Student's t-distribution

Student’s t
Probability density function
Cumulative distribution function
Parameters	$\nu$ > 0 degrees of freedom (real)
Support	x ∈ (−∞; +∞)
PDF	$\textstyle\frac{\Gamma \left(\frac{\nu%2B1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1%2B\frac{x^2}{\nu} \right)^{-\frac{\nu%2B1}{2}}\!$
CDF	$\begin{matrix} \frac{1}{2} %2B x \Gamma \left( \frac{\nu%2B1}{2} \right) \cdot\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu%2B1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \end{matrix}$ where ₂F₁ is the hypergeometric function
Mean	0 for $\nu$ > 1, otherwise undefined
Median	0
Mode	0
Variance	$\textstyle\frac{\nu}{\nu-2}$ for $\nu$ > 2, ∞ for 1 < $\nu$ ≤ 2, otherwise undefined
Skewness	0 for $\nu$ > 3
Ex. kurtosis	$\textstyle\frac{6}{\nu-4}$ for $\nu$ > 4
Entropy	$\begin{matrix} \frac{\nu%2B1}{2}\left[ \psi \left(\frac{1%2B\nu}{2} \right) - \psi \left(\frac{\nu}{2} \right) \right] \\[0.5em] %2B \log{\left[\sqrt{\nu}B \left(\frac{\nu}{2},\frac{1}{2} \right)\right]} \end{matrix}$ ψ: digamma function, B: beta function
MGF	undefined
CF	$\textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}\|t\|)(\sqrt{\nu}\|t\| \right)^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}}$ for $\nu$ > 0 $K_{\nu}$ (x): Bessel function^[1]

In probability and statistics, Student’s t-distribution (or simply the t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It plays a role in a number of widely-used statistical analyses, including the Student’s t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student’s t-distribution also arises in the Bayesian analysis of data from a normal family.

The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student’s t-distribution is a special case of the generalised hyperbolic distribution.

Definition

Probability density function

Student's t-distribution has the probability density function given by

$f(t) = \frac{\Gamma(\frac{\nu%2B1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1%2B\frac{t^2}{\nu} \right)^{-\frac{\nu%2B1}{2}},\!$

where $\nu$ is the number of degrees of freedom and $\Gamma$ is the Gamma function. This may also be written as

$f(t) = \frac{1}{\sqrt{\nu}\, B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \left(1%2B\frac{t^2}{\nu} \right)^{-\frac{\nu%2B1}{2}}\!,$

where B is the Beta function.

For $\nu$ even,

$\frac{\Gamma(\frac{\nu%2B1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} = \frac{(\nu -1)(\nu -3)\cdots 5 \cdot 3} {2\sqrt{\nu}(\nu -2)(\nu -4)\cdots 4 \cdot 2\,}.$

For $\nu$ odd,

$\frac{\Gamma(\frac{\nu%2B1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} = \frac{(\nu -1)(\nu -3)\cdots 4 \cdot 2} {\pi \sqrt{\nu}(\nu -2)(\nu -4)\cdots 5 \cdot 3\,}.\!$

The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1.

The following images show the density of the t-distribution for increasing values of $\nu$ . The normal distribution is shown as a blue line for comparison. Note that the t-distribution (red line) becomes closer to the normal distribution as $\nu$ increases.

Density of the t-distribution (red) for 1, 2, 3, 5, 10, and 30 df compared to the standard normal distribution (blue).
Previous plots shown in green.

Cumulative distribution function

The cumulative distribution function can be written in terms of I, the regularized incomplete beta function. For t > 0,^[2]

$\int_{-\infty}^t f(u)\,du = 1- \frac{1}{2} I_{x(t)}\left(\frac{\nu}{2},\frac{1}{2}\right) ,$

with

$x(t) = \frac{\nu}{{t^2%2B\nu}}.$

Other values would be obtained by symmetry. An alternative formula, valid for $t^2 < \nu$ , is^[2]

$\int_{-\infty}^t f(u)\,du = \frac{1}{2} %2B t \frac{\Gamma \left( \frac{\nu%2B1}{2} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \, _2F_1 \left ( \frac{1}{2},\frac{\nu%2B1}{2};\frac{3}{2}; -\frac{t^2}{\nu} \right)$

where ₂F₁ is a particular case of the hypergeometric function.

