140 THEORY OF STATISTICS. 
Means and Standard Deviations of the Distributions of Pauperism (Percentage 
of the Population in receipt of Poor-law Relief) in the Unions of England 
and Wales since 1850. (From Yule, Jour. Roy. Stat. Soc., vol. lix., 
1896, figures slightly amended.) 
Percentage of the Population 
in receipt of Relief. 
Foor. 
Arithmetic Standard 
Mean. Deviation. 
1850 6°51 2:50 
1860 5:20 2:07 
1870 5°45 2:02 
1881 3:68 1-36 
1891 3°29 1:24 
8. In the table given on p. 141 (Example iii.), the calculation of 
the standard deviation is similarly shown for the distribution of 
the statures of adult males in the British Isles, the work being 
continued from the stage which it reached for the calculation of 
the mean in Example ii. of Chap. VII. The steps of the arith- 
metic hardly call for further explanation, but it may be noted that 
the class-interval being a unit in this case, no conversion of 
the standard deviation from class-intervals to units is required. 
9. The student must remember, as in the case of the calculation 
of the mean, that the treatment of all values within each class- 
interval as if they were identical with the mid-value of the interval 
is an approximation and no more (¢f. Chap. VII. § 11), though, 
for a distribution of the symmetrical or moderately asymmetrical 
type with a class-interval not greater than one-twentieth or so 
of the range, the approximation may be a very close one. But 
while the value of the arithmetic mean may be either increased 
or decreased by grouping, in the case of distributions which are 
not more than slightly asymmetrical, the standard deviation of 
such distributions tends to be increased, and the increase is the 
greater the cruder the grouping. We give an approximate 
correction for this effect later (Chap. XI. § 4). The student is 
recommended to test for himself the effect of grouping in two 
or three cases. 
10. Tt is a useful empirical rule to remember that a range of 
six times the standard deviation usually includes 99 per cent. or 
more of all the observations in the case of distributions of the 
symmetrical or moderately asymmetrical type. Thus in Example 
~ No 
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