5 THEORY OF STATISTICS. 
measurement. Instead of assigning to any observation its true 
value X, we assign to it the value X, corresponding to the centre 
of the class-interval, thereby making an error 3, where 
xX; = X Zs 0. 
To deduce from this equation a formula showing the nature of 
the influence of grouping on the standard-deviation we must know 
the correlation between the error 6 and X or X;. If the original 
distribution were a histogram, X; and & would be uncorrelated, 
the mean value of 8 being zero for every value of Xj : further, the 
square of the standard-deviatioh of 6 would be ¢2/12, where c is 
the class-interval (Chap. VIII § 12, eqn. (10)). Hence, if 0 be the 
standard-deviation of the grouped values X; and o the standard- 
deviation of the true values JX, 
Bam 0 
go =g< 15s 
But the true frequency distribution is rarely or never a 
histogram, and trial on any frequency distribution approximating 
to the symmetrical or slightly asymmetrical forms of fig. 5, p. 89, 
or fig. 9 (a), p. 92, shows that grouping tends to increase rather 
than reduce the standard-deviation. If we assume, as in § 3, that 
the correlation between 8 and X, instead of 6 and X, is appreciably 
zero and that the standard-deviation of § may be taken as ¢?/12, 
as before (the values of 8 being to a first approximation uniformly 
distributed over the class-interval when all the intervals are 
considered together), then we have 
Sg 
g 01 19 . (4) 
This is a formula of correction for grouping (Sheppard’s correc- 
tion, refs. 1 to 4) that is very frequently used, and that trial 
(ref. 1) shows to give very good results for a curve approximating 
closely to the form of fig. 5, p. 89. The strict proof of the 
formula lies outside the scope of an elementary work : it is based 
on two assumptions: (1) that the distribution of frequency is 
continuous, (2) that the frequency tapers off gradually to zero 
in both directions. The formula would not give accurate results 
in the case of such a distribution as that of fig. 9 (8), p. 92, or 
fig. 14, p. 97, neither is it applicable at all to the more divergent 
forms such as those of figs. 15, et seq. 
5. If certain observations be repeated so that we have in every 
case two measures 2; and x, of the same deviation #, it is possible 
to obtain the true standard-deviation o, if the further assumption 
is legitimate that the errors 6, and §, are uncorrelated with each 
other. On this assumption 
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