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  independent measurements , let

(1)

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

(2)

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

The Student's -distribution with  degrees of freedom is implemented in the Wolfram Language as StudentTDistribution[n].

If ,  and the distribution becomes the normal distribution. As  increases, Student's -distribution approaches the normal distribution.

Student's -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)
			(7)
			(8)
			(9)

where

(10)

is the number of degrees of freedom, ,  is the gamma function,  is the beta function,  is a hypergeometric function, and  is the regularized beta function defined by

(11)

The mean, variance, skewness, and kurtosis excess of Student's -distribution are

			(12)
			(13)
			(14)
			(15)

The characteristic functions  for the first few values of  are

			(16)
			(17)
			(18)
			(19)
			(20)

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

The following table gives confidence intervals, i.e., values of  such that the distribution function  equals various probabilities for various small values of the numbers of degrees of freedom . 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.

	90%	95%	97.5%	99.5%
1	3.07768	6.31375	12.7062	63.6567
2	1.88562	2.91999	4.30265	9.92484
3	1.63774	2.35336	3.18245	5.84091
4	1.53321	2.13185	2.77645	4.60409
5	1.47588	2.01505	2.57058	4.03214
10	1.37218	1.81246	2.22814	3.16927
30	1.31042	1.69726	2.04227	2.75000
100	1.29007	1.66023	1.98397	2.62589
	1.28156	1.64487	1.95999	2.57588

A multivariate form of the Student's -distribution with correlation matrix  and  degrees of freedom is implemented as MultivariateTDistribution[r, m] in the Wolfram Language package MultivariateStatistics` .

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:

(21)

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

(22)

is distributed as Student's  with  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." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.

Shaw, W. "New Methods for Managing 'Student's' T-Distribution." Submitted to J. Comput. Finance. https://www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf.

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.