1922.] P. C. Mahai,anobis : Analysis of Stature. 11 



Once the typical curve is built up we can proceed to compa- 

 rison with other general populations as represented by their own 

 typical formulae. Goring observes " no valid comparison between 

 two series of statistics is possible until the constants of each series 

 have been determined." J 



But even then, no conclusion can be safely asserted from the 

 comparison, until a certain condition has been fulfilled. u Before 

 drawing conclusions from the comparison of statistics, we must be 

 certain that we are dealing with strictly random samples of the 

 same homogeneous material^ (italics mine). 



This introduces the second part of our work. For valid 

 comparison we must investigate the homogeneity (or otherwise) of 

 our material. I have discussed the statistical tests of homo- 

 geneity in section III, and the application of these tests in 

 section V. 



We then pass on to the question of comparison with other 

 data. In section VI, I have considered the nature 01 the material 

 for comparison and in the next section (section VII) I have in- 

 vestigated the question of comparative homogeneity in great 

 detail. 



In section VIII, I have added a preliminary note on the 

 variation of stature with age. I shall discuss the question of age 

 correlation and growth in a later paper. 



1 Cf. Goring, p. 33. " In order that complex groups such as two series of 

 measurements, may be compared, these have to be reduced to a simple form, to 

 the genius, as it were, of the series, i.e. certain values, called constants (the 

 mean, mode, standard deviation, etc.), have to be extracted; and the groups 

 compared through the medium of their constants. These values, however, are 

 only themselves comparable in certain conditions. First, we must know that the 

 statistics they represent are not chaotic in their distribution that the sequence of 

 their frequencies have been determined by law. And, secondly, we must know 

 the range of error to be discounted before any actual differences between the 

 constants compared may be regarded as significant. Before we can assert that 

 one series of measurements inherently differs from another, we must predict and 

 allow for a certain amount of difference or arithmetical inexactness, which, 

 according to the law of probability, is bound to appear in limited samples of the 

 same homogeneous material. This predicted amount of insignificant difference 

 is called, as we have already said, the probable error of the constants under 

 consideration." 



" Briefly resumed the matter stands thus: we must compare, .... not this 

 or that particular measurement, but the whole series of measurements obtained 

 from a random sample of (one population) with a similar whole series obtained 

 from a random sample of (another) population In order to make this compari- 

 son two things will be necessary: we must extiact from each series its statistical 

 constants, the mean, the standard deviation, etc.," of the series : and by the 

 theory of probability, we must determine for each constant obtained, its probable 

 error. These constants, with their probable errors, will be the representatives 

 of the series, which, through their medium, become comparable with each other. 

 If the differences between the results compared are not greater than the probable 

 errors of these results, such differences may be regarded as insignificant : if the 

 difference is not greater than twice the probable error, it may be regarded as 

 probably insignificant ; and if it is not greater than three times the probable error, 

 it may be regarded as possibly insignificant. On the other hand, if any differ- 

 ence found is greater than three times the probable error, it is reasonable to 

 assume that the difference is due to some definite influence over and above those 

 causes which are inherent in the sampling process." 



