Karl Pearson and David Heron 
297 
and standard deviation 2, and let the selected population be n, mean m, and 
standard deviation a. Then the second curve will always pass outside the 
first if 
flit' 
(m-My + 2(Xs-**) log, [ Wa )>0. 
This will happen of course if % = o", Mr Yule's first case. In this case the 
curves cut on one side only of the means in the point 
i / „> loffc N — log,, n „ 
<B = ^ + M ^ + ~ m -M 
In the case of blindness we should have m>M*, and all it would signify 
would be that after this value, x, of badness of sight, the older age would have 
for each grade of bad sight more individuals than at the lesser age ; since the 
two curves both extend to infinity there is no question, as Mr Yule suggests, of 
persons being "much more blind" than at the younger ages. Mr Yule is simply 
confusing in his own mind or in the minds of his readers two senses of " more 
blind," i.e. more blind persons of each grade, and blind men of greater degree 
of blindness than actually can occur at the younger age. If he merely means 
to say that the Gaussian extends to infinity in both directions, that is a very 
old objection on the theoretical side to the curve ; it has little value in practical 
statistics, where there is a reasonable approach to normality. 
If m = M, then we have 
x = M±a\/ 
/2V- \[og e (aN)-log e (nZ)} 
It is necessary therefore for real roots that a > S, if aN be > This is 
Mr Yule's second case, and this would give increasing numbers of persons in 
each grade of good sight beyond the value of x from M for increasing old age. 
We should have thought that this was a very improbable state of affairs, as it 
is almost certain that sight deteriorates with age after childhood at least. If we 
take the 1891 census dataf used by Mr Yule, we have : 
For ages : 
45—55 
55— G5 
Males in general . . . 
1,191,789 
770,124 
Blind males 
1,752 
1,905 
If we suppose in these cases m = M, we find A r /— = 2 - 974 and xja = 2810, and 
since the dichotomy at " blind " is the same for both curves, we must have X = x, 
* We have taken positive axis towards worse sight. 
f Vol. in. pp. v. and lvii. Mr Yule has clubbed together those blind from childhood and the 
numbers, 12 times as great for these years, not blind from childhood. That so many acquire blindness 
indicates what a range of graduated sight there must be unless we suppose blindness to arise in- 
stantaneously. 
Biometrika ix 38 