Special cases

Certain values of $\nu$ give an especially simple form.

$\nu=1$

Distribution function:

$F(x) = \frac{1}{2} %2B \frac{1}{\pi}\arctan(x).$

Density function:

$f(x) = \frac{1}{{\pi}(1%2Bx^2)}.$

See Cauchy distribution

$\nu=2$

Distribution function:

$F(x) = \frac{1}{2}\left[1%2B\frac{x}{\sqrt{2%2Bx^2}}\right].$

Density function:

$f(x) = \frac{1}{\left(2%2Bx^2\right)^{\frac{3}{2}}}.$

$\nu=3$

Density function:

$f(x) = \frac{6\sqrt{3}}{\pi\left(3%2Bx^2\right)^2}.$

How the t-distribution arises

Let x₁, ..., x_n be the numbers observed in a sample from a continuously distributed population with expected value μ. The sample mean and sample variance are respectively

$\bar{x} = \frac{x_1%2B\cdots%2Bx_n}{n}\text{ and }s^2=\frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2.$

The resulting t-value is

$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}.$

(This is not traditionally called a t-statistic unless the value of μ is known. An example is when the value of μ specified by a null hypothesis is used.)

The t-distribution with n − 1 degrees of freedom is the sampling distribution of the t-value when the samples consist of independent identically distributed observations from a normally distributed population.

History and etymology

In statistics, the t-distribution was first derived as a posterior distribution by Helmert^[3]^[4]^[5] and Lüroth.^[6]^[7]^[8] In the English literature, a derivation of the t-distribution was published in 1908 by William Sealy Gosset^[9] while he worked at the Guinness Brewery in Dublin. One version of the origin of the pseudonym Student, is that Gosset's employer forbade members of its staff from publishing scientific papers, so he had to hide his identity. Another version is that Guinness did not want their competition to know that they were using the t-test to test the quality of raw material.^[10] The t-test and the associated theory became well-known through the work of R.A. Fisher, who called the distribution "Student's distribution".^[11]^[12]

Characterization

As the distribution of a test statistic

Student's t-distribution can be defined as being the distribution of the random variable which is the ratio, T, defined as^[2]^[13]

$T=\frac{Z}{\sqrt{V/\nu}} = Z \sqrt{\frac{\nu}{V}} ,$

where

Z is normally distributed with expected value 0 and variance 1;
V has a chi-squared distribution with $\nu$ ("nu") degrees of freedom;
Z and V are independent.

A different distribution is defined as that of the random variable defined, for a given constant μ, by $\left(Z%2B\mu\right)\sqrt{\frac{\nu}{V}}$ . This random variable has a noncentral t-distribution with noncentrality parameter μ. This distribution is important in studies of the power of Student's t test.

Derivation

Suppose X₁, ..., X_n are independent values that are normally distributed with expected value μ and variance σ². Let

$\overline{X}_n = \frac{1}{n}(X_1%2B\cdots%2BX_n)$

be the sample mean, and

$S_n^{\;2} = \frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2$

be an unbiased estimate of the variance from the sample. It can be shown that the random variable

$V = (n-1)\frac{S_n^2}{\sigma^2}$

has a chi-squared distribution with n − 1 degrees of freedom (by Cochran's theorem). It is readily shown that the quantity

$Z = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{\sigma}$

is normally distributed with mean 0 and variance 1, since the sample mean $\scriptstyle \overline{X}_n$ is normally distributed with mean $\mu$ and variance $\scriptstyle\frac{\sigma^2}{n}$ . Moreover, it is possible to show that these two random variables (the normally distributed one and the chi-squared-distributed one) are independent. Consequently the pivotal quantity,

$T \equiv \frac{Z}{\sqrt{V/\nu}} = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{S_n},$

which differs from Z in that the exact standard deviation σ is replaced by the random variable S_n, has a Student's t-distribution as defined above. Notice that the unknown population variance σ² does not appear in T, since it was in both the numerator and the denominators, so it canceled. Gosset's work showed that T has the probability density function stated above, with $\nu$ equal to n − 1.

