The probable error of a variate from a normal distribution is the half-width of the symmetric interval
about its mean containing probability
1/2. For the sample mean 
of 
independent observations from a normal
distribution with mean 
and standard deviation 
, where 
is a positive integer,
the probability 
that 
is within 
standard errors of
is
 |
(1)
|
where 
is the normal distribution function
in the convention used here. The standardized sample mean
is
 |
(2)
|
The probable error measured in standard errors is the positive value 
such that 
, i.e.,
 |
(3)
|
which is given by
 |  |  |
(4)
|
 |  |  |
(5)
|
(OEIS A092678; Kenney and Keeping 1962, p. 134). Here, 
is the inverse erf function. The probability
of a deviation from the population
mean at least as great as the probable error is therefore 1/2. In the original
units, the probable error of 
is 
.
