XL.—CORRELATION : MISCELLANEOUS THEOREMS. 221 
often happens in the case of statements as to market prices), take 
the arithmetic mean (28s. 4d.) as the general average. But if we 
know that 23,930 qrs. were sold at 4, only 26 qrs. at B, and 3933 
qrs. at C, it will be better to take the weighted mean 
(29s. 1d. x 23,930) + (27s. 7d. x 26) + (28s. 4d. x 3933) 99 
~ 27889 oT 
to the nearest penny. This is appreciably higher than the 
arithmetic mean price, which is lowered by the undue importance 
attached to the small markets B and C. 
In the case of index-numbers for exhibiting the changes in 
average prices from year to year (¢f. Chap. VIL. § 25), it may 
make a sensible difference whether we take the simple arithmetic 
mean of the index-numbers for different commodities in any one 
year as representing the price-level in that year, or weight the 
index-numbers for the several commodities according to their 
importance from some point of view ; and much has been written 
as to the weights to be chosen. If, for example, our standpoint 
be that of some average consumer, we may take as the weight for 
each commodity the sum which he spends on that commodity in 
an average year, so that the frequency of each commodity is 
taken as the number of shillings or pounds spent thereon instead 
of simply as unity. 
Rates or ratios like the birth-, death-, or marriage-rates of a 
country may be regarded as weighted means. For, treating the 
rate for simplicity as a fraction, and not as a rate per 1000 of the 
population, 
Birth-rate of whole country = el 
_ =(birth-rate in each district x population in that district) 
- S (population of each district) 
i.e. the rate for the whole country is the mean of the rates in the 
different districts, weighting each in proportion to its population. 
We use the weighted and unweighted means of such rates as 
illustrations in §17 below. 
16. It is evident that any weighted mean will in general differ 
from the unweighted mean of the same quantities, and it is 
required to find an expression for this difference. If » be the 
correlation between weights and variables, o,, and o, the standard- 
deviations, and @ the mean weight, we have at once 
3(W.X)=NMw+ro,o,), 
whe... M=M+ ro, 
“noe 15)