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 $n$ independent measurements $x_i$, let

$$t\equiv {\bar x-\mu\over s/\sqrt{n}},$$ (1)

where $\mu$ is the population Mean, $\bar x$ is the sample Mean, and $s$ is the Estimator for population Standard Deviation (i.e., the Sample Variance) defined by

$$s^2\equiv {1\over N-1} \sum_{i=1}^n (x_i-\bar x)^2.$$ (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 $\sigma$. If $\sigma=s$, $t=z$ and the distribution becomes the Normal Distribution. As $N$ increases, Student's $t$-distribution approaches the Normal Distribution.

Student's $t$-distribution is arrived at by transforming to Student's z-Distribution with

$$z\equiv {\bar x-\mu\over s}.$$ (3)

Then define

$$t\equiv z\sqrt{n-1}.$$ (4)

The resulting probability and cumulative distribution functions are

$$f_r(t) = {\Gamma[{\textstyle{1\over 2}}(r+1)]\over\sqrt{r\pi}\,\Gamma({\textstyle{1\over 2}}r)\left({1+{t^2\over r}}\right)^{(r+1)/2}}$$

$$= {\left({r\over r+t^2}\right)^{(1+r)/2}\over \sqrt{r}\,B({\textstyle{1\over 2}}r, {\textstyle{1\over 2}})}$$ (5)

$$F_r(t) = \int_{-\infty}^t {\Gamma[{\textstyle{1\over 2}}(r+1)]\over \sqrt{... ...\,\Gamma({\textstyle{1\over 2}}r)\left({1+{t'^2\over r}}\right)^{(r+1)/2}}\,dt'$$

$$= {1\over\sqrt{r}B({\textstyle{1\over 2}}r, {\textstyle{1\over 2}})\left({1+{t^2\over r}}\right)^{(r+1)/2}}$$

$$= {1\over 2}+{1\over 2}\left[{I(1; {\textstyle{1\over 2}}r, {\texts... ...{r\over r+t^2}, {\textstyle{1\over 2}}r, {\textstyle{1\over 2}}}\right)}\right]$$

$$= 1-{1\over 2} I\left({{r\over r+t^2}, {\textstyle{1\over 2}}r, {\textstyle{1\over 2}}}\right),$$ (6)

where

$$r\equiv n-1$$ (7)

is the number of Degrees of Freedom, $-\infty<t<\infty$, $\Gamma(z)$ is the Gamma Function, $B(a,b)$ is the Beta Function, and $I(z;a,b)$ is the Regularized Beta Function defined by

$$I(z; a, b) = {B(z; a, b)\over B(a,b)}.$$ (8)

The Mean, Variance, Skewness, and Kurtosis of Student's $t$-distribution are

$$\mu = 0$$ (9)

$$\sigma^2 = {r\over r-2}$$ (10)

$$\gamma_1 = 0$$ (11)

$$\gamma_2 = {6\over r-4}.$$ (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 $r$ and several common confidence intervals.

$r$	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
$\infty$	1.28	1.65	1.96	2.58

The so-called $A(t\vert n)$ distribution is useful for testing if two observed distributions have the same Mean. $A(t\vert n)$ 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:

$$A(t\vert n) = {1\over\sqrt{n}\,B({\textstyle{1\over 2}}, {\t... ...}n)} \int_{-t}^t \left({1+{x^2\over n}}\right)^{-(1+n)/2}\,dx.$$ (13)

Let $X$ be a Normally Distributed random variable with Mean 0 and Variance $\sigma^2$, let $Y^2/\sigma^2$ have a Chi-Squared Distribution with $n$ Degrees of Freedom, and let $X$ and $Y$ be independent. Then

$$t\equiv {X\sqrt{n}\over Y}$$ (14)

is distributed as Student's $t$ with $n$ Degrees of Freedom.

The noncentral Student's $t$-distribution is given by

$$P(x) = {n^{n/2} n!\over 2^n e^{\lambda^2/2}\Gamma({\textstyle{1\over 2}}n)}$$

$$\phantom{=} \times \left\{{\sqrt{2}\,\lambda x(n+x^2)^{-(1+n/2)} {}_1F_1\left... ...a^2 x^2\over 2(n+x^2)}}\right)\over \Gamma[{\textstyle{1\over 2}}(1+n)]}\right.$$

$$\phantom{=} \left.{+{e^{(\lambda^2 x^2)/[2(n+x^2)]}\sqrt{\pi}(n+x^2)^{-(n+1)/... ...x^2\over 2(n+x^2)}}\right)} \over \Gamma[{\textstyle{1\over 2}}(1+n)]}\right\},$$ (15)

where $\Gamma(z)$ is the Gamma Function, ${}_1F_1(a;b;z)$ is a Confluent Hypergeometric Function, and $L_n^m(x)$ 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 $n-1$.'' 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.