HIERARCHICAL CRITERIA 357

Illustrative Results.—The value of these various criteria will be
more evident if I give one or two brief instances of their application.

(a) First-moment Criteria.—First of all, it is instructive to note
that, when such criteria are applied to data obtained in psychology
either by correlating tests or by correlating persons, the results
obtained from different tables evince (with a few explicable excep-
tions) a somewhat remarkable uniformity. Let us begin with the

simplest criterion of all, namely, $\£ = -. This is a useful and

n

common device for estimating the importance of any given factor.
Its application at once leads to a suggestive result. Provided
neither the battery of tests nor the sample of persons is the outcome
of specialized selection, and provided n is not so large as to disturb
the method of comparison, it appears that the relative contributions
of the factors to the total variance diminish in a fairly regular
fashion. The general proportions are indicated in the first column
of Table VII.1 When n is exceptionally large and the tests are more
heterogeneous than usual, the proportions incline towards the lower
of the values shown ; and, when the probable error is high, the steady
diminution of the variances begins, as we have already seen, to get
arrested as they approach figures of the order of the probable error.
This conclusion would seem to be of some theoretical importance.
Writers with such opposite views as Spearman and Thomson have
emphasized the apparent fact that nearly all the correlation tables
obtained by psychologists show a marked hierarchical character :
" the tendency to zero-tetrad differences," we are told, " is very
strong in mental measurements " ; " in a complete family of correla-
tion coefficients the rank of the correlation matrix tends towards
unity and a random sample from this family will show the same

1 These proportions are based mainly on an early review of tables of
correlations between tests or traits carried out at my suggestion by Miss
Jefferson. A similar review for tables of correlations between persons has
been more recently undertaken by Miss Davies ; the proportions are much
the same, viz. -48, -n, -05, -03, . . . ([130], p. 412). Similar proportions
are obtained with physical measurements, when reasonably comparable.

Taking these proportions and the above criteria, we can reach a rough
notion of the size of the sample that would be required to establish 2, 3, or
more significant factors by analysing a table of correlations between tests.
To obtain evidence that is reasonably conclusive we should require (in round
numbers)T-for I factor at least 20 persons, for 2 factors at least 60, for 3
factors at least 250, and for 4 factors about 1,000 ; but for merely suggestive
figures smaller numbers may serve. However, as the fuller criteria clearly
imply, the significance of the factors will depend, not only upon the
number of persons tested, but also upon the number of tests employed.