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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

 Q_delta=2Phi(|delta|),
(1)

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

 t=(x^_-mu)/(sigma/(sqrt(N))).
(2)

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

 Phi(delta^*)=1/4,
(3)

which is given by

delta^*=sqrt(2)erf^(-1)(1/2)
(4)
=0.674489750...
(5)

(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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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.

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Probable Error

Cite this as:

Weisstein, Eric W. "Probable Error." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProbableError.html

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