u n d e r a curve w h e n o n e or b o t h e n d s go to infinity m a y t r o u b l e those of you not a c q u a i n t e d with calculus. F o r t u n a t e l y , however, this is not a great c o n ceptual stumbling block, since in all the cases that we shall e n c o u n t e r , the tail

I KillKL 5.1 A p r o b a b i l i t y d i s t r i b u t i o n of ;i c o n t i n u o u s variable.

76

chapter 4 /

introduction

to

probability distributions

of t h e curve will a p p r o a c h the Y axis rapidly e n o u g h t h a t the p o r t i o n of t h e a r e a b e y o n d a certain p o i n t will for all practical p u r p o s e s be zero a n d t h e frequencies it represents will be infinitesimal. W e m a y fit c o n t i n u o u s frequency d i s t r i b u t i o n s t o s o m e sets of meristic d a t a (for example, t h e n u m b e r of teeth in an organism). In such cases, we h a v e r e a s o n t o believe t h a t u n d e r l y i n g biological variables t h a t cause differences in n u m b e r s of the s t r u c t u r e a r e really c o n t i n u o u s , even t h o u g h expressed as a discrete variable. W e shall n o w proceed t o discuss the m o s t i m p o r t a n t p r o b a b i l i t y density f u n c t i o n in statistics, the n o r m a l frequency distribution.

5.2 Derivation of the normal distribution T h e r e a r e several ways of deriving the n o r m a l frequency d i s t r i b u t i o n f r o m elem e n t a r y a s s u m p t i o n s . M o s t of these require m o r e m a t h e m a t i c s t h a n we expect of o u r readers. W e shall therefore use a largely intuitive a p p r o a c h , which we have f o u n d of heuristic value. S o m e inherently meristic variables, such as c o u n t s of blood cells, range into the t h o u s a n d s . Such variables can, for practical p u r poses, be treated as t h o u g h they were c o n t i n u o u s . Let us consider a b i n o m i a l distribution of t h e familiar f o r m ( ρ + qf in which k becomes indefinitely large. W h a t type of biological situation could give rise to such a b i n o m i a l distribution? An e x a m p l e might be one in which m a n y f a c t o r s c o o p e r a t e additively in p r o d u c i n g a biological result. T h e following h y p o t h e t i c a l case is possibly not t o o far r e m o v e d f r o m reality. T h e intensity of skin p i g m e n t a t i o n in an a n i m a l will be d u e to the s u m m a t i o n of m a n y factors, s o m e genetic, o t h e r s e n v i r o n m e n t a l . As a simplifying a s s u m p t i o n , let us state t h a t every factor can occur in t w o states only: present or absent. W h e n the factor is present, it c o n t r i b u t e s o n e unit of p i g m e n t a t i o n to skin color, but it c o n t r i b u t e s n o t h i n g to p i g m e n t a t i o n w h e n it is absent. Each factor, regardless of its n a t u r e or origin, has the identical effect, a n d the effects are additive: if three o u t of five possible factors are present in an individual, the p i g m e n t a t i o n intensity will be three units, or the s u m of three c o n t r i b u t i o n s of o n e unit each. O n e final a s s u m p tion: E a c h f a c t o r has an equal probability of being present or a b s e n t in a given individual. T h u s , ρ = = 0.5, the probability t h a t the factor is present; while q = P [ / ] = 0.5, the probability that the factor is absent. With only o n e factor (k = 1), e x p a n s i o n of the binomial (p +
/ } 0.5} 0 }

p i g m e n t a t i o n classes (probability space) expected frequency p i g m e n t a t i o n intensity

Half the a n i m a l s would have intensity I, the o t h e r half 0. W i t h k = 2 factors present in the p o p u l a t i o n (the factors arc a s s u m e d to occur i n d e p e n d e n t l y of each other), the distribution of p i g m e n t a t i o n intensities would be represented by

5.4 / a p p l i c a t i o n s o f t h e n o r m a l

distribution

77

Ten factors

0.4 0.3

Λ,ι

0.2

0.1 0 0

1

2

3

4

5

6

7

8 ! )

10

V FIGURE 5 . 2

H i s t o g r a m b a s e d o n relative expected frequencies resulting f r o m e x p a n s i o n of b i n o m i a l (0.5 + 0.5) 1 0 . T h e 7 axis m e a s u r e s t h e n u m b e r of p i g m e n t a t i o n f a c t o r s F.

the e x p a n s i o n of the b i n o m i a l ( ρ + q)2'· {FF, {0.25, {2,

Ff, 0.50, 1,

jf } 0.25} 0 }

p i g m e n t a t i o n classes (probability space) expected frequency p i g m e n t a t i o n intensity

O n e - f o u r t h of the individuals w o u l d have p i g m e n t a t i o n intensity 2; one-half, intensity 1; a n d the r e m a i n i n g f o u r t h , intensity 0. T h e n u m b e r of classes in the binomial increases with the n u m b e r of factors. The frequency distributions a r e symmetrical, a n d the expected frequencies at the tails b e c o m e progressively less as k increases. T h e b i n o m i a l d i s t r i b u t i o n for k— 10 is g r a p h e d as a h i s t o g r a m in F i g u r e 5.2 (rather t h a n as a b a r d i a g r a m , as it should be drawn). We note that the g r a p h a p p r o a c h e s t h e familiar bellshaped outline of the n o r m a l frequency distribution (seen in Figures 5.3 a n d 5.4). W e r e we to e x p a n d the expression for k = 20, o u r h i s t o g r a m w o u l d be so close to a n o r m a l frequency distribution that we could not show the difference between the t w o on a g r a p h the size of this page. At the beginning of this procedure, we m a d e a n u m b e r of severe limiting a s s u m p t i o n s for the sake of simplicity. W h a t h a p p e n s when these a r e removed? First, when ρ φ q, the distribution also a p p r o a c h e s n o r m a l i t y as k a p p r o a c h e s infinity. This is intuitively difficult to see, because when ρ / q, the h i s t o g r a m is at first asymmetrical. However, it can be s h o w n that w h e n k, ρ, a n d q are such that kpq > 3, the n o r m a l distribution will be closely a p p r o x i m a t e d . Second, in a m o r e realistic situation, factors w o u l d be permitted to o c c u r in m o r e t h a n two s t a t e s — o n e state m a k i n g a large c o n t r i b u t i o n , a second state a smaller c o n t r i b u t i o n , a n d so forth. However, it can also be s h o w n that the m u l t i n o m i a l (p + q + r + · · · + z)k a p p r o a c h e s the n o r m a l frequency distribution as k a p p r o a c h e s infinity. T h i r d , different factors m a y be present in different frequencies a n d m a y have different q u a n t i t a t i v e effects. As long as these a r e additive a n d independent, n o r m a l i t y is still a p p r o a c h e d as k a p p r o a c h e s infinity. Lifting these restrictions m a k e s the a s s u m p t i o n s leading to a n o r m a l distribution c o m p a t i b l e with i n n u m e r a b l e biological situations. It is therefore not surprising that so m a n y biological variables are a p p r o x i m a t e l y normally distributed.

