A normalized form of the cumulativenormal distribution function giving the probability that a variate assumes a value in the range,

(1)

It is related to theprobability 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 norerf can be expressed in terms of finite additions, subtractions, multiplications, androot 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 theconfidence interval.

For small values, a good approximation to is obtained from theMaclaurin series forerf,

(10)

(OEISA014481). For large values, a good approximation is obtained from the asymptotic series forerf,

(11)

(OEISA001147).

The value of for intermediate can be computed using thecontinued fraction identity

(12)

A simple approximation of which is good to two decimal places is given by

(13)

Abramowitz and Stegun (1972) and Johnsonet al.(1994) give other functionalapproximations. 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 theprobable error of a normally distributed variate.

See also

Explore with Wolfram|Alpha

More things to try:

• normal distribution function
• erf(x), erf'(x), erf''(x)
• plot Re(erf(x + I y)), Im(erf(x + I y))

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Normal Distribution Function."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NormalDistributionFunction.html

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