1922.] P. C. MahaiA-NObis : Analysis of Stature. 21 



which gives us (dividing by 200) : — 



V= 79 



"2'= 45 4i -36 



V = 3 48 44 "39 

 v 4 ' = 72 02 14 30 -38 



For purposes of comparison it is necessary to reduce all 



moments to the same unit. 50 mm. was chosen as the standard unit. 1 



Let Mn be any moment in units of grouping h y let M„ be the 



h 

 corresponding moment in standard units h , let p = — 



Then M n — p n . fx n> is the formula of reduction to standard unit. 



12^ 

 For h =50 mm., P = — , — , - and 2 successively for units 



50 5 5 



of 1 mm., 20 mm., 30 mm. and 100 mm. respectively. 



The annexed table gives the Frequency Constants for the 

 different units of grouping. I have added the probable errors in 

 each case. 



For the purpose of studying the effect of grouping it is natural 

 to take the c< ungrouped " constants as our standard. We have 

 accordingly assumed that the 1 mm. constants are the " true ' 

 constants. 



Different Values of Mean Stature. 



Unit of Grouping. 



16 5679 + 3*23 mm. 

 16 56-85 ± 3-23 „ 



16 56'35 ± 3*23 „ 

 16 56-25 + 3-22 „ 

 16 59-50 + 3-09 „ 



When the unit of grouping is so large as 100 mm. (and the 

 total record is divided only info 5 groups), there is considerable 

 difference in the Mean. But this difference of 2*71 mm. is less than 

 the probable error of over 3 mm. Thus even with 100 mm. group- 

 ing, the Mean is stable within the limits of its own probable error. 



The agreement is almost perfect when we omit the 100 mm. 

 group. The maximum li error " due to grouping amounts to only 

 \54 mm., which is considerably less than the unit of measurement 

 itself and is about ^ of the probable error. 



L,et us consider a very large sample of 7,500 individuals. It 

 is not likely that the Standard Deviation will exceed 70 mm. The 

 P.E. of Mean will be about '55 mm. The maximum observed 

 difference in the present case, due to grouping, is thus of the same 

 order as the random P.E. of the Mean in a sample of 7,500. We 

 conclude therefore that for samples of 200 , the effect of grouping on the 

 Mean up to 50 mm. is quite negligible. 



1 For reasons explained on pp. 39-40. 



I 



mm 



20 



>> 



30 



5 > 



50 



)> 



00 



5 > 



