4So APPENDIX I

2. Rearrange the whole table in the order indicated by these
totals, beginning with the largest (2.34) and ending with the smallest
(i'5<D). Since the correlation of each test with itself is assumed to
be unknown, the leading diagonal is left empty.

3. Begin by making a provisional estimate of these unknown self-
correlations. The simplest device is to consider by direct inspection
what figures will best fit the general pattern. Thus, when the
table has been arranged to form a descending hierarchy, we first
insert in the space at the top of the first column a figure (*8o? say)
somewhat larger ! than the largest intercorrelation in that column
(-72) ; and then in the space at the bottom of the last column a
figure (-25, say) somewhat smaller than the smallest intercorrelation
in that column (-30) 2 ; and finally fill in the intervening spaces
down the central diagonal with figures descending by regular or
irregular intervals corresponding with the regularity or irregularity
of the successive intervals between the totals.8 If these first trial
estimates are likely to require correction, it will be better to write
them, not in the body of the table, but below the totals, as has been
done in Table I. Round figures (-80, -65, -50, . . ., -25) will be
sufficient for the first trial.

4. Add each estimated figure to the total of the corresponding
column (2-34 + -80 = 3-14, etc.).

5. Add these augmented or completed totals, to obtain their
grand total (3-14 •+- 2-81 + , . . + 1-75 = 12-25).

6. Find the square root of this grand total (\/l2'2$ = 3'Jo).

7. Divide the augmented totals of each column by this square
root (3-14 —• 3*50 =5 -897, etc.). The quotients will yield a first
approximation to the saturation coefficients required,

8. As a check on the arithmetic, add the estimated saturation
coefficients. Their sum (3-50) should be equal to the square root
already obtained in step 6.

9. As a check on the estimation, square the estimated saturation
coefficients (-8972 = -805, 8o32 = -645, etc.). Their squares should

* The beginner, who does not trust his powers of smoothing the general pattern
at sight, generally asks : how much larger ? If a mechanical rule is desired, he can
take the proportions between the two adjacent coefficients (or between the two
column totals for the same tests) as a guide : e.g.

1 Thurstone, after considering half a dozen different procedures, recommends
always filling in the gap with the highest correlation in the corresponding column
([84], pp. 89, 233). In the present case this would lead to a very erroneous result.
In the first column the self-correlations should evidently be larger than the largest
intercorrelation ; in the last column (assuming there are no group-factors) it should
evidently be, not as large as the largest, but smaller than the smallest.

* Here again, if the beginner requires a mechanical rule, it is the equality or
inequality between the differences of the intervals that should be the guide.