No CrossRef data available.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 15 September 2014
Researches of recent years have shown that the intercorrelations which exist among the measures of mental qualities are connected in a particular way, from which it has been sought to deduce facts concerning the nature of the mental structure on which the various qualities depend. If we take any four such qualities, there are six correlation coefficients, r12, r13, r14, etc., and it is found that the quantity F = r13r24 — r14r23 is in practice approximately zero. Since the observed correlations are vitiated by errors due to measuring a sample of the population instead of the whole it is legitimate to think that the value of F throughout a set of mental qualities is truly zero.
page 16 note * Spearman, , The Abilities of Man, Appendix, pp. iiGoogle Scholar, iii.
page 17 note * Spearman, loc. cit.; Maxwell Garnett, Proc. Roy. Soc., A, xcvi, pp. 91 et seq.
page 17 note † Proc. Boy. Soc., A, 1919, xcv, p. 400.
page 17 note ‡ Thomson, , “A worked-out Example of the Possible Linkages of Four Correlated Variables on the Sampling Theory,” British Journ. Psychol., 1927, 18, 68Google Scholar; Mackie, , “The Probable Value of the Tetrad-difference on the Sampling Theory,” British Journ. Psychol., 1928, 19, 65.Google Scholar
page 17 note § See Garnett's article referred to.
page 18 note * For a discussion of this geometrical representation of qualities and of the correlation between them see Garnett, , “On Certain Independent Factors in Mental Measurements,” Proc. Roy. Soc., A, 1919, xcvi.Google Scholar
page 20 note * It has been pointed out to me that the same result may be obtained thus. The volume of an N-dimensional sphere is V = ∫ ∫ … ∫dx 1dx 2 … dx N. Let N new variables be defined by equations (7), where the l's are replaced by x's, along with r 2 = x 12 + x 22 + x 32 + … + x N2. The integral is transformed into V = ∫ ∫ … ∫ J(x 1, …, x N)drdθ1 … dθN–1, where J is the Jacobian 
The surface area is 
= ∫ ∫ … ∫J dθ1 … dθN–1, so that the surface element is 
dθ1dθ2 … dθN–1. When this is evaluated, and r put equal to 1, the result (8) is obtained.
page 22 note * In essence this is a geometrical theorem. E.g., if there are only two variables x and y, and the lines OQ1, OQ2, OQ3, OQ4 make the angles α, β, γ, δ with OX, we have cos (γ – α) cos (δ – β) – cos (γ – β) cos (δ – α) = (cos α sin β – cos β sin α)(cos γ sin δ - cos δ sin γ), a result easily proved by elementary trigonometry.
page 27 note * The above analysis supposes that n is an even number. If n is odd, there will be throughout slight modifications in the various integrations, but the final result as expressed in equation (19) will be the same.
page 28 note * This is because the distribution of every l is the same. If we take the (n + l)th axis as our example, we have from equations (12) 1l 2l = sin θn sin фn. Multiplying by the frequency and integrating, we have ∫ ∫, … ∫ cos θ2 cos2 θ3 … cosn–2 θn–1 cosn–1 θn sin θndθ1dθ2 … dθn, × a similar expression in ф 
as before.
page 31 note * See Brown, and Thomson, , The Essentials of Mental Measurement (1925), chap. x.Google Scholar
page 31 note † Mackie, loc. cit.
page 32 note * Mackie, loc. cit., equation (21).
page 34 note * Mackie, loc. cit.
page 34 note † Brit. Journ. Psychol., 1924, xv, 18, 19.