A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range ,
|
(1) |
It is related to the probability integral
|
(2) |
by
|
(3) |
Let so . Then
|
(4) |
Here, erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range is therefore given by
|
(5) |
Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so must be either computed numerically or otherwise approximated.
Note that a function different from is sometimes defined as "the" normal distribution function
|
(6) | |||
|
(7) | |||
|
(8) | |||
|
(9) |
(Feller 1968; Beyer 1987, p. 551), although this function is less widely encountered than the usual . The notation is due to Feller (1971).
The value of for which falls within the interval with a given probability is a related quantity called the confidence interval.
For small values , a good approximation to is obtained from the Maclaurin series for erf,
|
(10) |
(OEIS A014481). For large values , a good approximation is obtained from the asymptotic series for erf,
|
(11) |
(OEIS A001147).
The value of for intermediate can be computed using the continued fraction identity
|
(12) |
A simple approximation of which is good to two decimal places is given by
|
(13) |
Abramowitz and Stegun (1972) and Johnson et al. (1994) give other functional approximations. An approximation due to Bagby (1995) is
|
(14) |
The plots below show the differences between and the two approximations.
The value of giving is known as the probable error of a normally distributed variate.