Probable Error

The probability Q_delta that a random sample from an infinite normally distributed universe will have a mean m within a distance |delta| of the mean mu of the universe is


where Phi(z) is the normal distribution function and delta is the observed value of


The probable error is then defined as the value delta^* of delta such that Q_delta=1/2, i.e.,


which is given by


(OEIS A092678; Kenney and Keeping 1962, p. 134). Here, erf^(-1)(x) 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.

See also

Inverse Erf, Normal Distribution, Significance, Standard Error, Standard Normal Distribution

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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.

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Probable Error

Cite this as:

Weisstein, Eric W. "Probable Error." From MathWorld--A Wolfram Web Resource.

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