The distribution of the test statistic, T, depends on $\nu$ , but not μ or σ; the lack of dependence on μ and σ is what makes the t-distribution important in both theory and practice.

As a maximum entropy distribution

Student's t-distribution is the maximum entropy probability distribution for a random variate X for which $E(\ln(\nu%2BX^2))$ is fixed.^[14]

Properties

Moments

The moments of the t-distribution are

$E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0<k< \nu\\ \frac{1}{\sqrt{\pi}\Gamma\left(\frac{\nu}{2}\right)}\left[\Gamma\left(\frac{k%2B1}{2}\right)\Gamma\left(\frac{\nu-k}{2}\right)\nu^{\frac{k}{2}}\right] & k \text{ even}, \quad 0<k< \nu\\ \text{undefined} & k \text{ odd},\quad 0<\nu\leq k\\ \infty & k\text{ even},\quad 0<\nu\leq k \end{cases}$

It should be noted that the term for 0 < k < $\nu$ , k even, may be simplified using the properties of the Gamma function to

$E(T^k)= \prod_{i=1}^{\frac{k}{2}} \frac{2i-1}{\nu - 2i}\nu^{\frac{k}{2}} \qquad k\text{ even},\quad 0<k<\nu.$

For a t-distribution with $\nu$ degrees of freedom, the expected value is 0, and its variance is $\nu$ /( $\nu$ − 2) if $\nu$ > 2. The skewness is 0 if $\nu$ > 3 and the excess kurtosis is 6/( $\nu$ − 4) if $\nu$ > 4.

Relation to F distribution

$Y \sim \mathrm{F}(\nu_1 = 1, \nu_2 = \nu)$ has an F-distribution if $Y = X^2\!$ and $X \sim \mathrm{t}(\nu)\!$ has a Student's t-distribution.

Monte Carlo sampling

There are various approaches to constructing random samples from the Student-t distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples; e.g., in the multi-dimensional applications basis of copula-dependency. In the case of stand-alone sampling, an extension of the Box–Muller method and its polar variation is easily deployed.^[15] It has the merit that it applies equally well to all real positive and negative degrees of freedom.

Integral of Student's probability density function and p-value

The function $\scriptstyle A(t|\nu)$ is the integral of Student's probability density function, ƒ(t) between −t and t. It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function $\scriptstyle A(t|\nu)$ can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of t and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in t-tests. For the statistic t, with $\scriptstyle\nu$ degrees of freedom, $\scriptstyle A(t|\nu)$ is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t > 0). It is defined for real t by the following formula:

$A(t|\nu) = \Pr \left \{ \left | \frac{Z}{\sqrt{V/\nu}} \right | \le t \right \} = \frac{1}{\sqrt{\nu} B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \int\limits_{-t}^t \left (1%2B\frac{x^2}{\nu}\right )^{-\frac{\nu %2B1}{2} }\, dx$

where B is the Beta function. For t > 0, there is a relation to the regularized incomplete beta function I_x(a, b) as follows:

$A(t|\nu) = 1 - I_{\frac{\nu}{\nu %2Bt^2}}\left(\frac{\nu}{2},\frac{1}{2}\right).$

For statistical hypothesis testing this function is used to construct the p-value.

