Eleanor Paiiiman and Karl Pearson 
247 
It will be seen that the Sheppard's corrections alone give very unsatisfactory 
results, and that while the full corrections for the first three moments are statistically 
satisfactory, the approximation of the fourth moment-coefficient would not for 
certain investigations be ade(|uate. We are in fact using only five groups and 
trusting to these for the accuracy of our abruptness coefficients. We will accordingly 
now test what improvement aidses when we divide our terminal groups into five sub- 
groups and calculate the abruptness coefficients on these smaller subranges. Thus 
we have h, = 'b, = hp = •!, and therefore p,, = pp = 5. Our subgroups are : 
Hi = 17142 therefore «/ = ■1623,8<>52, 173 therefore = -0016,3886, 
% = 14979 < = -1418,9900, np_, = 124 n'p.^ = '0011,7468, 
«3= 12959 = -1227,6314, 88 = -0008,3364, 
= 11099 . »; = -1051,4300, = 61 = -0005,7786, 
11,= 9412 H,' = -0891,6172, 7ip =^ n'p =-0004,0735. 
65591 489 
Whence 
aj = - -1730,3848 ; 
ao=-(- -0216,6599 
a, = - -0010,4679 
a, = - -0002,8683 
a, = + -0000,3789 
Determining the abruptness functions from these values, we have 
< - + W20 = - -8629,6462, W - ^^6/ + ^^,^6/ = + -0015,2171, 
"a'-ile"/ - -5475,2345, k'-^fyft/ = + -0032,2164, 
ai - (ft < + iiio< = - -8543,1522, h/ - + ^i^b,' = + -0007,8021, 
a.J-Jjja,' = -5604,7508, 6/-^\^6/ = H- -0054,8656. 
Working out the corrective terms for abruptness we find them 
-•071,787, --003,368, -h -031,252, -|- -071,446 
in working units, leading to 
/i/ = -476,922, yu,' = -395,458, /i/ = -438,5735, /i/ = -.585,957, 
or transferring to the mean we have 
ya/ = -476,922, = -168,003, /Z3 = -089,722, /^^ = -133-783, 
as against the actual values 
-476,930, /io= -168,02.5, ((^3 = -089,730, yt^^ = -133,743, 
an eminently satisfactory agreement. It is thus clear that when possible it is 
desirable to obtain the abruptness corrections by small subranges — in this case of 
the standard deviation. Hence any terminal small range groupings such as are 
frequently provided in statistical data are useful from this standpoint. In fact if 
ft/ 
= --8651,9242, 
6j= -0003,5810 and V = 
-0017,9049, 
aJ 
= + -5416,4969, 
h., = -0000,5368 h: = 
-0013,4204, 
a,' 
= --1308,4851, 
h,= -0001,5157 = 
•0189,4639, 
= --1480,1868, 
6, = --0000,7579 = 
- -0473,6598, 
= + -1184,1494, 
b, = -0000,3789 6; = 
•1184,1494. 
