1922.] P. C. Mahalanobis : Analysis of Stature. 33 



This condition ensures that the sub-samples will not differ 

 significantly from the general sample. 1 



The above three tests are purely formal and have no reference 

 to the nature of the material. We can proceed further by taking 

 into consideration our previous experience of similar material. 



Let us take the case of stature as an example. In all known 

 cases stature distribution is either approximately Gaussian or is of 

 Type IV or Type I. Consider the frequenc}^ distribution of some 

 unknown sample. If we find that the curve though homotypic is 

 J or U shaped, we are naturally suspicious about the homogeneity 

 of the material. The curve may be smooth, it may successfully 

 resist dissection, its sub-samples m^ agree quite well, yet in view 

 of our previous experience we would, in the absence of other 

 evidence, hesitate to call it homogeneous. 



IV. Our fourth criterion is that the general nature of the 

 sampled frequency should be the same as that of known homogeneous 

 material. 



This criterion is quite empirical in character and its practical 

 utility depends upon what exact significance we can attach to the 

 concept of "general nature of known frequency constants.' : 

 Though somewhat vague this condition is by no means useless. 



Let us suppose that the given sample is really heterogeneous in character. 

 Consider a " random " subsample of the given sample. Now if this subsample is 

 to be representative in character, it must include the same degree of heterogeneity 

 as is present in the sample itself, that is, in order that k may be a " fair" as well as 

 a " random ' : subsample, it is necessary that it should be sufficiently large. 

 Samples which are large enough to be "fair" will obviously agree among them- 

 selves. Thus the agreement of large fair subsamples cannot reveal the want of 

 homogeneity of the given sample. 



Now consider a subsample which is again " random " but which is not suffici- 

 ently large to include the same degree of heterogeneity as is present in the sample. 

 Not being representative in character, it will not be surprising if these fail to 

 agree. Thus want of agreement on the part of subsamples on account of 

 their smallness of size will not necessarily prove the existence of heterogeneity in 

 the material. The lower limit of agreement of random subsamples may however 

 be locked upon as a measure of homogeneity. 



In any case however, agreement of random subsamples does show that these 

 subsamples are large enough to be representative in character. The given sample, 

 being larger than its own subsamples, will obviously be large enough to be 

 representative in character. Thus the agreement of subsamples is a test of the 

 •representative character of the sample, rather than any evidence of the homo- 

 geneity of the material. 



An example may help. Consider an ordinary black and white chess board. 

 Let us look at this chessboard through a sighting hole. The size of this sighting 

 hole determines the size of the sample. If this size is larger than the size of one 

 of the squares then each sample will show a mixed patch. In this case subsamples 

 would agree. On the other hand, if the size of the sighting hole is only a fraction 

 of the size of a square, then some samples will show white, some black and others 

 mixed patches. The lower limit, up to which samples agree is evidently a 

 measure of the size of the discontinuities. Agreement of subsamples of 100 shows 

 that 200 is large enough to be representative in character in the present case. 



1 This implication serves as the basis of Pearson's discussion of P.E. ot 

 sub-samples for comparison with the general sample. K. Pearson : " Note on 

 the Significant or Non-significant Character of a sub-Sample drawn from a 

 Sample." Bicmetrika Vol. 5 (1906), pp. 181 — 183. 



