The abilities of man : their nature and measurement / by C. Spearman.

  • Charles Spearman
Date:
1927
Catalogue details

Licence: In copyright

Credit: The abilities of man : their nature and measurement / by C. Spearman. Source: Wellcome Collection.

    451/474 page 437
    The problem is to discover what values to assign to the w*s in order to make rtfJ as large as possible, where t denotes the measurement supplied by the whole team. By the formula for correlations of sums (present writer, Brit. J. Psych, v. 1913, p. 419), rig=S(wu . rau9)l[S(wv2) +2SS(wu . wv . rauav)f, (26) where S indicates summing over all values of u, and SS over all different values for u and v. The value on the right of the preceding equation will reach its maximum when its total differential, or that of its log, =0. Differentiating, then, the term on the right of (26) with respect to wu and equating the result to zero, there ensues after some arithmetic ____S(wu . rug)_ Wa yag • S i^u • ^ug) S (wJS) 3 2 S S [Wy, . Wy . Vug • Yvg) where S' indicates summing over all values except a. But, by symmetry, the term on the left of (27) = a similar term with a replaced by any other letter. In this way we get, with some arithmetic, quite generally, o'. (I ~ Vug) I Vug = ~’ ^vg) frvg • (^8) That is to say, the weight wu of any correlation rug has to be made proportional to ~ ^ug) • (^9) Let us proceed to ascertain what this ensuing maximum value of rtg amounts to. We have by the formula for correlations of sums quoted above (remembering that all our test-scores have been reduced to equal standard deviations) Ytg = S(Wu . Yng^j /\S (wi 2SS (W(i . Wv . Yyg . Cyg) ] , and simplified = 1/(1 -1 /S)*, where S denotes S(wu . Vug) (3°) 7. To expYess the team measuYement in absolute value.—The preceding equations have shown us how the single tests should be weighted relatively to one another. But this will only serve to compare the different individuals submitted to this same team of tests ; it will be of no use towards comparing individuals thus tested with those who have been submitted to a different team of tests. For the latter purpose, the standard deviations for the different teams must be equalized (or at least shown to have some common unit). S.A.M. 2 E 2
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