114 THEORY OF STATISTICS. 
interval. In distributions of such a type the intervals must be 
made very small indeed to secure an approximately accurate value 
for the mean. The student should test for himself the effect of 
different groupings in two or three different cases, so as to get 
some idea of the degree of inaccuracy to be expected. 
12. If a diagram has been drawn representing the frequency- 
distribution, the position of the mean may conveniently be 
indicated by a vertical through the corresponding point on the 
base. Thus fig. 21 (a reproduction of fig. 10) shows the frequency- 
polygon for our first illustration, and the vertical MJ indicates 
the mean. In a moderately asymmetrical distribution at all of 
this form the mean lies, as in the present example, on the side of 
the greatest frequency towards the longer tail” of the distribu- 
Mo Mid 
Fie. 22.—Mean JM, Median M7, and Mode Mo, of the ideal moderately 
asymmetrical distribution. 
tion: Min fig. 22 shows similarly the position of the mean in 
an ideal distribution. In a symmetrical distribution the mean 
coincides with the centre of symmetry. The student should mark 
the position of the mean in the diagram of every frequency dis- 
tribution that he draws, and so accustom himself to thinking of 
the mean, not as an abstraction, but always in relation to the 
frequency-distribution of the variable concerned. 
13. The following examples give important properties of the 
arithmetic mean, and at the same time illustrate the facility of its 
algebraic treatment :— 
(a) The sum of the deviations from the mean, taken with their 
proper signs, is zero. 
This follows at once from equation (4): for if M and 4 are 
identical, evidently =(7.£) must be zero,