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Student's t-Distribution

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StudentsTDistribution

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 x_i, let

t=(x^_-mu)/(s/sqrt(N)),
(1)

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

s^2=1/(N-1)sum_(i=1)^N(x_i-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.

The Student's t-distribution with n degrees of freedom is implemented in the Wolfram Language as StudentTDistribution[n].

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 can be derived by transforming Student's z-distribution using

z=(x^_-mu)/s,
(3)

and then defining

t=zsqrt(n-1).
(4)

The resulting probability and cumulative distribution functions are

f_r(t)=(Gamma[1/2(r+1)])/(sqrt(rpi)Gamma(1/2r)(1+(t^2)/r)^((r+1)/2))
(5)
=((r/(r+t^2))^((1+r)/2))/(sqrt(r)B(1/2r,1/2))
(6)
F_r(t)=1/2+1/2[I(1;1/2r,1/2)-I(r/(r+t^2),1/2r,1/2)]sgn(t)
(7)
=1/2-(itB(-(t^2)/r;1/2,1/2(1-r))Gamma(1/2(r+1)))/(2sqrt(pi)|t|Gamma(1/2r))
(8)
=1/2+(tGamma(1/2(r+1))_2F_1(1/2,1/2(r+1);3/2;-(t^2)/r))/(sqrt(pir)Gamma(1/2r)),
(9)

where

r=n-1
(10)

is the number of degrees of freedom, -infty<t<infty, Gamma(z) is the gamma function, B(a,b) is the beta function, _2F_1(a,b;c;z) is a hypergeometric function, and I(z;a,b) is the regularized beta function defined by

I(z;a,b)=(B(z;a,b))/(B(a,b)).
(11)

The mean, variance, skewness, and kurtosis excess of Student's t-distribution are

mu=0
(12)
sigma^2=r/(r-2)
(13)
gamma_1=0
(14)
gamma_2=6/(r-4).
(15)
StudentsTCharacteristics

The characteristic functions phi_n(t) for the first few values of n are

phi_1(t)=e^(-|t|)
(16)
phi_2(t)=sqrt(2)|t|K_1(sqrt(2)|t|)
(17)
phi_3(t)=e^(-sqrt(3)|t|)(1+sqrt(3)|t|)
(18)
phi_4(t)=2t^2K_2(2|t|)
(19)
phi_5(t)=1/3e^(-sqrt(5)|t|)(3+3sqrt(5)|t|+5t^2),
(20)

and so on, where K_n(x) is a modified Bessel function of the second kind.

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

r90%95%97.5%99.5%
13.077686.3137512.706263.6567
21.885622.919994.302659.92484
31.637742.353363.182455.84091
41.533212.131852.776454.60409
51.475882.015052.570584.03214
101.372181.812462.228143.16927
301.310421.697262.042272.75000
1001.290071.660231.983972.62589
infty1.281561.644871.959992.57588

A multivariate form of the Student's t-distribution with correlation matrix r and m degrees of freedom is implemented as MultivariateTDistribution[r, m] in the Wolfram Language package MultivariateStatistics` .

The so-called A(t|n) distribution is useful for testing if two observed distributions have the same mean. A(t|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|n)=1/(sqrt(n)B(1/2,1/2n))int_(-t)^t(1+(x^2)/n)^(-(1+n)/2)dx.
(21)

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=(Xsqrt(n))/Y
(22)

is distributed as Student's t with n degrees of freedom.

See also

Bessel's Statistical Formula, Noncentral Student's t-Distribution, Paired t-Test, Student's z-Distribution

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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 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.Shaw, W. "New Methods for Managing 'Student's' T-Distribution." Submitted to J. Comput. Finance. http://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.

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Student's t-Distribution

Cite this as:

Weisstein, Eric W. "Student's t-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Studentst-Distribution.html

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