WORKING METHODS FOR COMPUTERS 451

be nearly identical with the trial figures originally proposed for
insertion in the diagonal (-80, -65, etc.).

10. If the agreement is not sufficiently close, repeat the calcula-
tions with new trial estimates. Take the squares of the saturation
coefficients, first slightly readjusting them wherever necessary—
e.g. if, like -805, they are larger than the first estimate (*8o), make
them larger still (say -81) ; if, like -645, they are smaller, make them
smaller still (say -64), Add the adjusted squares to the original
column totals (2-34 + -81 = 3-15, etc.) ; and proceed as before.
If a more exact estimate is required, the process of successive
approximation must be continued, until figures of the required
accuracy are obtained; but, after a little experience, one or two
repetitions can be made to suffice.1

11. Calculate the hierarchy of theoretical correlations resulting
from this factor by multiplying each of the saturation coefficients
with the rest, as shown in the original construction of Table I
(ist row : -90 X -80 = -72, -90 X -70 = -63, etc. 2nd row : -80 X
•70 =-56, etc.). '

12. Subtract these theoretical values from the observed correla-
tions, and test the significance of the deviations by the standard
error of the observed coefficient or by ^2, as suggested above
(p. 339). Here, of course, the differences are zero.

B. WEIGHTED SUMMATION

Formula: Saturation coefficient rag =

j i ' ' i "

If the observed table is not an exact hierarchy, the method of
weighted summation will produce a better fit; in fact, for a complete
table (i.e. with known figures for the diagonal) it gives the best
possible fit as judged by the method of least squares. With an
ordinary set of correlations, where a unique determination of values
for the diagonal is rather doubtful, the gain may seem scarcely worth
the additional labour : but with a table of variances and covariances,
weighted summation is much to be preferred. The ideal weights
are the saturation coefficients. When these are unknown, we must
take the figures already obtained by unweighted summation as pro-

1 If figures exact to more than three decimal places are required, the more elaborate
formula which dispense with the intercorrelations may prove to be quicker, e.g. the
'product formula/ calculated by logs, or Spearman's ^well-known 'summation for-
mula/ which, however, takes more time (see Marks of Examiners, pp. 285-6, eqs.
xxvi and xxvii, and comments). If only two or three digits are wanted, the method
in the text is the quickest; and, with tables deviating widely from the hierarchical
order, it yields results that are more exact.