78

chapter 5 / the normal probability

distribution

Let us s u m m a r i z e the c o n d i t i o n s t h a t tend to p r o d u c e n o r m a l f r e q u e n c y distributions: (1) t h a t there be m a n y factors; (2) t h a t these factors be i n d e p e n d e n t in occurrence; (3) t h a t t h e factors be i n d e p e n d e n t in e f f e c t — t h a t is, t h a t their effects be additive; a n d (4) t h a t they m a k e e q u a l c o n t r i b u t i o n s t o t h e variance. T h e f o u r t h c o n d i t i o n we a r e n o t yet in a position t o discuss; we m e n t i o n it here only for completeness. It will be discussed in C h a p t e r 7.

5.3 Properties of the normal distribution F o r m a l l y , the normal expression

probability

density

1 Z = — = e

function

2

"

can be represented by the

(5.1)

H e r e Ζ indicates the height of the o r d i n a t e of the curve, which represents the density of the items. It is the d e p e n d e n t variable in the expression, being a function of the variable Y. T h e r e are t w o c o n s t a n t s in the e q u a t i o n : π, well k n o w n to be a p p r o x i m a t e l y 3.141,59, m a k i n g \/yj2n a p p r o x i m a t e l y 0.398,94, a n d e, the base of the n a t u r a l logarithms, whose value a p p r o x i m a t e s 2.718,28. T h e r e are t w o p a r a m e t e r s in a n o r m a l probability density function. These are the p a r a m e t r i c m e a n μ a n d the p a r a m e t r i c s t a n d a r d deviation σ, which d e t e r m i n e the location a n d s h a p e of the distribution. T h u s , there is not j u s t one n o r m a l distribution, as might a p p e a r to the uninitiated w h o keep e n c o u n t e r i n g the same bell-shaped image in t e x t b o o k s . R a t h e r , there are an infinity of such curves, since these p a r a m e t e r s can a s s u m e a n infinity of values. This is illustrated by the three n o r m a l curves in Figure 5.3, representing the same total frequencies.

FKil IKh 5.3 I l l u s t r a t i o n of h o w c h a n g e s in t h e Iwo p a r a m e t e r s of t h e n o r m a l d i s t r i b u t i o n alTecl t h e s h a p e a n d l o c a t i o n of Ihe n o r m a l p r o b a b i l i t y d e n s i t y f u n c t i o n . (Α) μ = 4,
5.4

/ applications of t h e n o r m a l

distribution

79

Curves A a n d Β differ in their locations a n d hence represent p o p u l a t i o n s with different m e a n s . C u r v e s Β a n d C represent p o p u l a t i o n s t h a t h a v e identical m e a n s but different s t a n d a r d deviations. Since the s t a n d a r d deviation of curve C is only half t h a t of curve B, it presents a m u c h n a r r o w e r a p p e a r a n c e . In theory, a n o r m a l frequency distribution extends f r o m negative infinity to positive infinity a l o n g the axis of the variable (labeled Y, a l t h o u g h it is frequently the abscissa). This m e a n s t h a t a n o r m a l l y distributed variable can assume a n y value, however large or small, a l t h o u g h values f a r t h e r f r o m the m e a n t h a n plus or m i n u s three s t a n d a r d deviations are quite rare, their relative expected frequencies being very small. This can be seen f r o m Expression (5.1). W h e n 7 is very large or very small, the term (Υ — μ)2/2σ2 will necessarily b e c o m e very large. H e n c e e raised to the negative p o w e r of t h a t t e r m will be very small, a n d Ζ will therefore be very small. T h e curve is symmetrical a r o u n d the m e a n . Therefore, the m e a n , median, a n d m o d e of the n o r m a l distribution are all at the same point. T h e following percentages of items in a n o r m a l frequency distribution lie within the indicated limits: μ ± σ c o n t a i n s 68.27% of the items μ ±2σ c o n t a i n s 95.45% of the items μ ± 3σ c o n t a i n s 99.73% of the items Conversely, 50% of the items fall in the range μ ± 0.674σ 95% of the items fall in the range μ + 1.960σ 99% of the items fall in the range μ + 2.576σ These relations are s h o w n in F i g u r e 5.4. H o w have these percentages been calculated? T h e direct calculation of any p o r t i o n of the area u n d e r the n o r m a l curve requires an integration of the function s h o w n as Expression (5.1). F o r t u n a t e l y , for those of you w h o d o not k n o w calculus (and even for those of you w h o do) the integration has already been carried out a n d is presented in an alternative f o r m of the n o r m a l distribution: the normal distribution function (the theoretical cumulative distribution f u n c t i o n of the n o r m a l probability density function), also s h o w n in Figure 5.4. It gives the total frequency f r o m negative infinity u p to a n y point a l o n g the abscissa. We can therefore look u p directly t h e probability t h a t an observation will be less t h a n a specified value of Y. F o r example, Figure 5.4 shows that the total frequency u p to the m e a n is 50.00% a n d the frequency u p to a point o n e s t a n d a r d deviation below the m e a n is 15.87%. These frequencies are f o u n d , graphically, by raising a vertical line f r o m a point, such as — σ, until it intersects the c u m u l a t i v e distribution curve, a n d then reading the frequency (15.87%) off the ordinate. T h e probability that a n o b s e r v a t i o n will fall between t w o a r b i t r a r y points can be found by s u b t r a c t i n g the probability t h a t an observation will fall below the

chapter 5 / the normal probability

80

~ figure

'>5.45'ν W.7.V;

distribution

-

5.4

Areas u n d e r t h e n o r m a l p r o b a b i l i t y density f u n c t i o n a n d the c u m u l a t i v e n o r m a ! d i s t r i b u t i o n function

lower point f r o m the p r o b a b i l i t y t h a t an o b s e r v a t i o n will fall below the u p p e r point. F o r example, we can see f r o m F i g u r e 5.4 that the probability t h a t an o b s e r v a t i o n will fall between the m e a n a n d a point o n e s t a n d a r d deviation below the m e a n is 0.5000 - 0.1587 = 0.3413. T h e n o r m a l distribution f u n c t i o n is t a b u l a t e d in T a b l e II in A p p e n d i x A2, "Areas of the n o r m a l curve," where, for convenience in later calculations, 0.5 has been s u b t r a c t e d f r o m all of the entries. This table therefore lists the p r o p o r tion of the area between the m e a n a n d a n y p o i n t a given n u m b e r of s t a n d a r d deviations a b o v e it. T h u s , for example, t h e area between the m e a n a n d the point 0.50 s t a n d a r d deviations a b o v e the m e a n is 0.1915 of the total area of t h e curve. Similarly, the area between t h e m e a n a n d the point 2.64 s t a n d a r d deviations a b o v e the m e a n is 0.4959 of the curve. A point 4.0 s t a n d a r d deviations f r o m the m e a n includes 0.499,968 of the a r e a between it a n d the mean. H o w e v e r , since the n o r m a l distribution e x t e n d s f r o m negative to positive infinity, o n e needs

5.4 / a p p l i c a t i o n s o f t h e n o r m a l

81

distribution

t o g o a n infinite d i s t a n c e f r o m t h e m e a n t o r e a c h a n a r e a of 0.5. T h e use of the t a b l e of a r e a s of t h e n o r m a l c u r v e will b e i l l u s t r a t e d in t h e n e x t section. A s a m p l i n g e x p e r i m e n t will give y o u a "feel" f o r t h e d i s t r i b u t i o n of i t e m s sampled from a normal distribution. Experiment 5.1. You are asked to sample from two populations. The first one is an approximately normal frequency distribution of 100 wing lengths of houseflies. The second population deviates strongly from normality. It is a frequency distribution of the total annual milk yield of 100 Jersey cows. Both populations are shown in Table 5.1. You are asked to sample from them repeatedly in order to simulate sampling from an infinite population. Obtain samples of 35 items from each of the two populations. This can be done by obtaining two sets of 35 two-digit random numbers from the table of random numbers (Table I), with which you became familiar in Experiment 4.1. Write down the random numbers in blocks of five, and copy next to them the value of Y (for either wing length or milk yield) corresponding to the random number. An example of such a block of five numbers and the computations required for it are shown in the

