348 
MR. R. A. FISHER ON THE MATHEMATICAL 
in the neighbourhood of the terminus, then the chance of an observation falling within 
a distance x of the terminus is 
k 
rx + 1 
Z a+1 
LV +1 , 
and the chance of n observations all failing to fall in this region is 
(1 -fY 
or, when n is great, and / correspondingly small, 
e~ fn . 
Equating this to any finite probability, e ", we have 
k'x a+1 = 
a 
— ? 
n 
or, in other words, if we use the extreme observation as a means of locating the terminus, 
the error, x, is proportional to 
. i 
n 
when rx < 1, this quantity diminishes more rapidly than n~*, and consequently for large 
samples it is much more accurate to locate the curve by the extreme observation than 
by the mean. 
Since it might be doubted whether such a simple method could really be more accurate 
than the process of finding the actual mean, we will take as example the location of 
the curve (B) in the form of a rectangle, 
df = 
dx 
a 
a ^ a 
on -< x < on 4— 
9 9 
and 
df= 0, 
outside these limits. 
This is one of the simplest types of distribution, and we may readily obtain examples 
of it from mathematical tables. The mean of the distribution is m, and the standard 
deviation, the error m—m, of the mean obtained from n observations, when n is 
v 7 12 
reasonably large, is therefore distributed according to the formula 
dx. 
The difference of the extreme observation from the end of the range is distributed 
according to the formula 
n 7. 
— e a df; 
a 
