TABLE V. (b)

VARIABILITIES OF ARRAYS OF y RELATED TO LOW AND HIGH VALUES OF . IN TERMS OF A.D.

(Each case measured is recorded in three lines: the first line gives the values of ; the second line gives the variabilities of the related arrays of y; the third line gives the numbers of cases in the arrays. The volume and page numbers refer always to Biometrika.)

The gross variabilities often increase as we would expect with higher central tendencies, though by no means always. Seven out of ten do so, giving a median value of 109.2 instead of 100. The Pearson coefficient of variation makes too much of a deduction for an increase in the amount of the central tendency in all but three cases, giving a median value of 90.1 instead of 100. The square root deduction, with a median value of 97.5, makes the least error of any one single method. These facts alone disqualify the so-called 'coefficient of variation' as a means of comparing variabilities. But more detailed studies of the cases of length of finger, span and stature will be still clearer.

The facts for length of left middle finger are as given in Table VI.

TABLE VI.

RELATION OF AMOUNT OF VARIABILITY TO AMOUNT OF CENTRAL TENDENCY. FINGER LENGTH. (Biometrika, Vol. I., p. 216)

In the case of finger length increase in the amount of the central tendency does not imply an appreciable increase in the amount of variability. No allowance is needed.

In the case of span it would be equally absurd not to make an allowance and one as great or nearly as great as the Pearson method makes. For the preliminary study of the variability of span reported in Table V. is confirmed by the facts in the case of three other span series. These facts (given in Table VII.) abundantly prove that the influence of the amount of the central tendency on the amount of the variability follows totally different laws in the case of span and of finger length.

TABLE VII.

RELATION OF AMOUNT OF VARIABILITY TO AMOUNT OF CENTRAL TENDENCY. SPAN. (Biometrika, Vol. II.,

pp. 399-401)

As a final case let us take stature. Here the variability is slightly less as the amount of the central tendency increases. The facts are given in Table VIII. constructed on the same plan as Table VI.

TABLE VIII.

RELATION OF AMOUNT OF VARIABILITY TO AMOUNT OF CENTRAL TENDENCY IN GROUPS DIFFERING IN CENTRAL TENDENCY.

(Biometrika, Vol. I., p. 216)

STATURE.

MEASUREMENTS OF RELATIONSHIPS

The importance to any science of exact and convenient methods of measuring the relationships of the facts it studies should be obvious. It is therefore unfortunate that students of psychology and the social sciences have with few exceptions neglected both the theoretical problem of correlated variations and the careful measurement of such relationships as they have in fact found.

The failure to utilize the methods devised by Galton, Pearson, Sheppard, Spearman and others is due partly to an ignorant and partly to an intelligent suspicion aroused by the mathematical derivations of these methods. Ignorance of the rationale of their derivations cooperating with ignorance of the conditions which require their use and of the necessity of some such refined methods has caused the stupid suspicion and aversion. Inability to follow the mathematics of the derivation of formula, at least in detail, cooperating with the rational expectation that too abstract methods will fit the concrete cases imperfectly and with the equally rational confidence that proofs resting upon the assumption of close approximation of actual variations in mental and social facts to the probability curve distribution are always unsafe and, perhaps, usually misleading, has caused the intelligent suspicion.

It is probable that unless these methods are soon subjected to a review by some one who can both make perfectly clear their presuppositions to the rank and file of investigators in psychology and the social sciences and prove their applicability to actual cases of relations to be measured, there will be damage done in two ways. Many investigators will as in the past use hopelessly crude methods and misinterpret relationships; and also many investigators will learn off the formulas of the mathematical statisticians and apply them to cases where they are out of place and give inadequate and misleading results. To both of these errors the writer, for instance, confesses himself guilty in the past.

I am unable to make such a review but as no one of those who are able seems willing,' I have made a partial and inferior substitute for it which I hope may, in so far as it is sound, be instructive to students of mental measurements and, in so far as it is unsound,

1

Perhaps Mr. C. Spearman's article on 'The Proof and Measurement of Association between Two Things' (in the Am. J. of Psy., Vol. XV.) may be considered as filling the need, but I fear that it is too technical in parts and not inquisitive enough concerning the actual relations between (1) the individual relationships, from which all our computations ought to start, and (2) the general expressions or summaries of them. At all events I am not trying to do over again, for better or worse, what Mr. Spearman has done, but something which is needed as introductory and accessory to his work.

may provoke some capable student to give the adequate review that is so much needed.

This report will presuppose in the reader knowledge of the bare elements of the theory of measurement of variable facts such as is given for instance in the writer's Introduction to the Theory of Mental and Social Measurements. It will deal in order with the following topics:

I. What is actually measured by typical measures of the relationship between first and second member of a pair in a series of pairs of values, each first-member value being a deviation from the central tendency of one series and each second-member value being a related deviation from the central tendency of a second series?

II. What are the respective presuppositions of each of these typical measures?

III. What are the advantages and disadvantages of each of these typical measures?

The only original contributions which this discussion contains are (1) the investigation of certain artificially constructed cases of correlation and (2) a laborious but not very important experimental testing of the comparative reliability of different measures of relationship, and (3) a similar experimental testing of methods for correcting measures of relationship for the 'attenuation' due to inaccurate original data.

§ 5. I. What is actually measured by typical measures of the relationship between first and second member of a pair in a series of pairs of values, each first-member value being a deviation from the central tendency of one series and each second-member value being a related deviation from the central tendency of a second series? Consider the following series of paired values of A and B :

The average of ratios is valueless because it overweights positive values of 1, 2 pairs, etc.

Per cent. unlike signs = .267, r as calculated therefrom being .665.