274	THE  FACTORS  OF THE MIND
k N	N	k N
let    SS ~£? = £S x? = Q_m ;     2Z fo, — ^)^2 = Q_f;   and
* w   ^y '	*	^y *'
= 0(:    so    that    0, = O_m + Q_r.     Finally,   let

where (#£)/ and (^V)_r denote the ' number of degrees of
freedom ^5 ^1 on which the estimated variances V_m — Q_M/(N)_f
and V_r = Q_rl(Nk)_f are based. Then jp is the critical
ratio. Tables showing the probability that a given value
of F (or of z = \ log, F) may be reached or exceeded with
(N)_f and (NK)_f degrees of freedom are given in the text-
books of Snedecor and Fisher [50].
If we regard all the marks obtained by a single person as forming
an array or a class, then the F-ratio appears as an elaboration of a
ratio that is far more familiar to the student of psychology — the
correlation ratio^2 73. The relation between the two is given by
^u ^be ^noted ^that ^Qt ^is ^the ^value
^1 The * number of degrees of freedom ' is found by subtracting from the
number of measurements made the number of ^c constraints * or independent
linear relations obtaining between them. Here this means in effect that we
compute the ^c mean ' by dividing the corresponding sum, not by the number
of measurements summed, but by the number of independent comparisons
that can be made with a sample of that size. When the sample contains
more than 30, we can average in the more familiar way : e.g. we can take the
residual variance to be Q/Nk simply. Strictly, however, we should remember
that the residuals are deviations about the means of their respective columns.
Hence only N(k — i) are free to vary independently : in each of the N
columns the £th figure can be deduced from the rest. If the means of the
rows are also zero, one of the figures in each row can be deduced from the rest.
Hence the degrees of freedom are still further reduced to (N — i) (k — i).
If the standard deviations of each row are also equalized to each other and to
unity, yet a further limitation is imposed : and so on, with the calculation of
each additional parameter. It is the difference between these two methods
of estimation that gives rise to the difference between the biased and the
unbiased estimate of the intra-class correlation, referred to below.
* It should be noted that, since / (Here symbolizing the ;th test) does not
enter into the formula for calculating 07^, the so-called * correlation ' for
which Tftj is the corresponding * ratio ' does not necessarily indicate a correla-
tion of anything with / as a quantitative variable or rank in a fixed order,
though in the past the correlation ratio has generally been employed to
measure sucK a correlation.