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



Constants, for Data arranged according to Equidistant Divisions of 

 Scale," Proc. Lond. Math. Soc m) Vol. 29, pp. 353 — 380. 



In our notation the above correction (which is known as Shep- 

 pard's correction) is given by the following set of equations: — 



//, =v, 



H* = v% - tV W 



A is the length of the base unit, it is usually = 1 for working units. 



50 mm. unit of grouping. 



Making these corrections we find adjusted moments about 1655 

 to be 



/.«,' = -025, M ,/= 1-82 16 67, /'8 /== - '34 I2 5°» 



^' = 11-90 16 67, /V=-6-2_9 2I 8 7> /V = i47'33 97 83, 

 Now transferring to Mean we get 



H 2 = 1-82 10 42 /^=- '47 78 



a 4 =n«94 16 22 / , 5 =_778 23 



Hence we finally get ff corrected " constants : 



Mean= 1656-25 ±3'2i 7 mm. 



S.D.= 67-47 3 +2-61 62 

 Coeff. ofV, V= 4*07 38 ± -13 76 



/?',= -03 78 10+ -05 41 33 

 (3 2 = 3'6o 10 + 71 20 69 

 sk= 'oy 31 10+ *o6 22 32 

 d= 4*93 29 50 + 4*22 30 20 mm. 



Note. — Starting with 1430 as our base unit, we reach the same results, thus 

 the arithmetic is absolutely checked in this case. 



The Frequency Constants were next calculated (both with 

 and without Sheppard's correction) for widely different units of 

 grouping. We have 1 mm., 20 mm., 30 mm., 50 mm. and finally 100 

 mm. as our unit of grouping. It will be observed that the unit of 

 grouping is thus successively made the same, 10 times, 20 times, 

 50 times and finally 100 times the unit of measurement. 



With " ungrouped " (i.e, i mm.) measurements, the arithme- 

 tical labour is tremendous. In this case the maximum value of x 

 is —2io, which involves calculating (210) 4 for the fourth moment. 

 Hence it was not possible to go beyond the fourth moment. As 

 it is, the actual sum of fourth-products, i.e., S(x*'y) runs into n 

 figures. I quote actual results 



S(xy)= 1 58 



S(x*y) = 90 82 72 



S(A- 8 y)= -6 76 88 78 

 S( v *y) = 144 04 2c" 60 6^ 



