72 
On Errors of Random Sampling 
From these figures it is plain that even comparatively large values of m do not 
admit of the binomial being closely approximated to by a " normal " curve. 
Since, however, direct evaluation of the terms of the binomial is very tedious 
when m is at all large, we need a curve which bears the same relation to the skew 
binomial that the normal curve does to the symmetrical binomial. Such a function 
was provided years ago by Pearson*, viz. his Skew Curve of Type III, 
4 
where s = — : 1, 
\mpq mj 
2 
ry = - — = (taking the unit of measurement c = 1 as before), 
r(* + i) 
To use this curve with the rapidity possible in the case of a "normal" curve, 
w r e need tables not at present published. 
In any particular case, however, the curve may be calculated and the area 
between assigned ordinates approximated to with little labour. 
To sum up, we have the following rules for practical work when p is known or 
assumed. 
(1) When m is small, say less than 25, the binomial expansion should be 
directly evaluated. 
(2) When m is moderately large and p or q not small, say not less than "1, 
the ordinary method based on the " normal " curve can be trusted. 
(3) If m is moderately large and p or q less than '1, a skew curve of Type III 
should be fitted from the momental constants of the binomial and the areas 
between assigned ordinates estimated with the help of quadrature formulae. 
III. 
If in n trials an even t happened p times and failed q times, what is the probable 
distribution of successes and failures in m subsequent trials and what are the 
respective chances of 0, 1, 2, ... m successes in m trials, it being assumed that the 
occurrences are independent and that the " universe " of events is indefinitely greater 
than n + m ? 
This problem is of fundamental importance. We note at once that when the 
last condition is imperfectly fulfilled an important special case may arise, for we 
then have : — ■ 
n ~\~ m 
finite where N is the number of events comprising the " universe " 
* For a recent precis of the relevant facts, see Pearson, K. " On the Curves which are most suitable 
for describing the Frequency of Eandom Samples of a Population," Biometrika, 1906, Vol. v. p. 172. 
