percentile), is for a sampling of 100 or more of a series scored on a reasonably fine scale nearly as accurate as the measures that take account of the amount of every deviation. The facts for my series are given in Table II. In general the arguments that support the median as a measure of central tendency, support also the Quartile as a measure of variability in the case of large samplings from finely scaled series (say of 100 or more cases of a series with 20 or more steps). In the case of smaller samplings the calculation of Q. is often as long as that of the A.D.

If the report of an investigation gives somewhere the entire distributions the author may properly compute only the medians and Q.'s. If the average is used as the measure of central tendency the Q. is of course not very advantageous, since an approximate A.D. will have been calculated in getting the average.

Table III. gives the results of the individual sets drawn. It is not necessary to examine it to follow the discussion past or to come, but I insert it for the sake of any student who may wish to make calculations from its facts other than those which I have made in Table II.

§3. The Divergences of the Obtained from the True Measures by Theory and by Experiment

It is always interesting to compare the result of experiments in chance with the expectations derived from the theory of probability. Accordingly, I give the facts in Table IV. so as to save the reader interested in this matter the time of collation and calculation from the data of Tables II. and III.

The figures under Theory were calculated not from the A.D.'s of all the separate samplings, but once for all from the A.D. of the total series.

The figure under Theory is not in any case exactly the amount to be expected under strictly correct theory, but is the amount to be expected from the formula A.D. tr. - obt. Av.

=

A.D. dis
Vn

This formula, applicable to cases of random sampling from a distribution of the so-called normal type, will of course not suit exactly distributions limited in extent and irregular in form. Comparison with it is however the important matter practically, since it is the formula in universal use.

84. The Relation Between the Amount of a Central Tendency and the Amount of the Variability of the Group about the Central Tendency

In comparing groups with respect to variability allowance must be made for the fact that, in certain cases at least, the amounts of the central tendency influence the amounts of the variabilities. Thus the A.D. of men in weight is hundreds of times that of butterflies, yet the former are of course not really a hundred times as variable. Thus the A.D. of a group in a test of addition was, for trials of 40 seconds, 2.18; for trials of 80 seconds, 3.41; and for trials of 120 seconds, 5.18. It would obviously be silly if we had tested men with trials of 80 seconds and women with trials of 40 seconds, and obtained these results, to infer that men are 50 per cent. more variable in ability to add than are women.

In using the so-called coefficient of variation (proposed by Pearson) one makes allowance for the possible influence of the central tendencies' amounts by dividing through the gross variabilities each by the amount of its corresponding central tendency. I have elsewhere shown that for mental and social measurements no one such rule can be always or even often right and suggested that in any case a division through by the square root of the corresponding central tendency is more in accord with both theory and facts.1

In this section enough data will be presented to practically demonstrate both of these assertions. It is not important to investigate the matter exhaustively for the very reason that no one general rule for comparing groups with respect to variability can be found. All that is needed is a clear enough proof of the inadequacy of the practice of comparing groups after dividing through the gross variabilities by the corresponding means-clear enough to stop the spread of the practice and to warn readers against conclusions based on such comparisons.

If we take the arrays of y in a case where y is positively correlated with x we have a series of groups with central tendencies varying from lower to higher which are selected at random so far as concerns any influence on the variability except the influence of the amount of the mean. The differences in variability found for these arrays give, then, in connection with the differences in the amounts of their central tendencies, the answer to our problem for the case of comparisons of groups with respect to their variability in the same trait. If we find that even in such cases there is no constant relation of difference in central tendency to difference in Mental and Social Measurements, pp. 102–103.

variability, but that one law obtains for stature and another for span or finger length, then a fortiori no constant relation can be presupposed when the variability of a group in one trait is to be compared with its variability (or that of a second group) in a different trait.

The first facts to which I call the reader's attention are the comparison of arrays of y corresponding to very low values of x with arrays of y corresponding to very high values of x in the case of ten correlations chosen at random (so far as this issue is concerned) from Vols. I. and II. of Biometrika. The number of cases ranged from 49 to 319. The results are given in Table V. in the form of (1) the variability of arrays related to high central tendencies of (and consequently having high central tendencies of y) divided by the variability of arrays related to low central tendencies of x, under the heading 'Gross'; (2) the Pearson coefficient of variability for the former divided by the Pearson coefficient of variability for Gross the latter under the heading ; and (3) the similar ratio for the CT two variabilities each having been divided by the square root of the amount of the corresponding central tendency, under the heading Gross A perfect method would give values of 100 throughout. VC.T.

The detailed facts from which these ratios come are given in Table V. (b).