VIL. —AVERAGES. 109 
or, to express it more briefly by using the symbol 3 to denote 
“the sum of all quantities like,” 
1 
M= 7 2(&) 2 (1) 
The word mean or average alone, without qualification, is very 
generally used to denote this particular form of average: that 
is to say, when anyone speaks of ‘the mean” or “the average” 
of a series of observations, it may, as a rule, be assumed that the 
arithmetic mean is meant. It is evident that the arithmetic 
mean fulfils the conditions laid down in (a) and (8) of § 4, for it 
is rigidly defined and based on all the observations made. 
Further, it fulfils condition (c), for its general nature is readily 
comprehensible. If the wages-bill for & workmen is £P, the 
arithmetic mean wage, P/N pounds, is the amount that each 
would receive if the whole sum available were divided equally 
between them : conversely, if we are told that the mean wage 
is £M, we know this means that the wages-bill is VV. pounds. 
Similarly, if & families possess a total of C children, the mean 
number of children per family is C'/N—the number that each 
family would possess if the children were shared uniformly. 
Conversely, if the mean number of children per family is 27, the 
total number of children in A families is #.4#. The arithmetic 
mean expresses, in fact, a simple relation between the whole 
and its parts. 
7. As regards simplicity of calculation, the mean takes a high 
position. In the cases just cited, it will be noted that the mean 
is actually determined without even the necessity of determining 
or noting all the individual values of the variable: to get the 
mean wage we need not know the wages of every hand, but only 
the wages-bill ; to get the mean number of children per family 
we need not know the number in each family, but only the total. 
If this total is not given, but we have to deal with a moderate 
number of observations—so few (say 30 or 40) that it is hardly 
worth while compiling the frequency-distribution—the arithmetic 
mean is calculated directly as suggested by the definition, z.e. 
all the values observed are added together and the total divided 
by the number of observations. But if the number of observations 
be large, this direct process becomes a little lengthy. It may 
be shortened considerably by forming the frequency-table and 
treating all the values in each class as if they were identical with 
the mid-value of the class-interval, a process which in general 
gives an approximation that is quite sufficiently exact for prac- 
tical purposes if the class-interval has been taken moderately 
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