Eleanor P airman and Karl Pearson 
245 
It will be seen that now Sheppard's corrections are wholly inadequate and our 
corrections are essential, even in the case of the fourth moment-coefficient. This 
confirms the view of Sheppard himself, who insisted on the importance of high 
contact at the terminals, if they are to be used alone. It is a convincing illustration 
of the fallacy of those " proofs " of Sheppard's corrections which do not appeal to 
the principle of high terminal contact. 
We now propose to illustrate the degree of improvement in the exactness 
obtained, if we calculate the abruptness coefficients on smaller subranges. Accord- 
ingly we break up the terminal group 05591 on •5cr base into five groups each on 
■lo- base. These are " . 
•1622,5272 and a-i = - -1728,9281, 
•1417,7946 
■1226,5973 
•1050,5442 
•0890,8661 
17142 leading to t?/ = 
14979 
12959 
11099 
9412 
ns = 
n/ = 
a^= -0216,4780, 
= - -001 0,4603, 
a, = --0002,3651, 
-0000,3781, 
whence, remembering a/ = 
(a„) 
as = ^'as, 
we find 
rt/ =- -864,464, 
and 
- 677 + 
a/= -541,195, 
-Ttir«.') 
a/ = - -130,754, 
- (flips' + . 
«; = - -147,818, 
"~ s'lJ ) 
«/= -118,156. 
Thus the moment-coefficients become 
/x/ = -957,660 or in actual units 
1-606,102 
fi.;^ 3-647,389 
= 10^161,505 
Transferring to the mean we have 
On 
y-1 
5 subranges 
478,350 
172,587 
■098,801 
156,767 
On ■! subranges 
•478,830 
•172,248 
-098,705 
•156,516 
= - -004,559, 
= -021,351, 
= -004,398, 
•478,830, 
•401,526, 
•455,924, 
•635,094. 
Actual values 
•478,813 
•172,222 
•098,612 
•156,405 
While the first column of values would be amply adequate for most statistical 
purposes, the second makes a still closer approximation to the actual values, the 
differences being only 
-•000,017, +-000,026, +^000,093, + ^000,111 
as against -•000,463, + •000,365, +'000,189, + •000,:^62 
respectivel3^ The greatest improvements are in the mean and standard deviation. 
Accordingly it is well worth using smaller terminal subranges, if they are available 
as in the cases of cricket scores, wages, house values, infant mortality and other 
frequency material. 
