XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 277 
greater than 0-5, greater than is possible for positive errors. The 
assumption is not, however, likely as a rule to lead to a serious 
mistake ; as stated at the commencement of this paragraph, the 
point is of importance only when = is small, for when # is large the 
distribution tends to become sensibly symmetrical even for values 
of p differing considerably from 0-5. (CF. Chap. XV. for the 
properties of the limiting form of distribution.) 
2. In the second place, the student should note that, where we 
were unable to assign any a priori value to p, we have assumed 
that it is sufficiently accurate to replace p in the formula for the 
standard error by the proportion actually observed, say 
Where 7 is large so that the standard error of 2» becomes small 
relatively to the product pg the assumption is justifiable, and no 
serious error is possible. If, however, n be small, the use of the 
observed value = may lead to an under- or over-estimation of the 
standard error which cannot be neglected. To get some rough 
idea of the possible importance of such effects, the approximate 
standard error ¢ may first be calculated as usual from the 
observed proportion 7, and then fresh values recalculated, replac- 
ing 7 by 7+3e. It should be remembered that the maximum 
value of the product pg is given by »=¢=05, and hence these 
values, if within the limits of fluctuations of sampling, will give 
one limiting value for the standard error. The procedure is by 
no means exact, but may serve to give a useful warning. 
Thus in Example iii. of Chap. XIII. the observed proportion of 
tall plants is 29/68, or, say, 43 per cent. The standard error of 
this proportion is 6 per cent., and a true proportion of 50 per 
cent. is therefore well within the limits of fluctuations of sampling. 
The maximum value of the standard error is therefore 
i 
(20x50) = 606 per cent. 
On the other hand, the standard error is unlikely to be lower 
than that based on a proportion of 43 — 18 =25 per cent., 
i 
(BX) =5'25 per cent. 
3. The two difficulties mentioned in § 1 and 2 arise when n, 
the number of cases in the sample, is small. The interpretation 
of the value of the standard error is also more limited in this 
case than when = is large. Suppose a large number of observa- 
tions to be made, by means of samples of # observations each, on 
different masses of material, or in different universes, for each of 
which the true value of p is known. On these data we could