9 THEORY OF STATISTICS. 
that, in a proportion (7, —r,)/r; of all the pairs, the bones do 
not really belong to the same skeleton, and have been virtually 
paired at random. (For a more general form of the theorem cf. 
again ref. 20.) 
14. The Weighted Mean.—The arithmetic mean M of a series 
of values of a variable X was defined as the quotient of the sum 
of those values by their number #, or 
M=3(X)/N. 
If, on the other hand, we multiply each several observed 
value of X by some numerical coefficient or weight W, the 
quotient of the sum of such products by the sum of the weights 
is defined as a weighted mean of X, and may be denoted by 4", 
so that 
M=3(W.X)/3(D). 
The distinction between «weighted ” and ““ unweighted ” means 
is, it should be noted, very.often formal rather than essential, 
for the “weights” may be regarded as actual, estimated, or 
virtual frequencies. The weighted mean then becomes simply 
an arithmetic mean, in which some new quantity is regarded 
as the unit. Thus if we are given the means M;, My, MU, . . . . 
M, of r series of observations, but do not know the number 
of observations in every series, we may form a general average 
by taking the arithmetic mean of all the means, viz. 2(J/)/r, 
treating the series as the unit. But if we know the number 
of observations in every series it will be better to form the 
weighted mean Z(NM)/S(N), weighting each mean in proportion 
to the number of observations in the series on which it is based. 
The second form of average would be quite correctly spoken 
of as a weighted mean of the means of the several series: at 
the same time it is simply the arithmetic mean of all the 
series pooled together, z.e. the arithmetic mean obtained by 
treating the observation and not the series as the unit. 
(Chap. VIL § 13.) 
15. To give an arithmetical illustration, if a commodity is sold 
at different prices in different markets, it will be better to form 
an average price, not by taking the arithmetic mean of the several 
market prices, treating the market as the unit, but by weighting 
each price in proportion to the quantity sold at that price, if 
known, 7.e. treating the unit of quantity as the unit of frequency. 
Thus if wheat has been sold in market 4 at an average price of 
29s. 1d. per quarter, in market B at an average price of 27s. 7d., 
and in market C at an average price of 28s. 4d., we may, if no 
statement is made as to the quantities sold at these prices (as very 
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