TABLE 5.1 Populations of wing lengths and milk yields. Column I. R a n k n u m b e r . Column 2. L e n g t h s (in m m χ 1 ( T ' ) of 100 wings of houseflies a r r a y e d in o r d e r of m a g n i t u d e ; / / = 45.5. σ2 = 15.21, σ = 3.90; d i s t r i b u t i o n a p p r o x i m a t e l y n o r m a l . Column 3. T o t a l a n n u a l milk yield (in h u n d r e d s of p o u n d s ) of 100 two-year-old registered Jersey c o w s a r r a y e d in o r d e r of m a g n i t u d e ; μ = 66.61, a 2 = 124.4779, ο = 11.1597; d i s t r i b u t i o n d e p a r t s s t r o n g l y f r o m n o r m a l i t y .

(/)

(2)

li)

(/)

(2)

(3)

(/)

(2)

«

(0

(2)

(3)

(')

(2)

(3)

01 02 03 04 05 06 07 08 09 10

36 37 38 38 39 39 40 40 40 40

51 51 51 53 53 53 54 55 55 56

21 22 23 24 25 26 27 28 29 30

42 42 42 43 43 43 43 43 43 43

58 58 58 58 58 58 58 58 58 58

41 42 43 44 45 46 47 48 49 50

45 45 45 45 45 45 45 45 45 45

61 61 61 61 61 62 62 62 62 63

61 62 63 64 65 66 67 68 69 70

47 47 47 47 47 47 47 47 47 48

67 67 68 68 69 69 69 69 69 69

81 82 83 84 85 86 87 88 89 90

49 49 49 49 50 50 50 50 50 50

76 76 79 80 80 8! 82 82 82 82

II 12 13 14 15 16 17 18 19 20

41 41 41 41 41 41 42 42 42 42

56 56 57 57 57 57 57 57 57 57

31 32 33 34 35 36 37 38 39 40

43 44 44 44 44 44 44 44 44 44

58 59 59 59 60 60 60 60 60 61

51 52 53 54 55 56 57 58 59 60

46 46 46 46 46 46 46 46 46 46

63 63 64 65 65 65 65 65 67 67

71 72 73 74 75 76 77 78 79 80

48 48 48 48 48 48 48 49 49 49

70 72 73 73 74 74 74 74 75 76

91 92 93 94 95 96 97 98 99 00

51 51 51 51 52 52 53 53 54 55

83 85 87 88 88 89 93 94 96 98

Source: Column 2—Data adapted from Soka] and Hunter (1955). Column 3 —Data from Canadian government records.

82

chapter 5 / the normal probability

distribution

following listing, using the housefly wing lengths as an example:

Wing length Y

Random number

16 59 99 36 21

41 46 54 44 42 £ Y = £Y

2

227

= 10,413

y =

45.4

Those with ready access to a computer may prefer to program this exercise and take many more samples. These samples and the computations carried out for each sample will be used in subsequent chapters. Therefore, preserve your data carefully! In this experiment, consider t h e 35 variates for each variable as a single sample, r a t h e r t h a n b r e a k i n g them d o w n i n t o g r o u p s of five. Since the t r u e m e a n a n d s t a n d a r d deviation (μ a n d a) of the t w o d i s t r i b u t i o n s are k n o w n , you can calculate the expression (Υ, — μ)/σ for each variate Y,. T h u s , for the first housefly wing length s a m p l e d above, you c o m p u t e 41 - 45.5 - l 9 0 -

=

"

: 1 1 5 3 8

This m e a n s t h a t the first wing length is 1.1538 s t a n d a r d deviations below the true m e a n of t h e p o p u l a t i o n . T h e deviation f r o m the m e a n m e a s u r e d in s t a n d a r d deviation units is called a standardized deviate o r standard deviate. T h e a r g u m e n t s of T a b l e II, expressing distance f r o m the m e a n in units of σ, a r e called standard normal deviates. G r o u p all 35 variates in a frequency distribution; then d o t h e s a m e for milk yields. Since you k n o w the p a r a m e t r i c m e a n a n d s t a n d a r d deviation, you need not c o m p u t e each deviate separately, but can simply write d o w n class limits in terms of the actual variable as well as in s t a n d a r d deviation f o r m . T h e class limits for such a frequency distribution are s h o w n in T a b l e 5.2. C o m b i n e the results of y o u r s a m p l i n g with those of your classmates a n d study the percentage of the items in the distribution one, two, a n d three s t a n d a r d deviations t o each side of the m e a n . N o t e the m a r k e d differences in d i s t r i b u t i o n between the housefly wing lengths and the milk yields.

5.4 Applications of the normal distribution T h e n o r m a l frequency d i s t r i b u t i o n is the m o s t widely used distribution in statistics, a n d time a n d again we shall have recourse to it in a variety of situations. F o r the m o m e n t , wc may subdivide its a p p l i c a t i o n s as follows.

5.4 / a p p l i c a t i o n s o f t h e n o r m a l

t a b l e

83

distribution

5.2

Table for recording frequency distributions of standard deviates (¥, — μ)]ο for samples of Experiment 5.1. Wing lengths Variates falling between these limits

Milk

yields

Variates falling between these limits

f

— 00

— 00

— 3σ

-3σ

- 2 k - 2 a - i k — a - k μ = 45.5

36, 37

— 2(7

38, 39

- l k 51-55

40,41 42, 4 3

56-61

44, 4 5

62 - 6 6

μ = 66.61

67-72

46, 47

k a i k 2a 2 k

73-77

48, 49

Ik

78-83

52, 53

2σ

84-88

54, 5 5

2k

89-94

50, 51

3σ

3σ

+ GO

+ GO

95

98

1. We sometimes have to k n o w whether a given s a m p l e is normally distributed before we can apply a certain test to it. T o test whether a given sample is normally distributed, we have to calculate expected frequencies for a n o r m a l curve of the same m e a n a n d s t a n d a r d deviation using the table of areas of the n o r m a l curve. In this book we shall e m p l o y only a p p r o x i m a t e graphic m e t h o d s for testing normality. These are featured in the next section. 2. K n o w i n g whether a sample is normally distributed may confirm or reject certain underlying hypotheses a b o u t the n a t u r e of the factors affecting the p h e n o m e n o n studied. This is related to the c o n d i t i o n s m a k i n g for n o r m a l i t y in a frequency distribution, discusscd in Scction 5.2. T h u s , if we find a given variable to be normally distributed, we have no reason for rejecting the hypothesis t h a t the causal factors affecting the variable arc additive a n d independent a n d of equal variance. O n the o t h e r h a n d , when we find d e p a r t u r e from normality, this may indicate certain forces, such as selection, affecting the variable u n d e r study, f o r instance, bimodality may indicate a mixture

