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.

REFERENCES:

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 129 and 134, 1962.

Sloane, N. J. A. Sequence A092678 in "The On-Line Encyclopedia of Integer Sequences."

Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, p. 184, 1967.