Probability ellipses: *the origins (Galton 1886)

From

ANTHROPOLOGICAL MISCELLANEA

REGRESSION towards MEDIOCRITY in HEREDITARY STATURE

By FRANCIS GALTON, F.R.S. &C. 

[WITH PLATES IX AND X.] 

[Journal of the Anthropological Institute 15 (1886), 246–263.]

https://www.york.ac.uk/depts/maths/histstat/galton_reg.pdf

It is deduced from a large sheet on which I entered every

child's height, opposite to its mid-parental height, and in every case

each was entered to the nearest tenth of an inch. Then I counted

the number of entries in each square inch, and copied them out

as they appear in the table. The meaning of the table is best

understood by examples. Thus, out of a total of 928 children who

were born to the 205 mid- parents on my list, there were 18 of the

height of 69·2 inches (counting to the nearest inch), who were

born to mid-parents of the height of 70·5 inches (also counting to

the nearest inch). So again there were 25 children of 70·2 inches

born to mid-parents of 69·5 inches. I found it hard at first to

catch the full significance of the entries in the table, which had

curious relations that were very interesting to investigate. They

came out distinctly when I" smoothed" the entries by writing at

each intersection of a horizontal column with a vertical one, the

sum of the entries in the four adjacent squares, and using these to

work upon. I then noticed (see Plate X) that lines drawn through

entries of the same value formed a series of concentric and similar 

ellipses. Their common centre lay at the intersection of the

vertical and horizontal lines, that corresponded to 68!- inches.

Their axes were similarly inclined. The points where each

ellipse in succession was touched by a horizontal tangent, lay in a

straight line inclined to the vertical in the ratio of 2/3; those where

they were touched by a vertical tangent lay in a straight line

inclined to the horizontal in the ration of 1/3. These ratios confirm

the values of average regression already obtained by a different

method, of 1/3 from mid-parent to offspring, and of  2/3 from offspring 

to mid-parent, because it will be obvious on studying Plate X that

the point where each horizontal line in succession is touched by

an ellipse, the greatest value in that line must occur at the point

of contact. The same is true in respect to the vertical lines.

These and other relations were evidently a subject for mathe-

matical analysis and verification. They were all clearly dependent

on three elementary data, supposing the law of frequency of error 

to be applicable throughout; these data being ( 1) the measure of

racial variability, whence that of the mid-parentages may be inferred

as has already been explained, (2) that of co-family variability

(counting the offspring of like mid-parentages as members of the

same co-family), and (3) the average ratio of regression. I noted

these values, and phrased the problem in abstract terms such as a

competent mathematician could deal with, disentangled from all

reference to heredity, and in that shape submitted it to Mr. J. 

Hamilton Dickson, of St. Peter's College, Cambridge. I asked

him kindly to investigate for me the surface of frequency of error 

that would result from these three data, and the various particulars

of its sections, one of which would form the ellipses to which I

have alluded.