Preface v 



imagine that a sample of 10 to 15 with a fine mathematical theory will in some 

 manner be equivalent to a sample of several hundreds with a simple statement of 

 the standard errors of its constants. The student should also try to grasp what is the 

 nature of the assumptions, which have been marie as to the character of the unknown 

 parent-population from which the small sample is extracted, before he places great 

 confidence in what the most effective mathematical theory can deduce from sparse 

 material. Experimental work of a very useful kind has been started to discover 

 how far the present mathematical theory of small samples can be extended to 

 other than a single type of parent-population ; but it is too early yet to be dogmatic 

 as to the limits within which the application of such theory is valid. In particular 

 I hold that the so-called "z" test as usually applied to small samples, especially 

 when it is used to measure the probability or improbability of identity in the 

 constants of small correlated samples, really requires further consideration. I doubt 

 whether the user is always clear as to what hypotheses he is really testing. In 

 this respect I would further make a remark which applies not only to so-called 

 "presumptive values," but also to the variance or other constants of the distribu- 

 tion of samples from an unknown parent-population. Let GI, Cj,... c p be the known 

 constants in the individual sample to be tested, Cj, C t ,... C p those of the parent- 

 population, and F Cf be the other variance, or any other constant which helps to 

 describe the distribution of any c s in repeated samples, then mathematically we 

 often reach a formula 



where any / is a function of the (7's, and the series may be finite, or infinite and 

 converging. If we know the C population as in experimental work in the laboratory 

 no doubt useful ideas can be drawn from such formulae, but if the parent-population 

 be unknown, as it so frequently is, then all we can do is to use in some form or 

 another the c's of the single sample we have obtained, in order to measure F Cg . 

 But any c a will differ from the corresponding C g by terms of the order l/V'w. Hence 

 it is not legitimate to retain terms of the order I/n*, etc., when we are neglecting 

 terms of the order 1/Vn. In other words, our formula is not more legitimate when 

 we are dealing with an unknown parent-population than if we had merely taken 

 F Cf =fi /n, the value to which it tends as the size of the sample becomes large. I 

 am fully aware that this is a crude and rather vague way of stating the difficulty. 

 It can,, however, be illustrated by a very simple example. The astronomers, 

 using only too frequently very small samples, take as the standard error of the 

 standard-deviation ("square root mean square error" of their terminology), 



