A Distribution published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a
pseudonym, so he chose ``Student.'' Given independent measurements , let

(1) |

(2) |

Student's -distribution is arrived at by transforming to Student's *z*-Distribution with

(3) |

(4) |

(5) | |||

(6) |

where

(7) |

(8) |

The Mean, Variance, Skewness, and Kurtosis of Student's -distribution are

(9) | |||

(10) | |||

(11) | |||

(12) |

Beyer (1987, p. 514) gives 60%, 70%, 90%, 95%, 97.5%, 99%, 99.5%, and 99.95% confidence intervals, and Goulden (1956) gives 50%, 90%, 95%, 98%, 99%, and 99.9% confidence intervals. A partial table is given below for small and several common confidence intervals.

80% | 90% | 95% | 99% | |

1 | 3.08 | 6.31 | 12.71 | 63.66 |

2 | 1.89 | 2.92 | 4.30 | 9.92 |

3 | 1.64 | 2.35 | 3.18 | 5.84 |

4 | 1.53 | 2.13 | 2.78 | 4.60 |

5 | 1.48 | 2.01 | 2.57 | 4.03 |

10 | 1.37 | 1.81 | 2.23 | 4.14 |

30 | 1.31 | 1.70 | 2.04 | 2.75 |

100 | 1.29 | 1.66 | 1.98 | 2.63 |

1.28 | 1.65 | 1.96 | 2.58 |

The so-called distribution is useful for testing if two observed distributions have the same Mean.
gives the probability that the difference in two observed Means for a certain statistic with
Degrees of Freedom would be smaller than the observed value purely by chance:

(13) |

(14) |

The noncentral Student's -distribution is given by

(15) |

where is the Gamma Function, is a Confluent Hypergeometric Function, and is an associated Laguerre Polynomial.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 948-949, 1972.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 536, 1987.

Fisher, R. A. ``Applications of `Student's' Distribution.'' *Metron* **5**, 3-17, 1925.

Fisher, R. A. ``Expansion of `Student's' Integral in Powers of .'' *Metron* **5**, 22-32, 1925.

Fisher, R. A. *Statistical Methods for Research Workers, 10th ed.* Edinburgh: Oliver and Boyd, 1948.

Goulden, C. H. Table A-3 in *Methods of Statistical Analysis, 2nd ed.* New York: Wiley, p. 443, 1956.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
``Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution.'' §6.2 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England: Cambridge
University Press, pp. 219-223, 1992.

Spiegel, M. R. *Theory and Problems of Probability and Statistics.* New York: McGraw-Hill, pp. 116-117, 1992.

Student. ``The Probable Error of a Mean.'' *Biometrika* **6**, 1-25, 1908.

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1999-05-26