Related distributions

Three-parameter version

Student's t distribution can be generalized to a three parameter location/scale family^[16] that introduces a location parameter $\mu$ and a squared inverse scale parameter (i.e. precision) $\lambda$ , and has a density defined by

$p(x|\mu,\lambda,\nu) = \frac{\Gamma(\frac{\nu %2B 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left[1%2B\frac{\lambda(x-\mu)^2}{\nu}\right]^{-\frac{\nu%2B1}{2}} .$

Other properties of this version of the distribution are^[16]:

$\begin{align} \operatorname{E}(X) &= \mu \quad \quad \quad \text{for }\,\nu > 1 ,\\ \text{var}(X) &= \frac{1}{\lambda}\frac{\nu}{\nu-2}\, \quad \text{for }\,\nu > 2 ,\\ \text{mode}(X) &= \mu . \end{align}$

This distribution results from compounding a Gaussian distribution with mean $\mu$ and unknown precision (the reciprocal of the variance), with a gamma distribution with parameters $a = \nu/2$ and $b = \nu/2\lambda$ . In other words, the random variable X is assumed to have a normal distribution with an unknown precision distributed as gamma, and then this is marginalized over the gamma distribution. (The reason for the usefulness of this characterization is that the gamma distribution is the conjugate prior distribution of the precision of a Gaussian distribution. As a result, the three-parameter Student's t distribution arises naturally in many Bayesian inference problems.)

The noncentral t-distribution is a different way of generalizing the t-distribution to include a location parameter.

Discrete version

The "discrete Student's t distribution" is defined by its probability mass function at r being proportional to^[17]

$\prod_{j=1}^k \frac{1}{(r%2Bj%2Ba)^2%2Bb^2} \quad \quad r=\ldots, -1, 0, 1, \ldots .$

Here a, b, and k are parameters. This distribution arises from the construction of a system of discrete distributions similar to that of the Pearson distributions for continuous distributions.^[18]

Uses

In frequentist statistical inference

Student's t-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the t-distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of the t-distribution.

Confidence intervals and hypothesis tests are two statistical procedures in which the quantiles of the sampling distribution of a particular statistic (e.g. the standard score) are required. In any situation where this statistic is a linear function of the data, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's t-distribution. Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form.

Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student's t-distribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.

Hypothesis testing

A number of statistics can be shown to have t-distributions for samples of moderate size under null hypotheses that are of interest, so that the t-distribution forms the basis for significance tests. For example, the distribution of Spearman's rank correlation coefficient ρ, in the null case (zero correlation) is well approximated by the t distribution for sample sizes above about 20 .

Confidence intervals

Suppose the number A is so chosen that

$\Pr(-A < T < A)=0.9,\,$

when T has a t-distribution with n − 1 degrees of freedom. By symmetry, this is the same as saying that A satisfies

$\Pr(T < A) = 0.95,\,$

so A is the "95th percentile" of this probability distribution, or $A=t_{(0.05,n-1)}$ . Then

$\Pr \left (-A < {\overline{X}_n - \mu \over S_n/\sqrt{n}} < A \right)=0.9,$

and this is equivalent to

$\Pr\left(\overline{X}_n - A{S_n \over \sqrt{n}} < \mu < \overline{X}_n %2B A{S_n \over \sqrt{n}}\right) = 0.9.$

Therefore the interval whose endpoints are

$\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}$

is a 90-percent confidence interval for μ. Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the t-distribution to examine whether the confidence limits on that mean include some theoretically predicted value - such as the value predicted on a null hypothesis.

It is this result that is used in the Student's t-tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the t-distribution can be used to examine whether that difference can reasonably be supposed to be zero.

If the data are normally distributed, the one-sided (1 − a)-upper confidence limit (UCL) of the mean, can be calculated using the following equation:

$\mathrm{UCL}_{1-a} = \overline{X}_n %2B t_{a,n-1}\frac{S_n}{\sqrt{n}}.$

The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, $\overline{X}_n$ being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL_1−a is equal to the confidence level 1 − a.

Prediction intervals

The t-distribution can be used to construct a prediction interval for an unobserved sample from a normal distribution with unknown mean and variance.

Robust parametric modeling

The t-distribution is often used as an alternative to the normal distribution as a model for data.^[19] It is frequently the case that real data have heavier tails than the normal distribution allows for. The classical approach was to identify outliers and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the t-distribution is a natural choice of model for such data and provides a parametric approach to robust statistics.

Lange et al. explored the use of the t-distribution for robust modeling of heavy tailed data in a variety of contexts. A Bayesian account can be found in Gelman et al. The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.

