XIL—PARTIAL CORRELATION. _
three and the second for four variables. The introduction of
more variables does not involve any difference in the form of the
arithmetic, but rapidly increases the amount.
Example i.—The first illustration we shall take will be a
continuation of example i. of Chapter IX. in which the correla-
tion was worked out between (1) the average earnings of agri-
cultural labourers and (2) the percentage of the population in
receipt of Poor-law relief in a group of 38 rural districts. In
Question 2 of the same chapter are given (3) the ratios of the
numbers in receipt of outdoor relief to the numbers relieved in the
workhouse, in the same districts. Required to work out the partial
correlations, regressions, etc., for these three variables.
Using as our notation X, = average earnings, X, = percentage of
population in receiptof relief, X,; = out-relief ratio, the first constants
determined are—
M, = 15-9 shillings oy = 171 shillings ro=- 066
4',= 3°67 per cent. ¢,=1-29 per cent. r= -013
M, = 519 oy =3'09 ros = + 0°60
To obtain the partial correlations, equation (12) is used direct in
its simplest form—
on TT
== (T= (=r)
The work is best done systematically and the results collected
in tabular form, especially if logarithms are used, as many of the
logarithms occur repeatedly. First it will be noted that the
logarithms of (1-72)! occur in all the denominators ; these had,
accordingly, better be worked out at once and tabulated (col. 2 of
the table below). In col. 3 the product term of the numerator of
Z. Lo, 6. 7 8. o
—— Product |N Nl 1 ain rp
log LIFE Font Bima hl log I-72)
log. | Value.
rig=-066 T87580 -0:0780 -0'5820 I76492 1'89938 186554 rye5-073 I-83216
nz=-013 I'99620 -0'3960 +0-2660 142483 I77589 164599 riat0'4d T0567
23=+060 190309 +0°0858 +05142 T71113 187209 I'83904 rop.+0:69 I-85046
each partial coefficient is entered, 7.e. the product of the two other
coefficients on the remaining lines in col. 1 ; subtracting this from
the coefficient on the same line in col. 1 we have the numerator(col.
4) and can enter its logarithm. The logarithm of the denominator
(col. 6) is obtained at once by adding the two logarithms of (1 & #2)t
on the remaining lines of the table, and subtracting the logarithms
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