VARIANCE, COVARIANCE, AND CORRELATION 275

taken by Qm when Qr is zero, that is, the maximum value of Qm.
This gives an alternative method of determining Qt which is often
convenient.

If we proceed to correlate the several tests, we can esti-
mate the extent to which they agree by the average inter-

correlation, r = y-ry - r Sr,y (i 4= i'), i.e. by the mean
k(k i)

of the correlations of each test with every other, excluding
all self-correlations. (This device has more frequently
been used in the past where not tests, but persons, or
persons' estimates of traits, have been correlated.)

If the tests are all in standard measure, it is not necessary to
calculate each intercorrelation separately ; for when the means and
the standard deviations are equal, the average intercorrelation is
identical with what is called in the analysis of variance the ' intra-
class correlation ' (n»*)» that is, the correlation calculated by dis-
regarding the allocation of measurements to definite tests and taking
all possible pairs of measurements for one and the same individual.1
Shortened methods of computation are thus available. We have in

_ J

fact r = rint = — ^ - (an extremely useful formula which we can

K "'• '- I

call the * corrected square-sum ratio ?).2 Where the initial measure-

1 Fisher, loc. cit.9 pp. 198 et se.q. Yule and Kendall, loc. cit.9 pp. 254
ft seq,

* The equation given in the text corresponds with the formula usually
given for calculating the intra-class correlation (cf, Yule [no], p. 255 ;
Fisher [50], pp. 202, 211). This, however, yields a biased estimate, which is
in general somewhat too small ; from the equivalences given by Fisher
([50], p. 213) a more accurate estimate can be obtained by taking into account
the limitations in the degrees of freedom. The relation between the two
may be exhibited as follows :

m biased r-, - Q-IW»-Q' _*g«-g«
(i) bused rtnt _c______^__

(ft unbiased r- - «(* - ')Q«- (»~ l)Q- rm- F,

(u) unbased r1Ht - n(i _ ^ + (n _ ^ _ ^ - 7)B+(*-i)7,

In either case the second expression is the most convenient for computation.

The use of the average intercorrelation as a form of * saturation coefficient *

and of the grand average of all the intercorrelations as an indication of the

* fact or- variance * was an early device (e.g. Mental and Scholastic Tests, 1921,