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)
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where is the normal distribution function and is the observed value of
(2)
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The probable error is then defined as the value of such that , i.e.,
(3)
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which is given by
(4)
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(5)
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(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.