278 
Miscellanea 
1^3 = 
N 2 
giving : 
giving: 
From these we derive 
In tlie same way 
M3 
V 7r \7r 
= -0771 0-3. 
22 
M4- 
3 5 
7 + - 
■± TT 
11 
= -12770 
^l = -9906, /32 = 3-8692. 
Case of Samples of Three. 
= 2 = 
- 
3 ' 
\/ 6 
and 
giving: 
giving: 
and 
M4 
(3 "e 
-3782. 
(7 = -72360. 
-1431c 
6 V3 
•0342O-3. 
■0664o 
17'^ • 
12 J 
3i = -3983, ft., = 3-2451. 
Modal Values. 
In the case of n = 2, the mode of the theoretical curve is at the origin 2 = 0, but it is to be 
borne in mind that it is the areas of strips of the frequency-curve which are to be used to estimate 
the probability. In practice, therefore, seeing that all measurements must be made in discrete 
amounts and cannot be mathematically continuous, we can only assert that the most frequently 
occurring values of 2 are those which are nearest zero — not actually zero. The example given 
below will make this clear. 
For n = 3, the mode is obtained by differentiating the equation 
This gives, for the mode, 
0 
y d2 
thus the modal value 2 of 2 is given by 
2 = 
/3 
1 
2 
•5774c 
32 
2 ' 
Skewness. 
The skewness of the distributions is for w = 2, 1-3236, and for w = 3, -3867. 
Thus whether we consider the mean or the modal values of the distributions, it is evident that 
the probable error determined from a set of three observations is very untrustworthy and that when 
there are only two observations it is very much worse. 
