# Student's t-Distribution

A statistical distribution published by William Gosset in 1908. His employer, Guinness Breweries,
required him to publish under a pseudonym, so he chose "Student." Given *n*
independent measurements
,
let

(1) |

where
is the population mean,
is the sample mean, and *
s* is the estimator
for population standard
deviation (i.e., the
sample variance)
defined by

(2) |

Student's *t*-distribution is defined as the distribution of the random
variable *t* which is (very loosely) the "best" that we can do not knowing
.

If
,
*t = z* and the distribution becomes the
normal distribution.
As *N* increases, Student's *t*-distribution approaches the
normal distribution.

Student's *t*-distribution can be derived by transforming
Student's *
z*-distribution using

(3) |

and then defining

(4) |

The resulting probability and cumulative distribution functions are

(5) | |||

(6) |

where

(7) |

is the number of
degrees of freedom,
,
is the gamma function,
*B*(*a*,*b*) is the
beta function, and
is the regularized
beta function defined by

(8) |

The mean,
variance,
skewness, and
kurtosis of Student's
*t*-distribution are

(9) | |||

(10) | |||

(11) | |||

(12) |

The characteristic
functions
for the first few values of *n* are

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) |

and so on, where is a modified Bessel function of the second kind.

Beyer (1987, p. 571) 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 *r* and several
common confidence intervals.

r |
90% | 95% | 97.5% | 99.5% |
---|---|---|---|---|

1 | 3.07766 | 6.31371 | 12.7062 | 63.656 |

2 | 1.88562 | 2.91999 | 4.30265 | 9.92482 |

3 | 1.63774 | 2.35336 | 3.18243 | 5.84089 |

4 | 1.53321 | 2.13185 | 2.77644 | 4.60393 |

5 | 1.47588 | 2.01505 | 2.57058 | 4.03212 |

10 | 1.37218 | 1.81246 | 2.22814 | 3.16922 |

30 | 1.31042 | 1.69726 | 2.04227 | 2.74999 |

100 | 1.29007 | 1.66023 | 1.98397 | 2.62589 |

1.28156 | 1.64487 | 1.95999 | 2.57584 |

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
*t* with *n*
degrees of freedom
would be smaller than the observed value purely by chance:

(18) |

Let *X* be a
normally distributed
random variable with mean 0
and variance
,
let
have a chi-squared
distribution with *n*
degrees of freedom,
and let *X* and *Y* be independent. Then

(19) |

is distributed as Student's *t* with *n*
degrees of freedom.

**References**

Abramowitz, M. and Stegun, I. 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, pp. 536
and 571, 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."
ï¿½6.2 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.