TOPICS
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 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.
See also
Inverse Erf, Normal Distribution, Significance, Standard Error, Standard Normal DistributionExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Probable ErrorCite this as:
Weisstein, Eric W. "Probable Error." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProbableError.html