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.
