26G On 'Best' Values of Constmits in Frequency Distributions 
a was found equal to 2-282542 and the second formula of (3) gave the value 
2-341735 for o. As Ao- was found so large that the approximation could not be 
expected to be very good, the following values of ^ ^ were calculated from the 
da 
second formula of (4) : 
a 
da 
2-282542 
= 2-325581 
-43 
4-^ = 2-380952 
•42 
- 32-53 
- 11-81 
+ 8-06 
By interpolation in this table a = 2-355860 was found as the value for which 
— equals zero, and this is the a of the improved Gaussian given above. 
From X" ^^^^ ' goodness of fit ' P was found : 
P 
da 
Gaussian 
Impr. Gaussian 
10-833 
9-720 
0-211 
0-285 
- 32-53 
+ 0-20 
As will be seen the better fit is obtained by making a bigger than the Gaussian 
value, the improvement therefore cannot be looked upon as a correction for grouping. 
On the contrary the Sheppard correction would have given a = 2-264214 and 
have raised to 11-52. Thus we see that although the two methods give close 
values for P, the 'better value' is obtained as it should be from the lesser value 
of d {x^)/da. 
(3) Illustration II. Fit of a Normal Curve to Bilateral Data. 
For the next illustration I have used a table giving frequencies of cephalic 
index in Bavarian skulls*. Both a and ni have here been varied. As the formulae 
(3) are somewhat laborious to work with, the approximations were used roughly 
suggested by the process on p. 264, but the results were not satisfactory f and these 
d(v^) d(v^) 
approximate results are therefore not given here. But and for the two 
* J. Ranko, Beitrage zur jiliysi.icheji Antropologie. der Baiern, Miinehen, 1883. The table includes 
the material from Tables I-VI and VIII-X inclusive which may be treated as typically ' Alt-Baierisch.' 
t In fact the calculation of the exact value of — ^-„~ showed that the part of it neglected in 
da'- ° 
formula (2 i) was about -j'j of tlie wliole value. It essentially arose from the one tail group, this being 
of the whole neglected part. As ^"^ for this group was only as big as 1-0348, the approximate 
formula (2 h) cannot be expected to be of great value for the normal curve. 
