K. Pearson 
289 
the observations are known ; my fourth deals with the representation of vital 
statistics by Makeham's curve and my fifth with the deduction of the curve of 
errors from partial observations of frequency. 
(o) Illustration I. On the mean variability and distribution of fecundity in 
2000 thoroughbred broodmares. 
By fecundity of the mare is here meant the ratio of the number of yearling 
foals she has actually produced to the number of her opportunities. The base- 
elements were taken + on either side of 0, jL, j?^, f-^, ... ||, 1. Thus fecundity 
from 0 to 1 was divided into IG grades, respectively denoted by a, b, c, d, ... I, m, 
n, p, q. The data were extracted from the stud-books, every mare having had at 
least 8 or more coverings. 
I propose in this first illustration to go through the whole of the work of 
dealing with the frequency distribution as it may be unfamiliar to many of my 
readers, and yet it is really very simple. I shall first suppose the curve to have 
high contact at the terminals of the range, and work out the j^'s and deduce the yu's 
by Sheppard's corrections : see p. 287. This, however, is not in this case legitimate 
from mere inspection of the curve, and therefore we ought to start by using (xiv). 
Working out the /a's in the latter way also we can compare the results actually 
obtained. It will be sufficient to go as far as /u.4. 
Grade 
Frequency z 
X 
zx 
zx'^ 
zx-^ 
zx* 
a 
0 
-9 
0 
-1- 
0 
0 
+ 
0 
b 
2 
-8 
16 
+ 
128 
- 1024 
+ 
8192 
c 
7-5 
-7 
52-5 
-t- 
367-5 
- 2572-5 
+ 
18007-5 
d 
11-5 
-6 
69 
-1- 
414 
- 2484 
+ 
14904 
e 
21-5 
— 5 
107-5 
-f- 
537-5 
- 2687-5 
+ 
13437-5 
f 
55 
-4 
220 
-1- 
880 
- 3520 
14080 
9 
1045 
-3 
313-5 
4- 
940-5 
- 2821-5 
+ 
8464-5 
h 
182 
-2 
364 
-t- 
728 
- 14.56 
+ 
2912 
i 
271-5 
-1 
271-5 
+ 
271-5 
- 271-5 
+ 
271-5 
3 
315 
0 
1414 
- 16837 
k 
337 
+ 1 
+ 
337 
+ 
337 
+ 337 
+ 
337 
I 
293-5 
+ 2 
+ 
587 
-f- 
1174 
+ 2348 
4696 
m 
204 
+ 3 
+ 
612 
-1- 
1836 
+ 5508 
-t- 
16524 
n 
127 
+ 4 
+ 
508 
+ 
2032 
+ 8128 
+ 
32512 
P 
49 
+ 5 
+ 
245 
+ 
1225 
+ 6125 
+ 
30625 
1 
19 
+ 6 
+ 
114 
684 
-1- 4104 
+ 
24624 
2000 
-1-2403 
+ 11555 
+ 26550 
+ 189587 
1414 
- 16837 
-1- 989 
+ 9713 
"1' = 
= -4945 
"2' = 
:5-7775 
1^3' = 4-8565 
1/4' = 
94-7935 
Using (xiv) with the a's zero to find the moments we have 
/Lti'=-4945, = 5-694,167, /it/ = 4-732,875, /i/ = 91-933,917, 
Biometrika i 29 
