< THEORY OF STATISTICS. 
reduced, of course, from the original drawing, one of the squares 
shown representing a square inch on the original. The actual 
contour lines for the same frequencies are shown by the irregular 
polygons superposed on the ellipses, the points on these polygons 
having been obtained by simple graphical interpolation between 
the frequencies in each row and each column—diagonal interpola- 
tion between the frequencies in a row and the frequencies in a 
column not being used. It will be seen that the fit of the two 
lower contours is, on the whole, fair, especially considering the 
high standard errors. In the case of the central contour, y= 20, 
the fit looks very poor to the eye, but if the ellipse be compared 
carefully with the table, the figures suggest that here again we 
have only to deal with the effects of fluctuations of sampling. 
For father’s stature=66 in., son’s stature= 70 in., there is 
a frequency of 18:75, and an increase in this much less than the 
standard error would bring the actual contour outside the ellipse. 
Again, for father’s stature=68 in., son’s stature="71 in., there 
is a frequency of 19, and an increase of a single unit would give 
a point on the actual contour below the ellipse. Taking the 
results as a whole, the fit must be regarded as quite as good as 
we could expect with such small frequencies. It is perhaps of 
historical interest to note that Sir Francis Galton, working with- 
out a knowledge of the theory of normal correlation, suggested 
that the contour lines of a similar table for the inheritance of 
stature seemed to be closely represented by a series of concentric 
and similar ellipses (ref. 2): the suggestion was confirmed when 
he handed the problem, in abstract terms, to a mathematician, 
Mr J. D. Hamilton Dickson (ref. 4), asking him to investigate 
“the Surface of Frequency of Error that would result from 
these data, and the various shapes and other particulars of its 
sections that were made by horizontal planes” (ref. 3, p. 102). 
12. The normal distribution of frequency for two variables is 
an isotropic distribution, to which all the theorems of Chap. V. 
§§ 11-12 apply. For if we isolate the four compartments of the 
correlation-table common to the rows and columns centring 
round values of the variables w,, xy , xy we have for the ratio 
of the cross-products (frequency of #, #, multiplied by frequency 
of 2, 2), divided by frequency of », 2; multiplied by frequency of 
x, 7p), 
712 ’ ’ 
ya u)(# =) 
Assuming that 2; — 2; has been taken of the same sign as x; — x, 
the exponent is of the same sign as 7, Hence the association for 
398