Table of selected values

Most statistical textbooks list t distribution tables. Nowadays, the better way to a fully precise critical t value or a cumulative probability is the statistical function implemented in spreadsheets (Office Excel, OpenOffice Calc, etc.), or an interactive calculating web page. The relevant spreadsheet functions are TDIST and TINV, while online calculating pages save troubles like positions of parameters or names of functions. For example, a Mediawiki page supported by R extension can easily give the interactive result of critical values or cumulative probability, even for noncentral t-distribution.

The following table lists a few selected values for t-distributions with $\nu$ degrees of freedom for a range of one-sided or two-sided critical regions. For an example of how to read this table, take the fourth row, which begins with 4; that means $\nu$ , the number of degrees of freedom, is 4 (and if we are dealing, as above, with n values with a fixed sum, n = 5). Take the fifth entry, in the column headed 95% for one-sided (90% for two-sided). The value of that entry is "2.132". Then the probability that T is less than 2.132 is 95% or Pr(−∞ < T < 2.132) = 0.95; or mean that Pr(−2.132 < T < 2.132) = 0.9.

This can be calculated by the symmetry of the distribution,

Pr(T < −2.132) = 1 − Pr(T > −2.132) = 1 − 0.95 = 0.05,

and so

Pr(−2.132 < T < 2.132) = 1 − 2(0.05) = 0.9.

Note that the last row also gives critical points: a t-distribution with infinitely-many degrees of freedom is a normal distribution. (See Related distributions above).

The first column is the number of degrees of freedom.

One Sided	75%	80%	85%	90%	95%	97.5%	99%	99.5%	99.75%	99.9%	99.95%
Two Sided	50%	60%	70%	80%	90%	95%	98%	99%	99.5%	99.8%	99.9%
1	1.000	1.376	1.963	3.078	6.314	12.71	31.82	63.66	127.3	318.3	636.6
2	0.816	1.061	1.386	1.886	2.920	4.303	6.965	9.925	14.09	22.33	31.60
3	0.765	0.978	1.250	1.638	2.353	3.182	4.541	5.841	7.453	10.21	12.92
4	0.741	0.941	1.190	1.533	2.132	2.776	3.747	4.604	5.598	7.173	8.610
5	0.727	0.920	1.156	1.476	2.015	2.571	3.365	4.032	4.773	5.893	6.869
6	0.718	0.906	1.134	1.440	1.943	2.447	3.143	3.707	4.317	5.208	5.959
7	0.711	0.896	1.119	1.415	1.895	2.365	2.998	3.499	4.029	4.785	5.408
8	0.706	0.889	1.108	1.397	1.860	2.306	2.896	3.355	3.833	4.501	5.041
9	0.703	0.883	1.100	1.383	1.833	2.262	2.821	3.250	3.690	4.297	4.781
10	0.700	0.879	1.093	1.372	1.812	2.228	2.764	3.169	3.581	4.144	4.587
11	0.697	0.876	1.088	1.363	1.796	2.201	2.718	3.106	3.497	4.025	4.437
12	0.695	0.873	1.083	1.356	1.782	2.179	2.681	3.055	3.428	3.930	4.318
13	0.694	0.870	1.079	1.350	1.771	2.160	2.650	3.012	3.372	3.852	4.221
14	0.692	0.868	1.076	1.345	1.761	2.145	2.624	2.977	3.326	3.787	4.140
15	0.691	0.866	1.074	1.341	1.753	2.131	2.602	2.947	3.286	3.733	4.073
16	0.690	0.865	1.071	1.337	1.746	2.120	2.583	2.921	3.252	3.686	4.015
17	0.689	0.863	1.069	1.333	1.740	2.110	2.567	2.898	3.222	3.646	3.965
18	0.688	0.862	1.067	1.330	1.734	2.101	2.552	2.878	3.197	3.610	3.922
19	0.688	0.861	1.066	1.328	1.729	2.093	2.539	2.861	3.174	3.579	3.883
20	0.687	0.860	1.064	1.325	1.725	2.086	2.528	2.845	3.153	3.552	3.850
21	0.686	0.859	1.063	1.323	1.721	2.080	2.518	2.831	3.135	3.527	3.819
22	0.686	0.858	1.061	1.321	1.717	2.074	2.508	2.819	3.119	3.505	3.792
23	0.685	0.858	1.060	1.319	1.714	2.069	2.500	2.807	3.104	3.485	3.767
24	0.685	0.857	1.059	1.318	1.711	2.064	2.492	2.797	3.091	3.467	3.745
25	0.684	0.856	1.058	1.316	1.708	2.060	2.485	2.787	3.078	3.450	3.725
26	0.684	0.856	1.058	1.315	1.706	2.056	2.479	2.779	3.067	3.435	3.707
27	0.684	0.855	1.057	1.314	1.703	2.052	2.473	2.771	3.057	3.421	3.690
28	0.683	0.855	1.056	1.313	1.701	2.048	2.467	2.763	3.047	3.408	3.674
29	0.683	0.854	1.055	1.311	1.699	2.045	2.462	2.756	3.038	3.396	3.659
30	0.683	0.854	1.055	1.310	1.697	2.042	2.457	2.750	3.030	3.385	3.646
40	0.681	0.851	1.050	1.303	1.684	2.021	2.423	2.704	2.971	3.307	3.551
50	0.679	0.849	1.047	1.299	1.676	2.009	2.403	2.678	2.937	3.261	3.496
60	0.679	0.848	1.045	1.296	1.671	2.000	2.390	2.660	2.915	3.232	3.460
80	0.678	0.846	1.043	1.292	1.664	1.990	2.374	2.639	2.887	3.195	3.416
100	0.677	0.845	1.042	1.290	1.660	1.984	2.364	2.626	2.871	3.174	3.390
120	0.677	0.845	1.041	1.289	1.658	1.980	2.358	2.617	2.860	3.160	3.373
$\infty$	0.674	0.842	1.036	1.282	1.645	1.960	2.326	2.576	2.807	3.090	3.291

