= THEORY OF STATISTICS. 
The figures dealt with in this illustration are estimates of the 
weekly earnings of the agricultural labourers, <.e. they include 
allowances for gifts in kind, such as coal, potatoes, cider, etc. The 
estimated weekly money wages are, however, also given in the 
same Report, and we are thus enabled to make an interesting 
comparison of the dispersions of the two. It might be expected 
that earnings would vary less than wages, as his earnings and not 
the mere money wages he receives are the important matter to 
the labourer, and as a fact we find 
Standard deviation of weekly earnings . 20-5d. 
T > - wages . 260d. 
The arithmetic mean wage is 13s. 5d. 
6. If we have to deal with a grouped frequency-distribution, 
the same artifices and approximations are used as in the calculation 
of the mean (Chap. VIL §§ 8, 9, 10). The mid-value of one of 
the class-intervals is chosen as the arbitrary origin 4 from which 
to measure the deviations § the class-interval is treated as a 
unit throughout the arithmetic, and all the observations within 
any one class-interval are treated as if they were identical with 
the mid-value of the interval. If, as before, we denote the 
frequency in any one interval by f, these f observations con- 
tribute f¢2 to the sum of the squares of deviations and we 
have— 
1 
Boris 2 
$2 = 7 ( 72 
The standard deviation is then calculated from equation (4). 
7. The whole of the work proceeds naturally as an extension of 
that necessary for calculating the mean, and we accordingly use 
the same illustrations as in the last chapter. Thus in Example 
ii. below, cols. 1, 2, 3, and 4 are the same as those we have already 
given in Example i. of Chap. VIL for the calculation of the mean, 
Column 5 gives the figures necessary for calculating the standard 
deviation, and is derived directly from col. 4 by multiplying the 
figures of that column again by & Thus 90 x 5= 450, 192 x 4= 
768, and so on. The work is therefore done very rapidly. The 
remaining steps of the arithmetic are given below the table ; the 
student must be careful to remember the final conversion, if 
necessary, from the class-interval as unit to the natural unit 
of measurement. In this case the value found is 2:48 class 
intervals, and the class-interval being half a unit, that is 1-24 
per cent. 
138