- THEORY OF STATISTICS. 
That is to say, if the weights and variables are positively correlated, 
the weighted mean is the greater ; if negatively, theless. In some 
cases r is very small, and then weighting makes little difference, 
but in others the difference is large and important, » having a 
sensible value and o,0,/@ a large value. 
17. The difference between weighted and unweighted means 
of death-rates, birth-rates or other rates on the population in 
different districts is, for instance, nearly always of importance. 
Thus we have the following figures for rates of pauperism 
(Jour. Stat. Soc., vol. lix. (1896), p. 349). 
Percentages of the Population in 
receipt of Relief. 
January 1. 
Arithmetic Mean England and 
of Rates in Wales as a 
different Districts. whole. 
1850 6°51 5-80 
1860 5-20 4°26 
1870 545 4°77 
1881 368 3°12 
1891 3:29 2°69 
In this case the weighted mean is markedly the less, and the 
correlation between the population of a district and its pauperism 
must therefore be negative, the larger (on the whole urban) dis- 
tricts having the lower percentage in receipt of relief. On the 
other hand, for the decade 1881-90 the average birth-rate for 
England and Wales was 32:34 per thousand, the arithmetic 
mean of the rates for the different districts 30-34 only. The 
weighted mean was therefore the greater, the birth-rate being 
higher in the more populous (urban) districts, in which there is 
a greater proportion of young married persons. 
For the year 1891 the average population of a Poor-law district 
was found to be roughly 45,900 and the standard-deviation o, 
56,400 (populations ranging from under 2000 to over half a 
million). The standard-deviation o, of the percentages of the 
population in receipt of relief was 1:24. We have therefore, 
for the correlation between pauperism and population, 
3:29 — 2:69 459 
r= ———— X 
1-24 564 
amr 0 
222 
J.3G.