VIIL.—MEASURES OF DISPERSION, ETC. 145 
The new mean deviation is accordingly less than the old so long as 
m>31N. 
That is to say, if NV be even, the mean deviation is constant for 
all origins within the range between the &/2th and the (4/2 + 1)th 
observations, and this value is the least: if & be odd, the mean 
deviation is lowest when the origin coincides with the (& + 1)/2th 
observation. The mean deviation is therefore a minimum when 
deviations are measured from the median or, if the latter be 
indeterminate, from an origin within the range in which it lies. 
15. The calculation of the mean deviation either from the mean 
or from the median for a series of ungrouped observations is very 
simple. Take the figures of Example i. (p. 137) as an illustration. 
We have already found the mean (15s. 11d. to the nearest penny), 
and the deviations from the mean are written down in column 3. 
Adding up this column without respect to the sign of the devi- 
ations we find a total of 590. The mean deviation from the mean 
is therefore 590/38=15'53d. The mean deviation from the 
median is calculated in precisely the same way, but the median 
replaces the mean as the origin from which deviations are measured. 
The median is 15s. 6d. The deviations in pence run 63, 57, 50, 
36, and so on; their sum is 570; and, accordingly, the mean 
deviation from the median is 15d. exactly. 
16. In the case of a grouped frequency-distribution, the sum 
of deviations should be calculated first from the centre of the 
class-interval in which the mean (or median) lies, and then 
reduced to the mean as origin. Thus in the case of Example ii. 
the mean is 3:29 per cent. and lies in the class-interval centring 
round 3-5 per cent. We have already found that the sum of 
deviations in defect of 3-5 per cent. is 776, and of deviations in 
excess 509: total (without regard to sign) 1285,—the unit of 
measurement being, of course, as it is necessary to remember, the 
class-interval. If the number of observations below the mean is 
N, and above the mean N,, and M — 4 =d, as before, we have to 
add X,.d to the sum found and subtract N,d. In the present 
case N,=327 and N,=305, while d= — 0-42 class-intervals, 
therefore 
dN, -Ny)= -042x22=-92, 
and the sum of deviations from the mean is 1285 — 9:2 = 12758. 
Hence the mean deviation from the mean is 1275-8/632 =2019 
class-intervals, or 1-01 per cent. 
17. The mean deviation from the median should be found in 
precisely similar fashion, but the mid-value of the interval in 
which the median (instead of the mean) lies should, for con- 
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