132 THEORY OF STATISTICS. 
That is, if M, M,, M, be the respective means, 
M=0 5, (8) 
Evidently the form of this result is again quite general, so that 
if 
X=X +X + v hell ails +X, 
M=M, +t HY, +L... =H or {9 
As a useful illustration of equation (8), consider the case of 
measurements of any kind that are subject (as indeed all 
measures must be) to greater or less errors. The actual measure- 
ment X in any such case is the algebraic sum of the true 
measurement X; and an error X,. The mean of the actual 
measurements M is therefore the sum of the true mean M, and 
the arithmetic mean of the errors M, If, and only if, the 
latter be zero, will the observed mean be identical with the true 
mean. Errors of grouping (§ 11) are a case in point. 
14. The median.—The median may be defined as the middle- 
most or central value of the variable when the values are ranged 
in order of magnitude, or as the value such that greater and 
smaller values occur with equal frequency. In the case of a 
frequency-curve, the median may be defined as that value of the 
variable the vertical through which divides the area of the curve 
into two equal parts, as the vertical through M7 in fig. 22. 
The median, like the mean, fulfils the conditions (4) and (c) 
of § 4, seeing that it is based on all the observations made, and 
that it possesses the simple property of being the central or 
middlemost value, so that its nature is obvious. But the defini- 
tion does not necessarily lead in all cases to a determinate value. 
If there be an odd number of different values of X observed, say 
2n+1, the (n+ 1)th in order of magnitude is the only value 
fulfilling the definition. But if there be an even number, say 
2n different values, any value between the mth and (n+ 1)th 
fulfils the conditions. In such a case it appears to be usual to 
take the mean of the nth and (n+ 1)th values as the median, 
but this is a convention supplementary to the definition. It 
should also be noted that in the case of a discontinuous variable 
the second form of the definition in general breaks down: if we 
range the values in order there is always a middlemost value 
(provided the number of observations be odd), but there is not, as a 
rule, any value such that greater and less values occur with equal 
frequency. Thus in Table IIL, § 3 of Chap. VI, we see that 45 per 
cent. of the poppy capsules had 12 or fewer stigmatic rays, 55 
per cent. had 13 or more ; similarly 61 per cent. had 13 or fewer 
rays, 39 per cent. had 14 or more. There is no number of rays 
"1 
I)