Karl Pearson 
200 
Since X must be positive, these give us 
2-5X = -6851, /i= 1 0701, or 2-5X = 1-0701, /i = -6851. 
In order that the maximum fertility may be outside the range — 2-5 to 2'5 
we must take the first pair of roots. We easily find 
cr„ = 3-6490o-.^. A; = 3-9049 o-,., 
and from equation (xii) /=4'2613. 
Hence the fertility distribution is 
_ 1 fx-3-m9^^y 
Tx. J . 
y = 1e 
3-6490(r, 
The maximum fertility 7 occurs so far from the mean value of the character, 
that there are practically no parentages with a low value of the character and 
a low fertility, i.e. the number of individual parentages with an extremely low 
value of the character and a fertility under 7 is only 1 in the 20,000. We may 
therefore look upon our curve as one which sensibly describes a fertility constantly 
decreasing as the value of the character increases. 
Turning to the distribution of effective parentages we find : 
Wi = -2728(7.^, Si = -9644a^. 
Thus there is rather a greater shift than in Illustration I. of the average effective 
parentage, but the parentages are more variable. The distribution of effective 
parentages is : 
4-26ii\r _ V-;2728.,y 
W = =— e 2 V -96440-^ J . 
V27r X -96440-^ 
The number of effective parentages with low values of the character and fertility 
less than the maximum will be found from determining the area of this w curve 
for values of x greater than 3-9049o-,, - •2728o-,, = 3-6321o-.,,, or by looking 
up in the Tables = 3-6321/-9644 = 8-77 ; this gives 1 in 12,.500, a negligible 
frequency. 
In view of our previous results for Illustration I, it may be of interest to note 
the average fertilities of the top and bottom deciles of our individual parentages, 
they are to be found from : 
Nf ^j2TrJ / 
, , /i-Wi + 1-2816 --2728 
"^'^^^ ^= ="-^9644 ' 
whence by aid of the tables we find 2-28 for the average fertility of the top decile 
of the individual parentages, and 6-30 for the bottom decile. These values may 
be compared with the corresponding 2 and .5-34 of Illustration I. 
Bioiuetrika vii 35 
