KiRSTiNE Smith 
271 
Fitting the frequencies from the end by means of two moments we obtain the 
binomial 
(•6658208 + •3341792)12-126379^ 
the terms of which are given in the table above under the head Binomial. 
From these starting values of p and I we found by the equations (8) the constants 
of the improved binomial (a) f = -6674922 and I = 12-188945. 
A comparison between the coefficients of the two sets of formulae (8) and (9) 
gave the result that they only diverged by between 1-4 and 5 per mille of their 
^ 2 ^ 2 
value. As ^ _ ^ " for the tail group was as big as 5-44, we are from this justified 
in expecting the approximate formulae (9) to be useful for binomial data. 
Starting from the improved binomial {a) another improved binomial (6) was 
found by means of the formulae (9). As will be seen I only succeeded in 
diminishing by raising > a^^d came out with exactly the same value 
as by the former formula. The constants for the improved binomial (6) are 
p = -6675432 and I = 12-191141. 
The constants illustrating the ' goodness of fit ' were found as follows : 
TABLE IX. 
P 
dp 
dl 
Binomial 
11-643 
-390 
159-47 
- 8-15 
Improved Binomial (a) 
11-513 
-401 
26-02 
- -84 
11-513 
-401 
- -02 
+ 1-96 
It will be seen from the above illustrations that the probability of happening 
as determined by the test of ' goodness of fit ' being a maximum can always be 
made somewhat greater than the same probabihty deduced from a fit by the 
method of moments, which at any rate for the Gaussian curve is usually assumed 
to be the 'best.' 
(6) On the 'Best' Values of the Constants of Regression Curves. 
If we apply the test of ' goodness of fit' to regression curves as recently indicated 
by Pearson* modifying Slutsky's methods |, we shall experience the same divergence 
between the curves of regression found by the method of least squares and the 
curves calculated so as to make x^ ^ minimum, as we found when dealing with 
frequency distributions. 
In the paper cited x"^ for a regression curve is given as 
fn„ (m„ - 
.(10), 
* Bioinetrika, Vol. xi. pp. 239 et seq. 
f Journal of the Royal Statistical Society, Vol. Lxxvn. pp. 78-84. 
