496 Farther Evidence of Natural Selection in Man 
equal to mean and this (taking, as we have done, 'backward' diflferences) 
is given (the C's being the usual binomial coefficients) by 
— (A — Z_i + I.C.2, -'Y_2_ - ■ •_■) -'r{X,i^rp\Xn-\ + rG-i -^J!-2 ~ • • •) 
11 • , ' 
Now if we remenibei' that the X'a have chance values uncorrelated with each 
other then we shall have for the squared standard deviation of the mean 8,._,_iX, 
2cr/(l + ,.6V + ,A-^ + . . . + ,m 
^ mean 5,. , i A' 
|2r a,? 
r Ir 
Or, the probable error of the mean (r + l)th difference after the steady values 
have been reached 
At first sight this appears of no value, because ax is unknown, but Dr Anderson 
has given ag ^ in terms of when steady values have been reached*, i.e. 
12(r + l) 
= •67449 A/ 2 
r + 1 r + 1 
From this we deduce the probable error of a mean rth difference to be 
•67449 . / 
V 2r - 1 n 
when we assume steadiness reached. 
The values of the means of the differences with their probable errors on the 
assumption of steadiness are given in Table IV, and the ratio of the means to 
their differences in Table V. 
It will be seen that the positive and negative signs are not scattered quite as 
much at random as we might have hoped and that this is especially the case in 
the infantile mortality differencesf. If we take all the ratios of the means to 
their probable errors except the first difference, we find their average value 1"16 ; 
it should be of course 1"18. Of these ratios 33 are positive and 25 negative. If we 
omit the ratios for the first year of life, we find 24 negative and 20 positive, while 
the mean value = '98 as against 1"18, the theoretical ratio of the mean to its 
probable error. It is obvious that the infantile mortality differences are those 
which are anomalous. Otherwise the mean differences vary fairly satisfactorily 
* Biometrika, Vol. x. p. 272. 
t It may be noted that at the beginning of the period we have the disturbing influence of war and 
at the end of the period wholly changed conditions due to a great limitation of births. The means 
depend on differences of mortality under these conditions. 