84

chapter 5 / the normal probability

distribution

of o b s e r v a t i o n s f r o m t w o p o p u l a t i o n s . Skewness of milk yield d a t a m a y indicate t h a t these a r e r e c o r d s of selected c o w s a n d s u b s t a n d a r d milk c o w s h a v e n o t been included in the record. 3. If we a s s u m e a given d i s t r i b u t i o n t o be n o r m a l , we m a y m a k e p r e d i c t i o n s a n d tests of given h y p o t h e s e s based u p o n this a s s u m p t i o n . (An e x a m p l e of such a n a p p l i c a t i o n follows.) Y o u will recall the b i r t h weights of m a l e Chinese children, illustrated in Box 3.2. T h e m e a n of this s a m p l e of 9465 b i r t h weights is 109.9 oz, a n d its s t a n d a r d deviation is 13.593 oz. If y o u s a m p l e a t r a n d o m f r o m t h e b i r t h r e c o r d s of this p o p u l a t i o n , w h a t is y o u r c h a n c e of o b t a i n i n g a birth weight of 151 oz o r heavier? Such a birth weight is considerably a b o v e t h e m e a n of o u r sample, the difference being 151 — 109.9 = 41.1 oz. H o w e v e r , we c a n n o t consult t h e table of a r e a s of the n o r m a l curve with a difference in ounces. W e m u s t express it in standardized u n i t s — t h a t is, divide it by the s t a n d a r d deviation t o c o n v e r t it i n t o a s t a n d a r d deviate. W h e n we divide the difference by t h e s t a n d a r d deviation, we o b t a i n 41.1/13.593 = 3.02. This m e a n s t h a t a birth weight of 151 oz is 3.02 s t a n d a r d deviation units greater t h a n the m e a n . A s s u m i n g t h a t t h e birth weights a r e n o r m a l l y distributed, we m a y consult the table of areas of the n o r m a l curve (Table II), where we find a value of 0.4987 for 3.02 s t a n d a r d deviations. T h i s m e a n s t h a t 49.87% of the area of the curve lies between the m e a n a n d a point 3.02 s t a n d a r d deviations f r o m it. Conversely, 0.0013, or 0.13%, of the a r e a lies b e y o n d 3.02 s t a n d a r d deviation units a b o v e the m e a n . T h u s , a s s u m i n g a n o r m a l distribution of birth weights a n d a value of σ = 13.593, only 0.13%, or 13 o u t of 10,000, of the infants w o u l d have a birth weight of 151 oz or farther f r o m the m e a n . It is quite i m p r o b a b l e t h a t a single s a m p l e d item f r o m t h a t p o p u l a t i o n w o u l d deviate by so m u c h f r o m the m e a n , a n d if such a r a n d o m s a m p l e of o n e weight were o b t a i n e d f r o m the records of an unspecified p o p u l a t i o n , we m i g h t be justified in d o u b t i n g w h e t h e r the o b s e r v a t i o n did in fact c o m e f r o m the p o p u l a t i o n k n o w n to us. T h e a b o v e p r o b a b i l i t y was calculated f r o m o n e tail of the distribution. W e f o u n d t h e probability t h a t a n individual w o u l d be greater t h a n the m e a n by 3.02 o r m o r e s t a n d a r d deviations. If we are not c o n c e r n e d w h e t h e r t h e individual is either heavier o r lighter t h a n the m e a n but wish to k n o w only h o w different the individual is f r o m t h e p o p u l a t i o n m e a n , an a p p r o p r i a t e q u e s t i o n would be: A s s u m i n g t h a t the individual belongs to the p o p u l a t i o n , w h a t is the probability of observing a birth weight of a n individual deviant by a certain a m o u n t f r o m the m e a n in either direction? T h a t probability m u s t be c o m p u t e d by using b o t h tails of the distribution. T h e previous probability can be simply d o u b l e d , since the n o r m a l curve is symmetrical. T h u s , 2 χ 0.0013 = 0.0026. This, too, is so small t h a t we w o u l d c o n c l u d e t h a t a birth weight as deviant as 151 oz is unlikely t o have c o m e f r o m the p o p u l a t i o n represented by o u r s a m p l e of male Chinese children. W e can learn o n e m o r e i m p o r t a n t point f r o m this example. O u r a s s u m p t i o n has been t h a t the birth weights are n o r m a l l y distributed. Inspection of the

5.5 / d e p a r t u r e s f r o m n o r m a l i t y : g r a p h i c

methods

85

frequency d i s t r i b u t i o n in Box 3.2, however, shows clearly t h a t the d i s t r i b u t i o n is asymmetrical, t a p e r i n g t o the right. T h o u g h there are eight classes a b o v e the m e a n class, there a r e only six classes below the m e a n class. In view of this a s y m m e t r y , conclusions a b o u t o n e tail of the distribution w o u l d n o t necessarily p e r t a i n to the second tail. W e calculated t h a t 0.13% of the items w o u l d be f o u n d b e y o n d 3.02 s t a n d a r d deviations a b o v e t h e m e a n , which c o r r e s p o n d s to 151 oz. In fact, o u r s a m p l e c o n t a i n s 20 items (14 + 5 + 1) b e y o n d t h e 147.5-oz class, the u p p e r limit of which is 151.5 oz, a l m o s t the same as the single b i r t h weight. H o w e v e r , 20 items of the 9465 of the s a m p l e is a p p r o x i m a t e l y 0.21%, m o r e t h a n the 0.13% expected f r o m t h e n o r m a l frequency distribution. A l t h o u g h it w o u l d still be i m p r o b a b l e to find a single birth weight as heavy as 151 oz in the sample, conclusions based o n the a s s u m p t i o n of n o r m a l i t y m i g h t be in e r r o r if the exact p r o b a b i l i t y were critical for a given test. O u r statistical conclusions are only as valid as o u r a s s u m p t i o n s a b o u t the p o p u l a t i o n f r o m which the samples are d r a w n .

5.5 Departures from normality: Graphic methods In m a n y cases an observed frequency distribution will d e p a r t obviously f r o m normality. W e shall e m p h a s i z e t w o types of d e p a r t u r e f r o m n o r m a l i t y . O n e is skewness, which is a n o t h e r n a m e for a s y m m e t r y ; skewness m e a n s that o n e tail of the curve is d r a w n o u t m o r e t h a n the other. In such curves the m e a n a n d the m e d i a n will not coincide. C u r v e s a r e said to be skewed to the right o r left, d e p e n d i n g u p o n w h e t h e r the right or left tail is d r a w n out. T h e o t h e r type of d e p a r t u r e f r o m n o r m a l i t y is kurtosis, or " p e a k e d n e s s " of a curve. A leptokurtic curve has m o r e items near the m e a n a n d at the tails, with fewer items in t h e i n t e r m e d i a t e regions relative to a n o r m a l distribution with the s a m e m e a n a n d variance. A platykurtic curve has fewer items at the m e a n a n d at the tails t h a n the n o r m a l curvc but h a s m o r e items in i n t e r m e d i a t e regions. A b i m o d a l distribution is an extreme p l a t y k u r t i c distribution. G r a p h i c m e t h o d s have been developed that examine the s h a p e of an observed distribution for d e p a r t u r e s f r o m normality. These m e t h o d s also permit estimates of the m e a n a n d s t a n d a r d deviation of the distribution w i t h o u t computation. T h e graphic m e t h o d s are based on a c u m u l a t i v e frequency distribution. In F i g u r e 5.4 we saw t h a t a n o r m a l frequency distribution g r a p h e d in c u m u l a t i v e fashion describes an S - s h a p e d curve, called a sigmoid curve. In F i g u r e 5.5 the o r d i n a t e of the sigmoid curve is given as relative frequencies expressed as percentages. T h e slope of the c u m u l a t i v e curve reflects changcs in height of the frequency distribution o n which it is based. T h u s the steep m i d d l e segment of the c u m u l a t i v e n o r m a l curve c o r r e s p o n d s to the relatively greater height of the n o r m a l curvc a r o u n d its m e a n . T h e o r d i n a t e in Figures 5.4 a n d 5.5 is in linear scale, as is the abscissa in Figure 5.4. A n o t h e r possible scale is the normal probability scale (often simply called probability scale), which can be generated by d r o p p i n g p e r p e n d i c u l a r s

