VIL.—AVERAGES. 113 
It is evident that an absolute check on the arithmetic of any 
such calculation may be effected by taking a different arbitrary 
origin for the deviations: all the figures of col. (4) will be changed, 
but the value ultimately obtained for the mean must be the 
same. The student should note that a classification by unequal 
intervals is, at best, a hindrance to this simple form of calculation, 
and the use of an indefinite interval for the extremity of the 
distribution renders the exact calculation of the mean impossible 
(¢f. Chap. VI. § 10). 
11. We return again below (§ 13) to the question of the 
3 
5 
4. 
“30 
{ 20 
rE 
0 
0 1 WI 8 6,9... "87.8 Tio 
Percentage of the population in Ireceipt of relief. 
Fie. 21. —Showing the Arithmetic Mean JZ, the Median Mi, and the Mode Mo, 
by verticals drawn through the corresponding points on the base, for the 
distribution of pauperism of fig. 10, p. 92. 
errors caused by the assumption that all values within the same 
interval may be treated as approximately the mid-value of the 
interval. It is sufficient to say here that the error is in general 
very small and of uncertain sign for a distribution of the 
symmetrical or only moderately asymmetrical type, provided of 
course the class-interval is not large (Chap. VI. § 5). In the case 
of the “J-shaped” or extremely asymmetrical distribution, how- 
ever, the error is evidently of definite sign, for in all the intervals 
the frequency is piled up at the limit lying towards the greatest 
frequency, .e. the lower end of the range in the case of the illustra. 
tions given in Chap. VI,, and is not evenly distributed over the 
A