VIL.—AVERAGES. 121 
on the choice of the scale of class-intervals. It is no use making 
the class-intervals very small to avoid error on that account, for 
the class-frequencies will then become small and the distribution 
irregular. What we want to arrive at is the mid-value of the 
interval for which the frequency would be a maximum, if the 
intervals could be made indefinitely small and at the same time 
the number of observations be so increased that the class-frequen- 
cies should run smoothly. As the observations cannot, in a 
practical case, be indefinitely increased, it is evident that some 
process of smoothing out the irregularities that occur in the 
actual distribution must be adopted, in order to ascertain the 
approximate value of the mode. But there is only one smoothing 
process that is really satisfactory, in so far as every observation 
can be taken into account in the determination, and that is the 
method of fitting an ideal frequency-curve of given equation to 
the actual figures. The value of the variable corresponding to the 
maximum of the fitted curve is then taken as the mode, in 
accordance with our definition. fo in fig. 21 is the value of the 
mode so determined for the distribution of pauperism, the value 
2:99 being, as it happens, very nearly coincident with the centre 
of the interval in which the greatest frequency lies. The deter- 
mination of the mode by this—the only strictly satisfactory— 
method must, however, be left to the more advanced student. 
20. At the same time there is an approximate relation between 
mean, median, and mode that appears to hold good with surprising 
closeness for moderately asymmetrical distributions, approaching 
the ideal type of fig. 9, and it is one that should be borne in 
mind as giving—roughly, at all events—the relative values of 
these three averages for a great many cases with which the 
student will have to deal. It is expressed by the equation— 
Mode = Mean — 3(Mean — Median). 
That is to say, the median lies one-third of the distance from the 
mean towards the mode (compare figs. 21 and 22). For the dis- 
tribution of pauperism we have, taking the mean to three places of 
decimals,— 
Mean . . 3289 
Median 3-195 
Difference 0-094 
Hence approximate mode = 3:289 — 3 x 0-094 
= 3-007, 
or 3-01 to the second place of decimals, which is sufficient accuracy 
for the final result, though three decimal places must be retained 
for the calculation. The true mode, found by fitting an ideal