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