The probability 
that a random sample from an infinite normally distributed universe will have a mean
within a distance 
of the mean 
of the universe is
 |
(1)
|
where 
is the normal distribution function
and 
is the observed value of
 |
(2)
|
The probable error is then defined as the value 
of 
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 true population value at least as great as the probable error
is therefore 1/2.
