Normal Distribution Function

The normal distribution function in the convention used here is defined for a real number by

(1)

For , it gives the probability that a variate from the standard normal distribution lies in . For , the integral is negative and the probability of is . 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 standard normal distribution variate lies in , where , is therefore given by

(5)

Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions.

The cumulative distribution function of the standard normal distribution, also often denoted by , is

			(6)
			(7)
			(8)
			(9)

(Feller 1968; Beyer 1987, p. 551). The notation is due to Feller (1971).

For , the central interval containing a standard normal distribution variate with probability is determined by .

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).

For , the following continued fraction identity can be used to compute after rescaling the argument:

(12)

A simple approximation to for is given by

(13)

Abramowitz and Stegun (1972) and Johnson et al. (1994) give other functional approximations. An approximation for due to Bagby (1995) is

(14)

The plots above show the differences between and the two approximations.

The positive value of satisfying is known as the probable error of a standard normal distribution variate.