36 Records of the Indian Museum. [Vol. XXIII, 



variation of the frequency distribution determined from itself, then 

 we may also speak of it as a random sample from a general popula- 

 tion whose theoretical distribution differs only by quantities of the 

 order of the probable errors of the constants from the distribution 

 deduced from the observed sample- 

 Thus if a curve is a good fit to a sample, to the same fineness 

 of grouping it may be used to describe other samples from the same 

 population. If a curve serves to any degree, it will serve for all 

 rougher degrees, but it does not follow that it will suffice for 

 still finer groupings. A good fit for a large sample would be a 

 good fit for a smaller sample but not necessarily for a larger 

 one. 1 



I shall test the Goodness of Fit for different groupings. 

 I shall next compare the fit for the same grouping given by 

 the slightly different values of the Standard Deviation calculated 

 with different unit of grouping. This will test how the Goodness 

 of Fit is affected by different units of grouping adopted in calculat- 

 ing the frequency constants. 



Normal Curve. 



I have calculated the theoretical frequencies from the " raw ' ' 

 (i.e. uncorrected by Sheppard's adjustment) values of the S.D. 

 in some cases. For " if the ordinates of a normal curve be 

 calculated from the raw second moment value of the Standard 

 Deviation, these ordinates will more closely represent the actual 

 frequencies than do the ordinates of the true normal curve, which 

 have to be corrected by the factor 



I ,» Xy* — a' 1 



1 + __ h 2 — 



to obtain the actual frequencies.' ' 



If therefore our sole object is to compare observed and cal- 

 culated frequencies for definite series of groups, there are advant- 

 ages in using the " raw " second moment in the equation to 

 the curve. Such a curve has been termed by Sheppard a " spurious 

 curve of frequency "*. 



1 For a discussion of another test of ( ioodness of Fit proposed by Prof. 

 Edgeworth see a Note 1>\ L. Isserliss: "On the Representation of Statisti al 

 I )ata " Biometrika 1 t>i 7 . pp, 41s — 425. 



* Editorial No,te: "On an Elementary Prool of Sheppard's Formulae 

 for correcting Raw Moments and on other Allied Points," Biom. Vol. 3 (1904), 

 jti. 



