SECTION II. EFFECT OF GROUPING ON THE 

 FREQUENCY CONSTANTS. 



Frequency constants and probable errors. 



The object of the enquiry contained in this section may be 

 best explained in Karl Pearson's words. 1 



"It is well known that if the distribution of errors follows 

 the normal law, the " best" method of rinding the mean is to 

 add up all the errors and divide by their number, the ' best ' 

 method of rinding the square of the standard deviation is to form 

 the squares of the deviations from the mean and divide by their 



number These "best" methods become far too laborious 



in practice when the deviations run into hundreds or even thou- 

 sands. The deviations are then grouped together, each group con- 

 taining all deviations falling within a certain small range of quan- 

 tity, and the means, standard deviations, and correlations are 

 deduced from these grouped observations. If the means, stand- 

 ard deviations, and correlations be calculated from the grouped 

 frequencies as if these frequencies were actually the frequency of 

 deviations coinciding with the midpoints of the small ranges 

 which serve for the basis of the grouping, we do not obtain the 

 same values as in the cases of the ungrouped observations. It 

 becomes of some importance what corrective terms ought to be 

 applied to make the grouped and ungrouped results accord. This 

 point has been considered by Mr. W. F. Sheppard (who has pro 

 posed certain corrections). Thus corrected the values of the con- 

 stants of the distribution as found from the ungrouped and grouped 

 deviations will nearly, but not of course absolutely, coincide.' ' 



In this section I have calculated both ungrouped and grouped 

 constants with widely differing units of grouping. The constants 

 as corrected by Sheppard 's formulae have also been calculated in 

 each case. By a comparison of the different constants we find 

 that within very wide limits the effect of grouping is negligible. 



The Stature list was classified into groups of 50 mm. The 

 base number is taken to be 1655 mm. and the moment coefficients 

 were calculated as shown below.' 2 



We get the following table for u raw " moments about 1655 : — 



j 



Karl Pearson : "On the Mathematical Theory of Errors of Judgment and 

 on the Personal Equation," Phil. Trans. Roy. Soc, Vol. 198A, 1902, pp. 249, 



Kor details, see K. Pearson: "On the Systematic Fitting oi Curves, etc." 

 Pan I, Biometrika, Vol. I, 1902, pp. 205 -303 and Vol. II, [902, pp. 1—24. Also 

 W. Palin Elderton " Frequency curves and Correlation," pp. 13 — 19. (C. and 

 E. Lay ton, \iji-j) and (i. UdneyYule: "Theory of Statistics " (Charles Griffin 



