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Student t-distribution

Calculates cumulative distribution function value and probability density function value for Student t-distribution. Quantile calculator evaluates Student quantiles for given probability and specified number of degrees of freedom.

Student t-distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small and population deviation is unknown. William Sealy Gosset developed the distribution in the beginning of XX century, who published his works under the pseudonym Student.

Probability density function

Probability density function has the following form:
$f(t) = \frac{\Gamma(\frac{n+1}{2})} {\sqrt{n\pi}\,\Gamma(\frac{n}{2})} \left(1+\frac{t^2}{n} \right)^{\!-\frac{n+1}{2}},\!$
where n - is degrees of freedom and $\Gamma$ - Gamma function

Cumulative distribution function

Cumulative distribution function can be expressed using Gamma and hypergeometric function:
$\tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}(n+1) \right)} {\sqrt{\pi n}\,\Gamma \left(\tfrac{n}{2}\right)} {}_2F_1 \left ( \tfrac{1}{2},\tfrac{1}{2}(n+1); \tfrac{3}{2}; -\tfrac{t^2}{n} \right)$

Stdent t-distribution

Digits after the decimal point: 5
Probability density function value

Cumulative distribution function value

PDF Graph
CDF Graph

Quantile function

$\alpha$-quantile Student is a number $t_{\alpha,n}$ which conforms to $F_n\left(t_{\alpha,n}\right) = 1- \alpha$, where Fn - Student-t cumulative distribution function.
Inverse cumulative distribution function (quantile function) doesn't have simple form, commonly we use pre-calculated values from the tables published by Gosset and other researchers.

The following calculator approximates quantile function value with aid of jStat statistics package:

Student t-distribution quantile function

Digits after the decimal point: 2
Quantile

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PLANETCALC, Student t-distribution