352 THE FACTORS OF THE MIND

Hence we can only make a broad estimate of the general region in
which factors begin to be non-significant.

Nevertheless, the ratio to which we have thus been led offers the
simplest way of describing the relative importance of the several
factors. If we take the upper limit, we are virtually assuming that
the test-variances are all equal to unity, so that their sum Zvf = n.
The contribution of the ktb factor to the total variance may then at once

be expressed by the fraction ^ = —. This method was systematic-
ally adopted by Miss Davies in comparing the importance of the
* general factor ' with that of other factors in a long list of researches
on correlating persons x; and is also incidentally employed by Hotel-
ling in his illustrative example ([79], p. 434)-

If we take the lower limit, with Zvf < n, then it will generally
be better to use the c corrected ' formula,
nrf— I
n — i

~~ (n — i) <>! + vr)

This gives a formula analogous to the intracolumnar correlation
and a figure analogous to the average intercolumnar correlation.
As before pj, unlike yf, can take values between o (for a perfectly
random distribution) and I (for a perfect hierarchical distribution).
With certain plausible assumptions, we can develop tests along the
usual lines for the significance of this expression, which, it will be
observed, takes n into account as well as N into account.

(iii) Second Moments. — But we also require to test the significance
of the differences between the several factor-variances themselves.
We may attempt this in two ways. First, we may take the differ-
ence between each pair separately ; and in particular we may
proceed (as Thurstone does) by comparing results from successive
residual tables. Then, for a rough approximation we may regard
these two tables as depending almost entirely on two independent
factors ; and we may consider the significance of a single correlation
which could be analysed into these two dominant factors. In such
a case n = 2 ; and the formula just described reduces to

script, and write, for instance, ^ and 1^12. (To call the former * reduced
eta? as I have previously done, proves a little misleading, since the ' reduction *
of the variance magnifies eta.)
1 Brit. 7. PsycboL, XXIX, p. 413.