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Probable Error
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 thenormal 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) |
(OEISA092678; Kenney and Keeping 1962, p. 134). Here,
is theinverse 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 DistributionExplore with Wolfram|Alpha
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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. SequenceA092678 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.Referenced on Wolfram|Alpha
Probable ErrorCite this as:
Weisstein, Eric W. "Probable Error." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/ProbableError.html