1922.] P. C. Mahalanobis : Analysis of Stature. ig 



The S.D. of any moment t> q in a sample of size n is given by l 



w - W = ^29 -/V 2 - 2(7/^ ,,;,-, + qW f x <l q _ ] 



P.E. of /x 2 = -67449 V - . /x 2 =-o6 74 49'm 2 



P.E- of /x 3 = -67449 / _ . a 8 =-n 68 16-0-8 



V ft 



P.E. of /i 4 = -67449 V ■§- . ^ 2 a = 27 96 56> a * 



AX 



For # = 5, we must find /^ . 



But Sheppard * has shown that for the normal curve (in our 

 present notation) 



fx is =(2S-l)(2S-3) 1.^* 



, 720 



Hence we get 2 U5 = — . ^</ 



Substituting in the above formula, we get 



Thus P.E. of /x 5 = -67 44 9 / 7 — . 0-5 = 1-27 96 56-^ 



V n 



We thus get : — 



/j 2 = 1-90 43 75 ±o-i2 84 31 



H=~ '47 78 43 77+ '30 69 85 

 fx^ 12-86 56 42 58 + 1-46 74 85 

 ju 5 =— 8*i8 04 93 98 + 6-40 45 50 



Sheppard's Correction. 



I shall now consider the question of corrections for grouping. 

 The theoretical work in this subject now consists of a good deal of 

 literature. I shall discuss this question from a purely practical 

 point of view. The fundamental memoir is W. F. Sheppard 8 : 

 <( On the Calculation of the most Probable Values of Frequency 



1 " On Probable Errors of Frequency Constants," Biometvika Vol. 2 (1903). 

 p. 276. 



2 W. F. Sheppard : Phil. Trans. Roy. Soc, 192A. 



3 (a) A summary of Sheppard's memoir (with some new results) is given in 

 an Editorial Note : " On an Elemementary Proof of Sheppard's Formulae for cor- 

 recting Raw Moments and on Other Allied Points " in Biom. Vol. 3, pp. 308 — 310. 



(b) In Pearson's paper: " On Systematic Fitting of Curves, etc.'' Biom. Vols. 

 1 and 2, this question has been discussed from a different standpoint. 



(c) Sheppard himself has given a simplified method of obtaining- certain cor- 

 rections in a later paper: "The Calculation of Moments of a Frequency-Distri- 

 bution," Biom. Vol. 5 (1907), pp. 450 — 459. 



(d) Eleanor Pairman and Karl Pearson have published a memoir: " On Cor- 

 rections for the Moment-coefficients of Limited Range Frequency Distributions 

 etc." in Biom. Vol. 12 (1919), pp. 231 — 338, which I shall have occasion to 

 discuss later on in greater detail. 



