458 APPENDIX I

(V) The numerical totals of the columns will help to decide the final
order: tests having the largest totals and sub-totals should be placed
towards the beginning or towards the end of the table, and generally
the absolute totals should first descend and then rise again :
(2 X 1-52, 2 X 1-14, . . ., 2 X o-io, 2 X 0-57, 2 X 171).

3. Now insert estimates for the self-correlations l in the leading
diagonal: (-64, -36, etc.). Usually the figures at either end of the
diagonal (-64, *8i) should be larger than the largest correlation in
the column, and those near the centre (-01, '09) nearly as small as,
if not smaller than, the smallest correlations. Being squares, or
the sums of squares, all the self-correlations will be positive. These
trial estimates will usually have to be checked and revised as
explained in the previous example.

4. Prefix appropriate weights, + * or — I, to each of the rows
in turn, + I in front of the upper rows (containing the north-east
block of negative correlations), — I in front of the lower rows
(containing the same block in the south-west corner).2 If there is
occasionally some doubt as to the appropriate sign (e.g. in the
middle rows), the criterion is simple : the sign of the weight for the
xih row should be the same as the sign of its weighted sum (i.e. of
the weighted sum for the xth column).

5. Multiply each row by the sign thus prefixed : i.e. reverse all
the signs in those rows which have negative weights. The effect
is shown in Table III by bracketing the original signs (nearest the
figure) and prefixing the new sign : (— '24 becomes + ^24 ; -f- -09,
although a self-correlation, becomes — ^09 etc.).

6. Using the new signs, add each column3 to find its algebraic
total: (+ 3-04, + 2-28, . . ., — 3-42). The sign of the total for
each column should agree with the sign employed to weight the
corresponding row : if not, the sign of the weight must be changed,
and the multiplication and addition (steps 5 and 6) repeated. The
method thus provides its own check.

7. Find the numerical or absolute total of these column-totals :
i.e. treat them all as positive, and add them up ; (3*04 + 2-28 +
. . . + 3-42 = H'44)- _

8. Find the square root of this grand total: (^14*44 = 3-80).

9. Divide the totals of the columns, with their original signs,
positive or negative as the case may be, by this square root:

1 With tables of residuals there will already be a figure in each diagonal cell, possibly
a negative figure : after using them for the check described in step i, they should,
particularly if merely based on previous estimates, be discarded for better estimates,

* It is algebraically indifferent whether we assign the negative values to the lower
or to the upper rows, so long as the signs are opposite: hence it may sometimes be more
appropriate to determine the choice by the psychological nature of the bipolar factor*

* Here (but not always) it will be sufficient merely to add the positive and negative