82 
PROFESSOR K. PEARSON AND DR. A. LEE ON 
variability we must have more than two classes. We have, in fact, in our tables 
preserved Mr. Galton’s eight eye-colour classes and the seventeen classes under which 
the coat-colour of thoroughbred horses is classified in Wetherby’s studbooks. Such a 
classification enables us at any rate approximately to ascertain relative variability, 
and, what is more, to reconstruct approximately the quantitative scale according to 
which the tints must be distributed in order that the frequency should be normal. 
For, in order to attain this result, we have to ascertain from a table of the areas of 
the normal curve the ratio of the length of the abscissa to the standard deviation which 
corresponds to any given increase of frequency. Let us suppose that three classes 
have been made—?q, n. 2 , n 3 , represented by the areas of the normal curve in the 
accompanying diagram so marked. Let p x and p 5 be the distances of the mean from 
the two boundaries of n 2 . Here p x nray be negative, or p 3 infinite, &c. Then if 
h l = Pi/cr, ho = _p 3 /cr, we find at once, if N = total frequency, 
ft, — ft 2 —ft s 
N 
ll'Y “f“ n o — 
N 
e } - x ~dx .... 
.... (i.) 
e~- x \lx .... 
.... (ii.). 
Now the integrals on the right are tabulated, and thus, since the left-hand side is 
a known numerical quantity, it follows thatpj/o- and p 3 /<x, and accordingly the range 
(p 3 — Pi)/cr of the class in terms of the standard deviation, are fully determined. 
Thus, if e be the range on the scale of tint or colour of the group of which the 
observed frequency is n„, we have e = p 3 — p x , and thus e/a = q say, is known. 
For a second series e/o-' = q. Hence cr ja = q'/q, an d accordingly the ratio of the 
variabilities of the two series is determined. 
Again, the ratio Pi/(p^ — P\) enables us to find the position of the mean in terms 
of the range on the scale occupied by the tint corresponding to the frequency n. 2 . 
As a rule we shall take this tint to be that in which the mean actually lies, in which 
case we shall have pj{p% + yq) as determining the ratio in which the mean divides 
the true quantitative range of this particular tint. 
(3.) Let v —- P\ (p 3 P\) = 
i = cry = -/q) 
Ir remains to find the probalile errors of these quantities, 
(iii.), 
(»-.)■ 