The number at the beginning of each row in the table above is $\nu$ which has been defined above as n − 1. The percentage along the top is 100%(1 − α). The numbers in the main body of the table are t_α,. If a quantity T is distributed as a Student's t distribution with $\nu$ degrees of freedom, then there is a probability 1 − α that T will be less than t_α,.(Calculated as for a one-tailed or one-sided test as opposed to a two-tailed test.)

For example, given a sample with a sample variance 2 and sample mean of 10, taken from a sample set of 11 (10 degrees of freedom), using the formula

$\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}.$

We can determine that at 90% confidence, we have a true mean lying below

$10%2B1.37218 \frac{\sqrt{2}}{\sqrt{11}}=10.58510.$

(In other words, on average, 90% of the times that an upper threshold is calculated by this method, this upper threshold exceeds the true mean.) And, still at 90% confidence, we have a true mean lying over

$10-1.37218 \frac{\sqrt{2}}{\sqrt{11}}=9.41490.$

(In other words, on average, 90% of the times that a lower threshold is calculated by this method, this lower threshold lies below the true mean.) So that at 80% confidence (calculated from 1 − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval

$\left(10-1.37218 \frac{\sqrt{2}}{\sqrt{11}}, 10%2B1.37218 \frac{\sqrt{2}}{\sqrt{11}}\right) = \left(9.41490, 10.58510\right).$

This is generally expressed in interval notation, e.g., for this case, at 80% confidence the true mean is within the interval [9.41490, 10.58510].

(In other words, on average, 80% of the times that upper and lower thresholds are calculated by this method, the true mean is both below the upper threshold and above the lower threshold. This is not the same thing as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method—see confidence interval and prosecutor's fallacy.)

For information on the inverse cumulative distribution function see Quantile function.