86

chapter 5 / the normal probability

1

5 10

30 50 70

90 95

distribution

99

C u m u l a t i v e p e r c e n t in p r o b a b i l i t y scale figure

5.5

T r a n s f o r m a t i o n of c u m u l a t i v e p e r c e n t a g e s into n o r m a l probability scale.

f r o m the c u m u l a t i v e n o r m a l curve, c o r r e s p o n d i n g to given percentages o n t h e o r d i n a t e , t o the abscissa (as s h o w n in F i g u r e 5.5). T h e scale represented by the abscissa c o m p e n s a t e s for the n o n l i n e a r i t y of the cumulative n o r m a l curve. It c o n t r a c t s the scale a r o u n d t h e m e d i a n a n d e x p a n d s it at the low a n d high c u m u l a t i v e percentages. This scale c a n be f o u n d on arithmetic or normal probability

graph

paper

( o r s i m p l y probability

graph

paper),

w h i c h is g e n e r a l l y

avail-

able. Such p a p e r usually h a s the long edge g r a d u a t e d in probability scale, while the short edge is in linear scale. N o t e that there are n o 0% or 100% p o i n t s on the o r d i n a t e . These p o i n t s c a n n o t be shown, since the n o r m a l frequency distrib u t i o n extends f r o m negative t o positive infinity a n d thus however long we m a d e o u r line we w o u l d never reach the limiting values of 0% a n d 100%. If we g r a p h a c u m u l a t i v e n o r m a l distribution with the o r d i n a t e in n o r m a l probability scale, it will lie exactly on a straight line. Figure 5.6A shows such a g r a p h d r a w n o n p r o b a b i l i t y p a p e r , while the o t h e r p a r t s of F i g u r e 5.6 s h o w a series of frequency d i s t r i b u t i o n s variously d e p a r t i n g f r o m normality. These are g r a p h e d b o t h as o r d i n a r y f r e q u e n c y d i s t r i b u t i o n s with density on a linear scale (ordinate not shown) a n d as c u m u l a t i v e d i s t r i b u t i o n s as they w o u l d a p p e a r on

5.5 / d e p a r t u r e s f r o m n o r m a l i t y : g r a p h i c m e t h o d s

Normal

figurk

87

distributions

5.6

E x a m p l e s of s o m e f r e q u e n c y d i s t r i b u t i o n s with their c u m u l a t i v e d i s t r i b u t i o n s p l o t t e d with the o r d i n a t e in n o r m a l p r o b a b i l i t y scale. (See Box 5.1 for e x p l a n a t i o n . )

probability p a p e r . T h e y a r e useful as guidelines for e x a m i n i n g the d i s t r i b u t i o n s of d a t a o n probability paper. Box 5.1 shows you h o w to use probability p a p e r to e x a m i n e a frequency distribution for n o r m a l i t y a n d to o b t a i n g r a p h i c estimates of its m e a n a n d s t a n d a r d deviation. T h e m e t h o d w o r k s best for fairly large s a m p l e s (η > 50). T h e m e t h o d d o e s not permit the plotting of the last c u m u l a t i v e frequency, 100%,

88

chapter 5 / the normal probability

distribution

BOX 5.1 Graphic test for normality of a frequency distribution and estimate of mean and standard deviation. Use of arithmetic probability paper. Birth weights of male Chinese in ounces, from Box 3.2.

V) Class mark Y 59.5

67.5 75.5 83.5 91.5 99.5 107.5 115.5 123.5 131.5 139.5 147.5 155.5 163.5 171.5

(2)

(β)

Upper class limit

f

63.5 71.5 79.5 87.5 95.5 103.5 111.5 119.5 127.5 135.5 143.5 151.5 159.5 167.5 175.5

2 6 39 385 888 1729 2240 2007 1233 641 201 74 14 5 1

(4) Cumulative frequencies F 2

8 47 432 1320 3049 5289 7296 8529 9170 9371 9445 9459 9464 9465

(5) Percent cumulative frequencies

0.02 0.08 0.50 4.6 13.9 32.2 55.9 77.1 90.1 96.9 99.0 99.79 99.94 99.99 100.0

9465 Computational

steps

1. Prepare a frequency distribution as shown in columns (1), (2), and (3). 2. Form a cumulative frequency distribution as shown in column (4). It is obtained by successive summation of the frequency values. In column (5) express the cumulative frequencies as percentages of total sample size n, which is 9465 in this example. These percentages are 100 times the values of column (4) divided by 9465. 3. Graph the upper class limit of each class along the abscissa (in linear scale) against percent cumulative frequency along the ordinate (in probability scale) on normal probability paper (see Figure 5.7). A straight line is fitted to the points by eye, preferably using a transparent plastic ruler, which permits all the points to be seen as the line is drawn. In drawing the line, most weight should be given to the points between cumulative frequencies of 25% to 75%. This is because a difference of a single item may make appreciable changes in the percentages at the tails. We notice that the upper frequencies deviate to the right of the straight line. This is typical of data that are skewed to the right (see Figure 5.6D). 4. Such a graph permits the rapid estimation of the mean and standard deviation of a sample. The mean is approximated by a graphic estimation of the median. The more normal the distribution is, the closer the mean will be to the median.

5.5 / d e p a r t u r e s f r o m n o r m a l i t y : g r a p h i c

methods

BOX 5.1 Continued The median is estimated by dropping a perpendicular from the intersection of the 50% point on the ordinate and the cumulative frequency curve to the abscissa (see Figure 5.7). The estimate of the mean of 110.7 oz is quite close to the computed mean of 109.9 oz. 5. The standard deviation can be estimated by dropping similar perpendiculars from the intersections of the 15.9% and the 84.1% points with the cumulative curve, respectively. These points enclose the portion of a normal curve represented by μ ± σ. By measuring the difference between these perpendiculars and dividing this by 2, we obtain an estimate of one standard deviation. In this instance the estimate is s = 13.6, since the difference is 27.2 oz divided by 2. This is a close approximation to the computed value of 13.59 oz.

<«.5

79.5

95.5

111.5 127.5 1 Ci.5 159.5 175.5

Kirth weights (if mail- Chinese (in oz.) hg((r!·: 5.7 G r a p h i c analysis of d a t a f r o m Box 5.1.

