A normalized form of the cumulative Gaussian Distribution function giving the probability that a variate
assumes a value in the range ,

(1) |

(2) |

(3) |

(4) |

(5) |

Note that a function different from is sometimes defined as ``the'' normal distribution function

(6) |

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,

(7) |

(8) |

(9) |

(10) |

(11) |

The first Quartile of a standard Normal Distribution occurs when

(12) |

**References**

Abramowitz, M. and Stegun, C. 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.

Johnson, N.; Kotz, S.; and Balakrishnan, N. *Continuous Univariate Distributions, Vol. 1, 2nd ed.* Boston, MA: Houghton Mifflin, 1994.

© 1996-9

1999-05-25