Notes

^ Hurst, Simon, The Characteristic Function of the Student-t Distribution, Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95
^ ^a ^b ^c Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition. Wiley, ISBN 0-471-58494-0 (Chapter 28)
^ Helmert, F. R. (1875). "Über die Bestimmung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler". Z. Math. Phys., 20, 300-3.
^ Helmert, F. R. (1876a). "Über die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und uber einige damit in Zusammenhang stehende Fragen". Z. Math. Phys., 21, 192-218.
^ Helmert, F. R. (1876b). "Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers director Beobachtungen gleicher Genauigkeit", Astron. Nachr., 88, 113-32.
^ Lüroth, J (1876). "Vergleichung von zwei Werten des wahrscheinlichen Fehlers". Astron. Nachr. 87 (14): 209–20. Bibcode 1876AN.....87..209L. doi:10.1002/asna.18760871402.
^ Pfanzagl, J.; Sheynin, O. (1996). "A forerunner of the t-distribution (Studies in the history of probability and statistics XLIV)". Biometrika 83 (4): 891–898. doi:10.1093/biomet/83.4.891. MR 1766040. http://biomet.oxfordjournals.org/cgi/content/abstract/83/4/891.
^ Sheynin, O (1995). "Helmert's work in the theory of errors". Arch. Hist. Ex. Sci. 49: 73–104. doi:10.1007/BF00374700.
^ Student [William Sealy Gosset] (March 1908). "The probable error of a mean". Biometrika 6 (1): 1–25. doi:10.1093/biomet/6.1.1. http://www.york.ac.uk/depts/maths/histstat/student.pdf.
^ Mortimer, Robert G. (2005) Mathematics for Physical Chemistry, Academic Press. 3 edition. ISBN 0125083475 (page 326)
^ Fisher, R. A. (1925). "Applications of "Student's" distribution". Metron 5: 90–104. http://digital.library.adelaide.edu.au/coll/special/fisher/43.pdf.
^ Walpole, Ronald; Myers, Raymond; Myers, Sharon; Ye, Keying. (2002) Probability and Statistics for Engineers and Scientists. Pearson Education, 7th edition, pg. 237 ISBN 8177584049
^ Hogg & Craig (1978, Sections 4.4 and 4.8.)
^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics (Elsevier): 219–230. http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf. Retrieved 2011-06-02.
^ Bailey, R. W. (1994). "Polar Generation of Random Variates with the t-Distribution". Mathematics of Computation 62 (206): 779–781. doi:10.2307/2153537.
^ ^a ^b Bishop, C.M. (2006). Pattern recognition and machine learning. Springer.
^ Ord, J.K. (1972) Families of Frequency Distributions, Griffin. ISBN 0-85264-137-0 (Table 5.1)
^ Ord, J.K. (1972) Families of Frequency Distributions, Griffin. ISBN 0-85264-137-0 (Chapter 5)
^ Lange, Kenneth L.; Little, Roderick J.A.; Taylor, Jeremy M.G. (1989). "Robust statistical modeling using the t-distribution". JASA 84 (408): 881–896. JSTOR 2290063.

References

Senn, S.; Richardson, W. (1994). "The first t-test". Statistics in Medicine 13 (8): 785–803. doi:10.1002/sim.4780130802. PMID 8047737.
Hogg, R.V.; Craig, A.T. (1978). Introduction to Mathematical Statistics. New York: Macmillan.
Venables, W.N.; B.D. Ripley, B.D. (2002)Modern Applied Statistics with S, Fourth Edition, Springer
Gelman, Andrew; John B. Carlin, Hal S. Stern, Donald B. Rubin (2003). Bayesian Data Analysis (Second Edition). CRC/Chapman & Hall. ISBN 1-584-88388-X. http://www.stat.columbia.edu/~gelman/book/.

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Student's t-distribution

Contents

Definition

Probability density function

Cumulative distribution function

Special cases

How the t-distribution arises

History and etymology

Characterization

As the distribution of a test statistic

Derivation

As a maximum entropy distribution

Properties

Moments

Relation to F distribution

Monte Carlo sampling

Integral of Student's probability density function and p-value

Related distributions

Three-parameter version

Discrete version

Uses

In frequentist statistical inference

Hypothesis testing

Confidence intervals

Prediction intervals

Robust parametric modeling

Table of selected values

See also

Notes

References

External links