89

Τ3 οa « = > a c "3 >< ·2 w £ uχ Η 3 'δ Ε 13 (Λ C Λ ϊΐ "S 'Q J= = •£ -ο ·° ε s « 52 » 2 ο · α -S θ § £ Μ Ο κ -J fi υ g ? -ο Ο ·— ° ·Κ £ ο •S ο ν; U Ο u. -S = β ξ «ο 2 J3 . 5 so u Η S3 ^ 5 < Έ Ι 8. 4> 00 S . s s Ξ * 2 U ο (Λ Ο ;

c

c

Ο

Ο

3

3

οVI

ο

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5.5 / d e p a r t u r e s f r o m n o r m a l i t y : g r a p h i c

methods

91

since t h a t c o r r e s p o n d s to an infinite distance f r o m t h e m e a n . If y o u a r e interested in plotting all observations, y o u c a n plot, instead of cumulative frequencies F, the q u a n t i t y F — j expressed as a p e r c e n t a g e of n. O f t e n it is desirable to c o m p a r e observed frequency d i s t r i b u t i o n s with their expectations w i t h o u t resorting to c u m u l a t i v e frequency distributions. O n e m e t h o d of d o i n g so w o u l d be t o s u p e r i m p o s e a n o r m a l curve o n the h i s t o g r a m of a n observed frequency distribution. Fitting a n o r m a l d i s t r i b u t i o n as a curve s u p e r i m p o s e d u p o n a n observed frequency distribution in t h e f o r m of a histog r a m is usually d o n e only when g r a p h i c facilities (plotters) a r e available. O r d i nates are c o m p u t e d by m o d i f y i n g E x p r e s s i o n (5.1) to c o n f o r m t o a frequency distribution: l /Υ-μ\ ζ Z =

(5.2) Syfln

In this expression η is the s a m p l e size a n d i is the class interval of the frequency distribution. If this needs t o be d o n e w i t h o u t a c o m p u t e r p r o g r a m , a table of ordinates of the n o r m a l curve is useful. In F i g u r e 5.8A we s h o w t h e frequency distribution of b i r t h weights of m a l e Chinese f r o m Box 5.1 with t h e o r d i n a t e s of the n o r m a l curve s u p e r i m p o s e d . T h e r e is an excess of observed frequencies at the right tail d u e t o the skewness of the distribution. You will p r o b a b l y find it difficult t o c o m p a r e the heights of bars against the arch of a curve. F o r this reason, J o h n T u k e y h a s suggested t h a t the bars of the h i s t o g r a m s be s u s p e n d e d f r o m t h e curve. T h e i r d e p a r t u r e s f r o m expectation c a n then be easily observed against the straight-line abscissa of the g r a p h . Such a h a n g i n g h i s t o g r a m is s h o w n in Figure 5.8B for the birth weight d a t a . T h e d e p a r t u r e f r o m n o r m a l i t y is n o w m u c h clearer. Becausc i m p o r t a n t d e p a r t u r e s are frequently noted in the tails of a curve, it has been suggested that s q u a r e r o o t s of expectcd frequencies should be c o m pared with the s q u a r e roots of observed frequencies. Such a " h a n g i n g r o o t o g r a m " is s h o w n in Figure 5.8C for the Chinese birth weight d a t a . N o t e the a c c e n t u a t i o n of the d e p a r t u r e f r o m normality. Finally, o n e can also use an a n a l o g o u s technique for c o m p a r i n g expected with observed histograms. Figure 5.8D shows the s a m e d a t a plotted in this m a n n e r . S q u a r e r o o t s of frequencies are again s h o w n . T h e excess of observed over expected frequencies in the right tail of the distribution is quite evident.

Exercises 5.1

U s i n g t h e i n f o r m a t i o n g i v e n in B o x 3.2, w h a t is t h e p r o b a b i l i t y o f o b t a i n i n g a n i n d i v i d u a l w i t h a n e g a t i v e b i r t h w e i g h t ? W h a t is t h i s p r o b a b i l i t y if w e a s s u m e t h a t b i r t h w e i g h t s a r e n o r m a l l y d i s t r i b u t e d ? A N S . T h e e m p i r i c a l e s t i m a t e is z e r o . If a n o r m a l d i s t r i b u t i o n c a n b e a s s u m e d , it is t h e p r o b a b i l i t y t h a t a s t a n d a r d n o r m a l d e v i a t e is less t h a n (0 - 1 0 9 . 9 ) / 1 3 . 5 9 3 = - 8 . 0 8 5 . T h i s v a l u e is b e y o n d t h e r a n g e of m o s t tables, a n d t h e p r o b a b i l i t y can be c o n s i d e r e d z e r o for practical purposes.

92

5.2 5.3

chapter 5 / the normal probability

distribution

C a r r y o u t t h e o p e r a t i o n s l i s t e d in E x e r c i s e 5.1 o n t h e t r a n s f o r m e d d a t a g e n e r a t e d i n E x e r c i s e 2.6. A s s u m e y o u k n o w t h a t t h e p e t a l l e n g t h of a p o p u l a t i o n of p l a n t s of s p e c i e s X is n o r m a l l y d i s t r i b u t e d w i t h a m e a n o f μ = 3.2 c m a n d a s t a n d a r d d e v i a t i o n o f σ = 1.8. W h a t p r o p o r t i o n o f t h e p o p u l a t i o n w o u l d b e e x p e c t e d t o h a v e a p e t a l l e n g t h ( a ) g r e a t e r t h a n 4 . 5 c m ? ( b ) G r e a t e r t h a n 1.78 c m ? (c) B e t w e e n 2 . 9 a n d 3.6 c m ? A N S . (a) = 0 . 2 3 5 3 , ( b ) = 0 . 7 8 4 5 , a n d (c) = 0 . 1 5 4 .

5.4

P e r f o r m a g r a p h i c a n a l y s i s o f t h e b u t t e r f a t d a t a g i v e n i n E x e r c i s e 3.3, u s i n g p r o b ability paper. In addition, plot the d a t a on probability p a p e r with the abscissa in l o g a r i t h m i c units. C o m p a r e t h e r e s u l t s of t h e t w o a n a l y s e s .

5.5

A s s u m e that traits A a n d Β are independent a n d normally distributed with p a r a m e t e r s μΛ = 2 8 . 6 , σΑ = 4 . 8 , μΒ = 16.2, a n d σΒ = 4.1. Y o u s a m p l e t w o i n d i v i d u a l s a t r a n d o m (a) W h a t is t h e p r o b a b i l i t y o f o b t a i n i n g s a m p l e s i n w h i c h b o t h i n d i v i d u a l s m e a s u r e l e s s t h a n 2 0 f o r t h e t w o t r a i t s ? (b) W h a t is t h e p r o b a b i l i t y t h a t a t l e a s t o n e o f t h e i n d i v i d u a l s is g r e a t e r t h a n 3 0 f o r t r a i t B ? A N S . (a) P{A < 20}P{B < 2 0 } = ( 0 . 3 6 5 4 ) ( 0 . 0 8 2 , 3 8 ) = 0 . 0 3 0 ; (b) 1 - (P{A < 3 0 } ) χ ( Ρ { Β < 30}) = 1 - (0.6147)(0.9960) = 0.3856.

5.6

P e r f o r m t h e f o l l o w i n g o p e r a t i o n s o n t h e d a t a o f E x e r c i s e 2.4. (a) If y o u h a v e not already d o n e so, m a k e a frequency distribution f r o m the d a t a a n d g r a p h the r e s u l t s i n t h e f o r m of a h i s t o g r a m , ( b ) C o m p u t e t h e e x p e c t e d f r e q u e n c i e s f o r e a c h o f t h e c l a s s e s b a s e d o n a n o r m a l d i s t r i b u t i o n w i t h μ = Ϋ a n d σ = s. (c) G r a p h t h e e x p e c t e d f r e q u e n c i e s in t h e f o r m o f a h i s t o g r a m a n d c o m p a r e t h e m w i t h t h e o b s e r v e d f r e q u e n c i e s , (d) C o m m e n t o n t h e d e g r e e of a g r e e m e n t b e t w e e n o b s e r v e d a n d expected frequencies.

