ANALYSIS OF ILLUSTRATIVE GROUP 395

a curious algebraic corollary,1 from which it follows that
(under certain not unreasonable conditions) identical

1 Marks of Examiners., p. 287. Or, as I put it in another paper, " g is
simply the average of the tests" ([102], p. 176. Cf. also pp. 330 L above).
This is my justification for determining c true marks' by simply averaging
ranks for the separate tests (e.g. in tests of aesthetic appreciation, etc.)
and the order of' general preference ' by simply averaging ranks for individual
preferences (e.g. in determining preferences for school subjects: cf. Burt,
ap. Board of Education's Report on the Primary School, 1931, pp. 277-8 ;
Pritchard, Brit. J. Educ. Psych^ V, 1935, pp. 15 et seq.) : the minor factors
included in such individual ranks or standard measurements are more or less
cancelled by the process of averaging. Stephenson raised strong objections
to this principle in his paper on " A New Application of Correlation to
Averages" (Brit. J. Educ. Psych., VI, pp. 43-57), urging that the opposite
types produced by the minor factors are not cancelled, but ignored. Here,
as elsewhere (so he insists), " a factor. . . should be clearly distinguished
from a mere average " ([98], p. 357). It is, however, easy to show that the
results obtained by formally eliminating the type-factors are virtually (if not
precisely) the same as those that are obtained by the simpler and commoner
procedure, and that these latter are usually as exact as the data will warrant
(cf. Davies' reply to Stephenson [30]).

As was indicated in both the passages just cited, the e corollary ' mentioned
in the text would seem to apply equally to Thurstone's f centroid method/
since his cardinal equation is the same as my own (loc. tit., eq. xxv) : but when
it comes to determining the factor-measurements, Thurstone himself, like
Spearman, employs the method of least squares. The differential weights,
however, derived by the method of least squares, although no doubt in theory
supplying results of superior accuracy and validity, make little practical
difference to the final averages or sums.

I may add that in the fuller version of my 1935 Memorandum I showed that
a more general form of the corollary could be reached if the variances of the
measurements to be summed are made equal, not to each other and to unity,
but to the communalities of the tests or traits, and if the diagonal elements are
made equal to the communalities instead of to unity. From a practical
standpoint, however, the calculation of type-factor measurements on this
basis would be more laborious and not less, and the gain in accuracy is all but
negligible.

To the mathematical reader an apology is perhaps owing for the long
attempt to prove in words a proposition which may seem obvious to him as
soon as it is expressed in algebraic form. My excuse is, first, that several
statistical investigators (including those who have used the summation for-
mula most frequently) have questioned the proposition ; and secondly, that a
verbal argument and an arithmetical example will make the result plausible to
numerous readers who either cannot follow the algebra or cannot accept the
algebraic premisses in the precise form which is necessary for the simple
mathematical proof.