32 Records of the Indian Museum. [Voi,. XXIII, 



From the statistical standpoint our first necessity is suitable 

 graduation of the given sample. This is necessary in order to 

 draw legitimate inferences about the general population from a 

 study of the given sample. 1 Our first condition is : — 



I. We should be able to graduate the given sample by a smooth 

 curve. That is } the given frequency distribution must be homotypic 2 

 in character . 3 



The goodness of fit can be tested by the Pearsonian Contin- 

 gency Coefficient.* 



Possibility of graduation by a smooth curve is thus a necessary 

 condition of statistical homogeneity. 



This is not however sufficient. All heterotypic curves are 

 excluded, but a homotypic frequency curve need not necessarily be 

 homogeneous. For example, it may well happen that a mixture 

 of two different homogeneous samples is amenable to graduation 

 by a homotypic curve. But even then if the given curve can be 

 split up into simpler components we get direct evidence of hetero- 

 geneity. 



II. Thus our second condition is that the sampled frequency 

 curve should not be capable of being analysed 6 into simpler real* 

 components. 



Pearson 7 has furnished us with a technical method for dissec- 

 tion into two components. But failure in dissection may also 

 imply that the curve is multi- complex in character, i.e. that it is 

 built up of more than two simple components. This second 

 condition (impossibility of analysis) again though necessary, is yet 

 not sufficient. 



The concept of functional equivalency provides us with 

 another test. If we consider any sub-sample ' it should be gener- 

 ally equivalent to another sub-«ample, that is, it should not differ 

 significantly from other sub-samples. Thus we get : — 



III. The frequency constants of different sub-samples should 

 agree within the limits of their own probable error? 



1 We assume throughout that all samples are random samples, that is, we 

 definitely exclude heterogeneity due to mere " bias " in sampling 1 . 



Homotypic curves will ordinarily include the Gaussian and the different 

 Pearsonian skew curves. Other smooth curves (Edgeworth, Charlier, Thiele 

 Kapteyn etc) may also be included. 



1 'he possibility of suitable graduation of the present material has been 

 ed in Section IV, pp. 35-40. 

 ♦ rhe original memoir was given in Phil. Mag. 1900, pp. 1=57 — 175. For a 

 dis< ussion of us use in testing goodness of lit see L. Lsserlis: " On the Represen- 

 tation of Statistical Data." Biometrika, Vol. XI (1917), pp. 418 — 425. 



The possibilit) ol dissection of the present material has been investigated in 

 n VI. & 



Negative and imaginary solutions are sometimes obtained; until we can 

 nsistent interpretation of these, it is perhaps safer to ignore such purely 

 ■ lumati< al solutions. 



Memoir on Dissection of Curves, already cited Phil. Trans. Roy. Soc, 

 \. ( 1 804) . 



i.tlv speaking, the agreement of subsampies is only an indirect test ot 

 homogeneity. \\ hat it ... tually docs serve to show is the representative character 



<»l tl n sample. 