5.7

L e t u s a p p r o x i m a t e t h e o b s e r v e d f r e q u e n c i e s in E x e r c i s e 2.9 w i t h a n o r m a l f r e q u e n c y distribution. C o m p a r e the observed frequencies with those expected w h e n a n o r m a l d i s t r i b u t i o n is a s s u m e d . C o m p a r e t h e t w o d i s t r i b u t i o n s b y f o r m i n g a n d superimposing the observed a n d the expected histograms a n d by using a h a n g i n g h i s t o g r a m . A N S . T h e e x p e c t e d f r e q u e n c i e s f o r t h e a g e c l a s s e s a r e : 17.9, 4 8 . 2 , 7 2 . 0 , 5 1 . 4 , 17.5, 3.0. T h i s is c l e a r e v i d e n c e f o r s k e w n e s s in t h e o b s e r v e d distribution. Perform a graphic analysis on the following measurements. Are they consistent w i t h w h a t o n e w o u l d e x p e c t in s a m p l i n g f r o m a n o r m a l d i s t r i b u t i o n ?

5.8

12.88 9.46

11.06 21.27

7.02 9.72

10.25

15.81 5.60

14.20

6.60

10.42

8.18

11.44

6.37

6.26 5.40 11.09

7.92

12.53

3.21 8.74

6.50

6.74 3.40

T h e f o l l o w i n g d a t a a r e t o t a l l e n g t h s (in c m ) o f b a s s f r o m a s o u t h e r n laki 29.9

40.2

37.8

19.7

19.1 41.4

34.7 13.6

33.5 32.2

18.3 24.3

17.2

13.3

37.7

12.6

29.7

19.4

39.2

24.7

20.4

16.2 33.3

39.6

24.6

38.2 23.8 18.6

36.8

19.1

27.3 37.4

33.1 20.1 38.2

30.0 19.4

18.0

31.6 33.7

C o m p u t e t h e m e a n , t h e s t a n d a r d d e v i a t i o n , a n d t h e coefficient of v a r i a t i o n . M a k e a h i s t o g r a m of t h e d a t a . D o t h e d a t a s e e m c o n s i s t e n t w i t h a n o r m a l d i s t r i b u t i o n o n t h e b a s i s o f a g r a p h i c a n a l y s i s ? If n o t , w h a t t y p e o f d e p a r t u r e is s u g g e s t e d ? A N S . F = 2 7 . 4 4 7 5 , s = 8 . 9 0 3 5 , V = 3 2 . 4 3 8 . T h e r e is a s u g g e s t i o n o f b i m o d a l i t y .

CHAPTER

Estimation Hypothesis

and Testing

In this c h a p t e r we provide m e t h o d s to a n s w e r t w o f u n d a m e n t a l statistical questions that every biologist must ask repeatedly in the c o u r s e of his or her work: (1) how reliable are the results I h a v e o b t a i n e d ? a n d (2) h o w p r o b a b l e is it that the differences between observed results a n d those expected on the basis of a hypothesis have been p r o d u c e d by c h a n c e alone? T h e first question, a b o u t reliability, is answered t h r o u g h the setting of confidencc limits to s a m p l e statistics. T h e second question leads into hypothesis testing. Both subjects belong to the field of statistical inference. T h e subject m a t t e r in this c h a p t e r is f u n d a mental to an u n d e r s t a n d i n g of a n y of the s u b s e q u e n t chapters. In Section 6.1 we consider the f o r m of the distribution of m e a n s a n d their variance. In Section 6.2 we examine the d i s t r i b u t i o n s a n d variances of statistics o t h e r t h a n the mean. This brings us to the general subject of s t a n d a r d errors, which a r e statistics m e a s u r i n g the reliability of an estimate. C o n f i d e n c e limits provide b o u n d s to o u r estimates of p o p u l a t i o n parameters. W e d e v e l o p the idea of a confidence limit in Section 6.3 a n d s h o w its application to samples where the true s t a n d a r d d e v i a t i o n is k n o w n . However, o n e usually deals with small, m o r e o r less normally distributed s a m p l e s with u n k n o w n s t a n d a r d deviations,

94

chapter 6 / estimation and hypothesis

testing

in w h i c h case t h e t d i s t r i b u t i o n m u s t be used. W e shall i n t r o d u c e the t dist r i b u t i o n in Section 6.4. T h e a p p l i c a t i o n of t t o t h e c o m p u t a t i o n of c o n f i d e n c e limits f o r statistics of s m a l l s a m p l e s w i t h u n k n o w n p o p u l a t i o n s t a n d a r d d e v i a t i o n s is s h o w n in S e c t i o n 6.5. A n o t h e r i m p o r t a n t d i s t r i b u t i o n , t h e c h i - s q u a r e d i s t r i b u t i o n , is e x p l a i n e d in S e c t i o n 6.6. T h e n it is a p p l i e d to s e t t i n g c o n f i d e n c e limits for t h e v a r i a n c e in S e c t i o n 6.7. T h e t h e o r y of h y p o t h e s i s t e s t i n g is i n t r o d u c e d in Section 6.8 a n d is a p p l i e d in S e c t i o n 6.9 to a variety of cases e x h i b i t i n g the n o r m a l o r t d i s t r i b u t i o n s . Finally, S e c t i o n 6.10 illustrates h y p o t h e s i s t e s t i n g for v a r i a n c e s by m e a n s of t h e c h i - s q u a r e d i s t r i b u t i o n .

6.1 Distribution and variance of means W e c o m m e n c e o u r s t u d y of t h e d i s t r i b u t i o n a n d v a r i a n c e of m e a n s with a s a m pling experiment. Experiment 6.1 You were asked to retain from Experiment 5.1 the means of the seven samples of 5 housefly wing lengths and the seven similar means of milk yields. We can collect these means from every student in a class, possibly adding them to the sampling results of previous classes, and construct a frequency distribution of these means. For each variable we can also obtain the mean of the seven means, which is a mean of a sample 35 items. Here again we shall make a frequency distribution of these means, although it takes a considerable number of samplers to accumulate a sufficient number of samples of 35 items for a meaningful frequency distribution. In T a b l e 6.1 we s h o w a f r e q u e n c y d i s t r i b u t i o n of 1400 m e a n s of s a m p l e s of 5 h o u s e f l y w i n g lengths. C o n s i d e r c o l u m n s (1) a n d (3) for the t i m e being. A c t u a l l y , t h e s e s a m p l e s w e r e o b t a i n e d not by b i o s t a t i s t i c s classes but by a digital c o m p u t e r , e n a b l i n g us t o collect t h e s e values with little elTort. T h e i r m e a n a n d s t a n d a r d d e v i a t i o n a r c given at the f o o t of the table. T h e s e v a l u e s are p l o t ted o n p r o b a b i l i t y p a p e r in F i g u r e 6.1. N o t e t h a t t h e d i s t r i b u t i o n a p p e a r s q u i t e n o r m a l , as d o c s t h a t of the m e a n s b a s e d o n 200 s a m p l e s of 35 w i n g l e n g t h s s h o w n in t h e s a m e figure. T h i s i l l u s t r a t e s a n i m p o r t a n t t h e o r e m : The means of samples from a normally distributed population are themselves normally distributed regardless of sample size n. T h u s , we n o t e t h a t t h e m e a n s of s a m p l e s f r o m the n o r m a l l y d i s t r i b u t e d housefly w i n g l e n g t h s a r e n o r m a l l y d i s t r i b u t e d w h e t h e r t h e y a r e b a s e d o n 5 or 35 i n d i v i d u a l r e a d i n g s . Similarly o b t a i n e d d i s t r i b u t i o n s of m e a n s of t h e heavily s k e w e d milk yields, as s h o w n in F i g u r e 6.2, a p p e a r t o be close t o n o r m a l d i s t r i b u t i o n s . H o w e v e r , t h e m e a n s based o n five milk yields d o n o t a g r e e with the n o r m a l nearly as well as d o the m e a n s of 35 items. T h i s illustrates a n o t h e r t h e o r e m of f u n d a m e n t a l i m p o r t a n c e in statistics: As sample size increases, the means of samples drawn from a population of any distribution will approach the normal distribution. This theorem, when rigorously stated (about sampling from populations with finite variances), is k n o w n as t h e central limit theorem. T h e i m p o r t a n c e of this t h e o r e m is that if η is l a r g e e n o u g h , it p e r m i t s us t o use the n o r m a l distri-

