A Statistical Discussion of Sets of Precise Astronomical Measurements Parallaxes

STATISTICS: WILSON AND LUYTEN

VOL. 10, 1924

If equations (2.6) are multiplied by of Tvpe II are given by

Ogik

gktk and summed,

129

the directions

+ Fkjl2 i = 0-

(3.6)

But from (3.2),

Fkjl2 tk = 0,

(3.7)

and the directions are given by We find

Ogik

= 0

Fl212= Ug22 F, tFl12 = -2glu F, F212 = - g12 F, and the equation (2.7) for 0 becomes 02 + 4gF2 = 0. Thus, if neither F nor g is zero (3.8) gives the null-directions.

(3.8)

{Fl112= jg12F,

(3.91)

A STA TISTICAL DISCUSSION OF SETS OF PRECISE ASTRONOMICAL MEASUREMENTS: PARALLAXES By EDWIN B. WILSON AND WILLZM J.- LUYTZN HARVARD SCHOOL OF PUBLIC H1ALTH AND HARVARD COLLZGU OBSERVATORY

Communicated February 29, 1924

In discussing the precision of the determination of the velocity of light Simon Newcomb remarked,' "SQ far as could be determined from the discordance of the separate measures the mean errors of Newcomb's result would be less than 10 km., but making allowance for the various sources of systematic error the actual probable error was estimated at .30 km." Thus did Newcomb give expression to a belief that the probable error as calculated from the data was likely to represent only a fraction of the -real probable error connected with the determination of a mean, and although he would have liked to claim the maximum precision for his value of the velocity of light, and probably did claim all he thought safe, his judgment led him to give as the actual probable error the triple of that calculated bv his formula. If Newcomb had had faith in this value -10 km. he would have asserted that the chance of the departure of the velocity of light from his mean by so much as 30 km. was less than one in twenty-two; instead, he said not more than one in two. In view of the ever widening application of statistical methods to fields of investigation less precise, less controlled, and less controllable than the

130

~~STA TISTICS:-

PROC, N. A. S.

WILSON A ND L UYTEN

best physical experimentation, and in vilew of the tendency manifest in some quarters to place a great deal of confidence in the probable error4 for mean values, derived by the application of mathematical formulas, there may be some value in discussing the question of how far an actual probable error may exceed the value calculated from the discordance of the separate measures. The discussion may be carried out theoretically, i.e., mathematically, by an analysis of the derivation of the formula for the probable error, or experimentally by the examination of the actual determined values of physical measurements and their probable errors. The latter method is the one selected here, and in applying it we naturally turn first to the sort of measurements in which precision is most important, and has been striven for by every known means: determination of parallax. The modern observations of trigonometric parallaxes may undoubtedly be classed among the most precise in astronomy. Parallaxes have been

.....

--

..

50~~~~~~~~~~~~.

.. ..

...

..

..

..

..

..

...MN........

{il t +~~~~~~~~~~~~~..... I;7

:7..........

FIGUR

1.FIURE.

Fig. 1. Plot on arithmetic-probability paper. Ordinates represent....percentages.of epane nth et xed tevlu ienb h str orwih h uatt abscissae.~~~lotonarihmeicproabiit paer

Fig. 2.

....

Ordnats.eprsetEj perenagejo

stars forwhich the difference in parallax. (Allegheny~~~1: relative.+.........Mc.ormic. relative)~~~~: exedvau sgie:yabcsae1U i O~O........) by photographic methods, employing..... rerctr r eletr determined oflong focal length,~...... at....th.Alghn.corik,Geewch.t.Wlsn Yerkes..Sprou an.Daror.osevtoie..f llth .ubisedreuls only those ofAlegen an.cCrik.aea.ufcin.ume.ftr incomo to allow a1rell.stifatoy. ttitialdicusin Les is consider::~::: wha reult wema eec.. Both.obsrvatorie L

L

- :;

:."

an McCormick deemn5hi0aalae (Allegheny~~~......

