Notices and Bibliogrcq)hf/ 
209 
an unsatisfactory result. In all the graduation.s we could doubtless improve the agreement by 
using a greater number of terms in the series, but we think a considerable increase in the 
number would be required to give what we should consider a satisfactory graduation. 
Size 
of 
family 
Observations 
Edgeworth's 
third 
Approximation 
Charlier Type A t 
<T — 2-92S ; /5..= - -1214 
/34= -0104 
CharUer 
Type B 
Pearson 
Type It 
-3 
— 
1* 
-2 
— 
— 
-2 
— 
9* 
4 
— 
— 
-1 
— 
30* 
15 
12 
2 
64 
64 
38 
64 
67 
1 
116 
102 
71 
104 
116 
2 
140 
1 AQ 
lUo 
lay 
loo 
3 
145 
135 
137 
134 
139 
4 
134 
130 
148 
128 
128 
5 
106 
111 
135 
116 
110 
6 
82 
92 
108 
93 
89 
7 
72 
73 
78 
73 
69 
8 
49 
53 
54 
53 
51 
9 
37 
36 
37 
36 
35 
10 
25 
20 
27 
25 
24 
11 
13 
10 
18 
14 
15 
12 
10 
4 
12 
10 
9 
13 
5 
7 
5 
5 
14 
2 
4 
2 
2 
15 
■4 
1 
1 
Totals 
1000 
1000 
1001 
1000 
1000 
* Approximation by help of diagram in Edgeworth (2). 
+ Notation of Charlier (G), mid-ordinates, found by Charliers tables, being used, 
t " Chances of Death," Vol. i. p. 74. 
To the actuary, influenced perliai>s by professional bias, the justification of a fornuila for 
graduating frequency distributions is its width of application ; to some extent we feel that such 
is also the justification of any theoretical conditions from which a curve is evolved. Edgeworth's 
series and Charlier's Type A will be found to give good graduations provided the distributions 
are not markedly skew but they become less satisfactory as the range of the obsers ations takes 
a definite limit. Charlier's Type B on the other hand is certainly capable of graduating some 
distributions having a I'ange limited in one direction but, though it can hardly be criticised 
fully at present, as the author states in (6) that his work on it is not yet complete, it may be well 
ti) point out that the solutions he gives are approximate and the choice of solution in any 
particular case seems somewhat arbitrary. The comi>arati\ely poor agreement reached above 
may be due to this approximate fitting and not to the failure of the curve itself A statistical 
criterion to show whether Type A or Type B should be used in any particular case is certainly 
needed before these types can be used extensively in practice, but even then it would seem 
impossible to graduate the U-shaped distributions or those that rise abruptly from the axis of x 
at both ends. 
One or two examples, besides that already mentioned, are given in (2;, while there is a plentiful 
supply of statistical examples in (6) and most of them show a close agreenrent between the 
theoretical and actual frequencies ; some are less satisfactory and fig. 9 of (6) gives so poor a fit 
that the odds against the graduation are moi-e than 50 to one. There are many other points of 
interest in (6) beside the main subject, such as a proof, on the basis of Type A, of the relative 
positions of the mode, mean and median, a method of checking the numerical calculation of 
Biometrika v 27 
