1 2 The Fundamental Problem of Practical Statistics 
(\ -i- 1 '//I + 1 „ , 
Ur, writing - H = e for brevity, 
Zi 71/ 
h = cn {e' - PQ)K 
b,^i\b + >ic{Q-F) e], 
h,= \[h- nc{Q-P) ej. 
Distance of mode from start of hypergeo metrical 
= c{P(m + l)-i}. 
Further distance from mode to mean 
(6,-^>,)/(» + 2) = c(Q-P) (2 + ,- 
m 
+ 2 
(6) Illustration. I owe an illustrative example of this to Mr E. C. Rhodes, and 
the accompanying diagram to Miss A. Davin. Suppose 20% of individuals in a 
sample of 1000 have been found to possess a given character, what will be the 
chances of such percentage arising in a further sample of 100 individuals? 
This is the type of problem which arises every day in statistical work and 
which I term the fundamental problem of statistics. Only too often the first sample 
is treated as indefinitely great, and the probabilities calculated from the probability 
integral of the normal curve on the hypothesis that - 0-,, = JmPQ . c. In Diagram I 
the results for the normal curve have been bettered by taking 
^(m + l)PQ(^ 
n 
It is still quite inadequate to provide the requisite percentages. These are plotted 
as the small full circles, and they are seen to lie most closely on the skew curve 
reckoned with the above constants. The integral of the curve was then obtained 
by the integraph, and the areas read off to the nearest tenth. The results are given 
in Table I, p. 13. See Diagram I. 
It will be seen that the skew curve gives frequencies never differing more than 
a fraction per cent, from the series, while the deviations of the Gaussian are much 
larger. A still better result would have been obtained had we fitted the series by 
moments rather than by the geometrical relation. This will be illustrated in the 
next example. 
(7) It may be of interest to see what does really happen when we suppose 11 
infinite or the population " known." 
Returning to p. 10 we have 
cr,' = PQ{m+ l)c-, 