6.1 / d i s t r i b u t i o n a n d v a r i a n c e o f

95

means

TABLE 6 . 1

Frequency distribution of means of 1400 random samples of 5 housefly wing lengths. ( D a t a f r o m T a b l e 5.1.) C l a s s m a r k s chosen t o give intervals of t o each side of the p a r a m e t r i c m e a n μ.

Class

mark Y (in mm χ 10~ ')

W Class mark (in ffy units)

39.832

_ il

40.704

z

(S) f 1

->4 4

41.576

11 19

4

42.448

- U

43.320

- U

64 128

3 4 41

44.192 , 45.064 μ = 45.5 -» 45.936

247 226

1

259

4 3 4

46.808

231

47.680

u

121

48.552

|3

*A

61

49.424

21

23

50.296

z

51.168

->4

6

4

3 1400

F=

45.480

s =

1.778

ffy =

1.744

b u t i o n to m a k e statistical inferences a b o u t m e a n s of p o p u l a t i o n s in which the items are not at all n o r m a l l y distributed. T h e necessary size of η d e p e n d s u p o n the distribution. (Skewed p o p u l a t i o n s require larger s a m p l e sizes.) T h e next fact of i m p o r t a n c e that we n o t e is that the r a n g e of the m e a n s is considerably less t h a n that of t h e original items. T h u s , the wing-length m e a n s range f r o m 39.4 to 51.6 in samples of 5 a n d f r o m 43.9 t o 47.4 in s a m p l e s of 35, but the individual wing lengths r a n g e f r o m 36 to 55. T h e milk-yield m e a n s range f r o m 54.2 to 89.0 in samples of 5 a n d f r o m 61.9 to 71.3 in samples of 35, but the individual milk yields range f r o m 51 t o 98. N o t only d o m e a n s s h o w less scatter than the items u p o n which they are based (an easily u n d e r s t o o d p h e n o m e n o n if you give s o m e t h o u g h t to it), but the range of t h e distribution of the m e a n s diminishes as the sample size u p o n which the m e a n s a r e based increases. T h e differences in ranges a r e reflected in differences in the s t a n d a r d deviations of these distributions. If we calculate t h e s t a n d a r d deviations of the m e a n s

Samples of 5

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1

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i

i

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- 3 - 2 - 1 0

i

i

i

i

1

i

2

i

1

3

i

l

4

H o u s e f l y w i n g l e n g t h s in σ γ units Samples of 35

0.1 . 1 ι 1 I I I I I I—I I 1 I I I - 3 - 2 - 1 0

I

2

3

4

H o u s e f l y w i n g l e n g t h s in (i v units figure

6.1

G r a p h i c analysis of m e a n s of 14(X) r a n d o m s a m p l e s of 5 housefly wing lengths (from T a b l e 6.1) a n d of m e a n s of 200 r a n d o m s a m p l e s of 35 housefly wing lengths.

Samples of 5

0.1 - 3 - 2 - 1 0

1

2

3

M i l k y i e l d s in ιτ,7 units S a m p l e s of 3 5

99.9

•S a

99

£-95

|

90

1 8 0 1.70

r 5 ft) s 0 8

40

X 30 ω 20 1

"0

3 ε3

5

υ

ι οι - 2 - 1 0

1

2

3

M i l k y i e l d s in <τν units FIGURE 6.2 , , „ ,„,. . m G r a p h i e analysis of m e a n s of 1400 r a n d o m s a m p l e s of 5 milk yields a n d of m e a n s of . 0 0 r a n d o m samples of 35 milk yields.

98

c h a p t e r 6 / estimation and hypothesis

testing

in t h e four distributions u n d e r c o n s i d e r a t i o n , we o b t a i n the following values:

Observed standard deviations of distributions of means η = 5 η =35

Wing lengths Milk yields

1.778 5.040

0.584 1.799

N o t e t h a t t h e s t a n d a r d deviations of the s a m p l e m e a n s based o n 35 items are considerably less t h a n t h o s e based on 5 items. This is also intuitively obvious. M e a n s b a s e d o n large s a m p l e s should be close to the p a r a m e t r i c m e a n , a n d m e a n s based o n large s a m p l e s will not vary as m u c h as will m e a n s based on small samples. T h e v a r i a n c e of m e a n s is therefore partly a f u n c t i o n of t h e s a m ple size o n which the m e a n s are based. It is also a function of the variance of t h e items in the samples. T h u s , in t h e text table above, the m e a n s of milk yields have a m u c h greater s t a n d a r d deviation t h a n m e a n s of wing lengths based o n c o m p a r a b l e s a m p l e size simply because the s t a n d a r d deviation of the individual milk yields (11.1597) is considerably greater t h a n that of individual wing lengths (3.90). It is possible ίο w o r k out the expected value of the variance of s a m p l e m e a n s . By expected value we m e a n the average value to be obtained by infinitely repeated sampling. T h u s , if we were t o t a k e samples of a m e a n s of η items repeatedly a n d were t o calculate t h e variance of these a m e a n s each time, the average of these variances would be t h e expected value. W e can visualize the m e a n as a weighted a v e r a g e of the η independently sampled o b s e r v a t i o n s with cach weight w, equal to 1. F r o m Expression (3.2) we o b t a i n v w _ in > i i

Σ Η,' for the weighted m e a n . W c shall state w i t h o u t proof t h a t the variance of the weighted s u m of independent items Σ" is V a r ( £ w , Y ^ = £vvraf

(6.1)

where nf is the variance of V^. It follows that η

Since the weights u·, in this case equal 1. Σ" »ν( = η, a n d we can rewrite the a b o v e expression as η V σΐ

6.1 / d i s t r i b u t i o n a n d v a r i a n c e o f

99

means

If we a s s u m e that the variances of a r e all e q u a l t o σ 2 , the expected variance of the m e a n is

a n d consequently, t h e expected s t a n d a r d deviation of m e a n s is σ

(6.2a) F r o m this f o r m u l a it is clear t h a t the s t a n d a r d deviation of m e a n s is a f u n c t i o n of the s t a n d a r d deviation of items as well as of s a m p l e size of means. T h e greater the sample size, the smaller will be the s t a n d a r d deviation of means. In fact, as s a m p l e size increases to a very