338 
MR. R. A. FISHER ON THE MATHEMATICAL 
9. Location and Scaling of Frequency Curves in General. 
The general problem of the location and scaling of curves may now be treated more 
generally. This is the problem which presents itself with respect to error curves of 
assumed form, when to find the best value of the quantity measured we must locate the 
curve as accurately as possible, and to find the probable error of the result of this process 
we must, as accurately as possible, estimate its scale. 
The form of the curve may be specified by a function <]>, such that 
df oc ® dg, when p 
x—m 
a 
In this expression p specifies the form of the curve, which is unaltered by variations 
of a and m. 
When a sample of n observations has been taken, the likelihood of any combination 
of values of a and m is 
L = 0 — a log ct t & ( (j) ) , 
whence 
since 
A = 8(A. A) = _ Is(.//), 
dm \dg dm1 a 
K = _i; 
a 
also 
since 
Differentiating a second time, 
cL 
ca 
a 
S (&>')■ 
n 
a 
cSL 
end 
= 4S (</>") ; 
a~ 
therefore 
9 
q~ 
"77/ ‘ 
n<p 
This expression enables us to compare the accuracy of error curves of different form, 
when the location is performed in each case by the method which yields the minimum 
error. 
Example : —The curve 
d /= I At 
7T 1 + f 
referred to in Section 5 has an infinite standard deviation, but it is not on that account 
an error curve of zero accuracy, for 
(/) — — log’ (1 + f"), 
2f 
1+f 2 ’ 
2(1 -f ) 
(l + f) 2 ' 
