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

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

by

Let so . Then

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

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

(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,

(OEIS A014481). For large values , a good approximation is obtained from the asymptotic series for erf,

(OEIS A001147).

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

A simple approximation to for is given by

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

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.

See also

Berry-Esséen Theorem, Confidence Interval, Distribution Function, Erf, Erfc, Fisher-Behrens Problem, Gaussian Integral, Hh Function, Normal Distribution, Owen T-Function, Probability Integral, Tetrachoric Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 931-933, 1972.Bagby, R. J. "Calculating Normal Probabilities." Amer. Math. Monthly 102, 46-49, 1995.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Bryc, W. "A Uniform Approximation to the Right Normal Tail Integral." Appl. Math. Comput. 127, 365-374, 2002. https://doi.org/10.1016/S0096-3003(01)00015-7.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. New York: Wiley, p. 45, 1971.Hastings, C. Approximations for Digital Computers. Princeton, NJ: Princeton University Press, 1955.Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, 2nd ed. New York: Wiley, 1994.Patel, J. K. and Read, C. B. Handbook of the Normal Distribution. New York: Dekker, 1982.Sloane, N. J. A. Sequences A001147/M3002 and A014481 in "The On-Line Encyclopedia of Integer Sequences."Whittaker, E. T. and Robinson, G. "Normal Frequency Distribution." Ch. 8 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 164-208, 1967.

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Normal Distribution Function

Cite this as:

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

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