204 THEORY OF STATISTICS.
if the two dispersions are approximately the same, the slope of
RR to the vertical is 7.
Plotting the medians of arrays on a diagram with the quartile
deviations as units, and measuring the slope of the line, was the
method of determining the correlation coefficient (‘‘Galton’s
function ”) used by Sir Francis Galton, to whom the introduction
of such a coeflicient is due. (Refs. 2-4 of Chap. IX. p. 188.)
(3) If s, be the standard deviation of errors of estimate like
x — b,.y, we have from Chap. IX. § 11—
8a = g2{L jE v2),
and hence
3,2
r= —_ a
But if the dispersions of arrays do not differ largely, and the
regression is nearly linear, the value of s, may be estimated from
the average of the standard-deviations of a few rows, and r determined—or
 rather estimated—accordingly. Thus in Table III,
Chap. IX., the standard-deviations of the ten columns headed
625-635, 63:5-64-5, etc., are—
2:56 2-26
2-11 2-26
2:55 2-45
2:24 2:33
2:23 - =
2-60 Mean 2-359
The standard-deviation of the stature of all sons is 2:75: hence
approximately
ni
ge “\ 275
=0514.
This is the same as the value found by the product-sum method
to the second decimal place. It would be better to take an
average by counting the square of each standard-deviation
once for each observation in the column (or * weighting”
it with the number of observations in the column), but in the
present case this would only lead to a very slightly different
result, viz. o=2'362, r=0'512.
20. The Correlation Ratio.—The method clearly would not
give an approximation to the correlation coefficient, however, in
the case of such tables as V. and VI. of Chap. IX., in which the
means of successive arrays do not lie closely round straight lines.