Ethel M. Elderton and Karl Pearson 
493 
these deathrates up to the sixth and, in a few cases, to the tenth were then 
formed. In our notation m,. is the deathrate in the ?'th year of life, i.e. from r — 1 
to r years of age, and hsVi,. is the sth difference of this deathrate. As we have five 
deathrates for each sex this involves 10 means, 10 standard deviations and 20 corre- 
lation coefhcients, but as we have used six successive differences these numbers 
must be multiplied by seven. The calculation of these differences and of upwards 
of 150 correlation coefficients has meant very strenuous labour. It must, indeed, be 
admitted that the application of the variate difference correlation method is not, 
even with small popidations, a light task, but the change from the high positive 
to low negative and then to high negative values of the correlation is of 
extraordinary interest, and indicates the stages by which the associations are 
freed from the spurious influence of the time-factor. 
(3) All our correlations are given in Table III (p. 497), but it is desirable to 
discuss in detail certain groups of them. We take first the correlations of the 
deathrates in successive years. They are : 
Male. Female. 
^m.vu +-398 ±-080 + -390 + -081 
+ ■859 ± -025 + -864 ± "024 
^,»,7», +-924 ±-014 +-928 + -013 
r,„^„,^^ + -911 + -016 + -917 ±-015 
All these are positive, all are significant and, the first excepted, are very high 
correlations. There is no significant difference between male and female. The 
least important is the relation between deaths in infancy and deaths in the first 
year of childhood. We have in these correlation coefficients the numerical 
expression of what is obvious in Tables I and II, i.e. as the deathrate in any year 
of age falls so does the deathrate of the same group in the following year. It is 
this fact which has led to the erroneous idea that natural selection plays no part 
in man. The fact, however, simply expresses the continuous change of environ- 
ment which has been in progress since 1860. During the half-century improved 
economic conditions, bettered sanitation, and developed medical care have lowered 
the deathrate at each age*. It is therefore impossible to deduce any argument 
as to natural selection in man from these correlations until we have removed this 
continuous influence of the time-factor. This is achieved by the variate difference 
correlation method. In every case a preliminary examination of Tables I and II 
shows that the correlation of the first differences of the deathrates of successive 
years is negative, and as we take higher and higher differences the intensity of this 
negative correlation increases, until with the sixth differences it reaches to the 
* As we have already remarked the infantile deathrate showed little of this improvement till 1905. 
It was about this same year that the absolute number of births iu England and Wales began to decline, 
so that while the population has increased by something like 3^ millions, that population produces 
about 76,000 fewer babies annually. 
63—2 
