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) |
(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.
See also Paired t-Test, Student's z-Distribution
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.
© 1996-9 Eric W. Weisstein