FOUNDATIONS OF THEORETICAL STATISTICS. 
349 
if £ is the difference at one end of the range and tl the difference at the other end, the 
joint distribution (since, when n is considerable, these two quantities may be regarded 
as independent) is 
2 ii *- 
n — i+% 7 _ 7 
-o e a df cltj. 
a 
Now if we take the mean of the extreme observations of the sample, our error is 
i-n. 
for which we write x ; writing also y for £ -j- )h we have the joint distribution of x and y, 
— 0 e a ° dx dy. 
a “ 
For a given value of x the values of y range from 2 j £c | to oo, whence, integrating with 
respect to y, we find the distribution of x to be 
2 n 
I s I 
dx, 
the double exponential curve shown in fig. 5. 
-1- ■■ ~ T I _ I _ 1 __|_|_|_ ----- 
-25 -20 -15 -10 -5 0 5 10 15 20 25 
Fig. 5. Double exponential frequency curve, showing distribution of 25 deviations. 
The two error curves are thus of a radically different form, and strictly no value for 
the efficiency can be calculated ; if, however, we consider the ratio of the two standard 
deviations, then 
2 2*2 n 
^ « . a _ 6 
a- 2 ,,, 2n 2 " 12 n n 
. 
when n is large, a quantity which diminishes indefinitely as the sample is increased. 
3 o 
yoL. ccxxn. 
a. 
