350 
MR. R. A. FISHER ON THE MATHEMATICAL 
For example, we have taken from Vega (14) sets of digits from the table of Natural 
Logarithms to 48 places of decimals. The last block of four digits was taken from the 
logarithms of 100 consecutive numbers from 101 to 200, giving a sample of 100 numbers 
distributed evenly over a limited range. It is sufficient to take the three first digits 
to the nearest integer ; then each number has an equal chance of all values between 0 
and 1000. The true mean of the population is 500, and the standard deviation 289. 
The standard error of the mean of a sample of 100 is therefore 28-9. 
Twenty-five such samples were taken, using the last five blocks of digits, for the 
logarithms of numbers from 101 to 600, and the mean determined merely from the highest 
and lowest number occurring, the following values were obtained :— 
Digits. 
1st hundred. 
2nd hundred. 
3rd hundred. 
4th hundred. 
5th hundred. 
Lowest. 
Highest. 
m — m. 
Lowest. 
Highest. 
m—m. 
Lowest. 
Highest. 
m— m. 
Lowest. 
Highest. 
771— m. 
Lowest. 
Highest. 
in - m. 
45-48 
24 978 + 1-0 
39 980 + 9-5 
1 999 0 
16 983 - 0-5 
18 994 -)-6-0 
41-44 
35-5 993 +14-0 
3 960 -18-5 
6 997 +1-5 
1 978 —10-5 
4 979 —8-5 
37-40 
9 988 - 1-5 
11 999 + 5-0 
31 984 +7-5 
4 978 - 9-0 
2 986 -6.0 
33-36 
7 995 + 1-0 
13 997 + 5-0 
4 998 +1-0 
0 994 - 3-0 
3 981 -8-0 
29-32 
1 988 - 5-5 
3 988 - 4-5 
4 992 —2-0 
1 996 - 1-5 
21 977 -1-0 
It will be seen that these errors rarely exceed one-half of the standard error of the 
mean of the sample. The actual mean square error of these 25 values is 6-86, while the 
calculated value, \/50, is 7 • 07. It will therefore be seen that, with samples of only 100, 
there is no exaggeration in placing the efficiency of the method of moments as low as 
6 per cent, in comparison with the more accurate method, which in this case happens 
to be far less laborious. 
Such a value for the efficiency of the mean in this case is, however, purely conven¬ 
tional, since the curve of distribution is outside the region of its valid application, and 
the two curves of sampling do not tend to assume the same form. It is, however, 
convenient to have an estimate of the effectiveness of statistics for small samples, and 
in such cases we should prefer to treat the curve of distribution of the statistic as an 
error curve, and to judge the effectiveness of the statistic by the intrinsic accuracy of 
the curve as defined in Section 9. Thus the intrinsic accuracy of the curve of distri¬ 
bution of the mean of all the observations is 
12n 
9 5 
a 