ymauig

VOL. 10, 1924

STATISTICS: WILSON AND L UYTEN1

131

at different epochs, the position of the parallax star relative to three or four comparison stars. The basis on which these stars are selected is as follows: 1° they should be reasonably close, and evenly distributed around the central parallax star, and 20 they should be of approximately the same photographic brightness as the reduced image of the comparison star. It therefore appears not unlikely that there may have been considerable overlapping in the stars used by both observatories (Allegheny and McCormick), and that a part of the systematic error arising from the comparison stars will be the same for both. There seems to be no a priori reason for assuming that any of the experimental technique would introduce a negative correlation between the results, and accordingly we may expect a positive, though probably small, correlation between the two series, which will result in the probable error of the difference between the values of the individual parallaxes being less than the root of the sumsquare probable error of the two: rA-M = pr4+ r -2prArM, p = coefficient of correlation. (1) If we take for each star the difference dr = parallax Allegheny -parallax McCormick, and divide by the quantity r = V(rA + rm) calculated from the probable errors reported, we should expect that the resulting series of values dxr/r = a should be distributed in an error curve, and that the parameter of this curve should be such as to make a trifle more than one-half of the values of a fall between a = 41. Or rather, this is what we must expect if we believe that the actual probable errors are those deduced from the discordances of the observations in the individual parallax determinations. We have 273 values of a at our disposal.2 As it appears from other investigations that the reduction from relative to systematically correct absolute parallax is +0'"'004 for Allegheny and +0"003 for McCormick, d7r was then taken to be 0.001 larger than the difference between the relative Allegheny and McCormick values. The distribution curve for the quotient a, plotted on arithmetic-probability paper is shown in figure 1. The accordance with the normal error curve is perfect. The probable error of the present curve (the probable error is 0.6745 a if a. is the standard deviation of the curve e_<x2/20: \/2xr oa) is, however, decidedly larger than unity, viz: 1.24 t0.06. The frequency of the difference dir itself also follows a normal error curve, as is shown in figure 2, with -0"0023 as mean value, and ±0f.'015 as its

probable error. We conclude that, on the whole, the McCormick parallaxes are 0.0033 ±0.'0010 larger than the Allegheny values, and that so far as may be judged by this comparison of the two sets of values, the actual probable errors of the individual determinations are on the average about 25% greater than those calculated from the observations.

132

STATISTICS: WILSON AND LUYTEN

PRoc. N. A. S.

If we now return to the general formula (1) and ask what coefficient of correlation p would explain the spread of the values of a as found, we have: A + M) rA-M = V(rr + rM - 2prArm) = 1.24 Hence p = .27 (YAI[7M + YM/7A) = -.54 (a.15) It will probably not seem so reasonable to attribute the extra dispersion to such a high negative correlation between the observations as to a relatively small systematic underestimation of the probable errors when calculated by the rule. It is a testimonial' to the judgment with which parallaxes determinations are reduced that the factor of amplification in passing from the calculated to the actual probable errors in this series of 237 Allegheny-McCormick values should be as low as 1.24. There are possibilities of correlation between these series that may have reduced this value; there is the probability that if the same series of stars were observed by a large number of obs,ervatories using a variety of methods we should have evidence of a somewhat larger factor, say 1.5, but the evidence is reasonably good that we should not have to use a figure anywhere nearly as high as the 3 sug. gested by Newcomb for his determination of the velocity of light. Of the other parallax series only Yerkes (Ye) has a sufficiently large number of stars in common with Allegheny (A) and McCormick (M) to be treated with success. The differences A-Ye and M-Ye were first obtained, and it was found that their arithmetical mean values were: A - Ye- 0"0017 -0"0022 (82 stars); M -Ye =-0.0042 ±t0.0020 (87 stars). After correcting the dir for these systematic differences the respective a's were formed. In both cases the distribution curve of a was as nearly normal as could be expected from such scanty material. It was found that the probable error of an individual difference A-Ye was 1.51 -0.12 larger, and that of M -Ye was 1.32 +0.10 larger than the resultant of the published probable errors.3 1 Encyclopedia Britannica, 16, p. 625 (llth edition).

2 Based on all published parallaxes and some unpublished values kindly put at our disposal by Dr. Curtis and Dr. Mitchel. For results based on 56 stars see van Rhijn, Pubi. Astron. Lab. Groningen, 1923, No. 34, p. 74. 3 The three differences A-M, M- Ye, and Ye-A do not add up to zero, in spite of the fact that some stars are common to all three series. They must, therefore, be considered to have significance for our work only insofar as their application reduces the values of the ratios between the presumptive and the published probable errors to a minimum