Student's t-Distribution 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)

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

Student's -distribution is arrived at by transforming to Student's z-Distribution with (3)

Then define (4)

The resulting probability and cumulative distribution functions are     (5)         (6)

where (7)

is the number of Degrees of Freedom, , is the Gamma Function, is the Beta Function, and is the Regularized Beta Function defined by (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)

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 (14)

is distributed as Student's with Degrees of Freedom.

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.