456 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XV, 



In this classification two things have been sought: (1) Smoothness of the resulting fre- 

 quencies, so that the values of x 2 may be as free as possible from irregularities in frequencies 

 that have no significance for the diagnostic values of the tests; (2) frequencies in all classes as 

 large as possible relative to the total for the distribution in order that the fundamental assump- 

 tion of the x 2 test may be satisfied. 



The following table, abbreviated and modified from Elderton, shows the approximate 

 odds against obtaining the corresponding values of x 2 by chance alone: 



It will be seen that the odds corresponding to a value of 15 are about 68 times those corre- 

 sponding to a value of 5. This ratio increases as larger and larger pairs of values are taken. 

 Interpolation in Elderton's tables gives a value of x 2 = 4.351 for exactly even chances. Thus 

 4.351 is the median value of x 2 for pairs of samples drawn from the same population, and differ- 

 ing, therefore, only in so far as purely chance factors have been operative, differences which 

 are just as often greater than the median value as less. Interpolation of the values of x 2 

 corresponding to odds of three to one (the upper quartile or probable error) and one to three (the 

 lower quartile or probable error), under the same circumstances, gives 2.600 and 6.623, respec- 

 tively. The differences between each of these values and the value for even chances, 4.351, 

 may thus be taken to indicate approximate magnitudes of the probable error of x 2 on either 

 side of its median, provided we assume that we may apply to pairs of samples from different 

 populations this criterion which has been developed for samples from the same population. 

 A further complication ensues when we compare two or more modes of measurement of the same 

 two samples, for the values are correlated if the variables measured are correlated. Now the 

 correlation of the fluctuations of sampling of the means of two sets of measuremants is the same 

 as the correlation of the variables measured. 



Suppose, then, we are comparing two samples — one, say, of the "poorest" infantry group 

 with another, say, of the standard group. We get a given value of x 2 for the difference between 

 the distributions of alpha, test 1, for the two samples, and we get another value of x 2 for the 

 corresponding difference in the case of alpha, test 2. But alpha, test 1, is correlated with 

 alpha, test 2. Hence, if we take a second sample for one group (say, the "poorest" infantry) 

 and compare it in the same manner with the original sample for the other group (the standard) 

 we may expect, in this second comparison, that a chance increase (or decrease) in the value 

 of x 2 for alpha, test 1 , will be accompanied by an increase (or decrease) for alpha, test 2. If 

 alpha, test 1, and alpha, test 2, were perfectly correlated, any changes in the difference between 

 these two groups would necessarily be reflected equally by both tests, and the tendency of both 

 tests to increase (or decrease) together would affect inversely the size of the difference between 

 the samples — i. e., the difference would depend on the degree of correlation. According, then, 

 as we take different samples of population we tend to change the values of x 2 for different tests 

 together in the same direction and approximately in the same amount. It follows that the 

 differences between two values of x 2 , altering under these conditions, would tend to remain 

 constant. In other words, the difference between two values of x 2 for different measurements 

 of the same pair of samples is not subject to much greater variability than is a single value 

 of x 2 ) provided the correlation between the variables measured is high. This last condition is 

 filled in the case of the alpha tests and for most of the beta tests. 



We have seen that the probable error of x 2 may be taken approximately as 2 (6.623 — 

 4.351=2.272 and 4.351-2.600 = 1.751, the distribution of x 2 is skewed). It seems reasonable, 

 therefore, to assume that, in the comparison of two groups, pairs of values x 2 differing by 10 



