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{{About|the mathematics of Student's ''t''-distribution|its uses in statistics|Student's t-test}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{More footnotes|date=February 2010}}
{{DISPLAYTITLE:Student's ''t''-distribution}}
{{Probability distribution |
  name      =Student's ''t''|
  type      =density|
  pdf_image  =[[Image:student t pdf.svg|325px]]|
  cdf_image  =[[Image:student t cdf.svg|325px]]|
  parameters =ν > 0 [[degrees of freedom (statistics)|degrees of freedom]] ([[Real number|real]])|
  support    =''x'' ∈ (−∞; +∞)|
  pdf        =<math>\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!</math>|
<!--For "<math>":
  --  Split formulas by "\begin{matrix}" with "\\[0.5em]" split --
  --  as 0.5em interline spacing; end with "\end{matrix}".      --
  --  Fractions have 2 brace-pairs. A centered dot is "\cdot". -->
  cdf        =<math>\begin{matrix}
    \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \times\\[0.5em]
    \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2};
          -\frac{x^2}{\nu} \right)}
    {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)}
    \end{matrix}</math><br/
    >where <sub>2</sub>''F''<sub>1</sub> is the [[hypergeometric function]]|
  mean      =0 for ν > 1, otherwise [[indeterminate form|undefined]]|
  median    =0|
  mode      =0|
  variance  =<math>\textstyle\frac{\nu}{\nu-2}</math> for ν > 2, ∞ for 1 < ν ≤ 2, otherwise [[indeterminate form|undefined]]|
  skewness  =0 for ν > 3, otherwise [[indeterminate form|undefined]]|
  kurtosis  =<math>\textstyle\frac{6}{\nu-4}</math> for ν > 4, ∞ for 2 < ν ≤ 4, otherwise [[indeterminate form|undefined]]|
  entropy    =<math>\begin{matrix}
        \frac{\nu+1}{2}\left[
            \psi \left(\frac{1+\nu}{2} \right)
              - \psi \left(\frac{\nu}{2} \right)
        \right] \\[0.5em]
+ \log{\left[\sqrt{\nu}B \left(\frac{\nu}{2},\frac{1}{2} \right)\right]}
\end{matrix}</math>
* ψ: [[digamma function]],
* ''B'': [[beta function]]|
|
  mgf        = undefined|
  char      =<math>\textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}|t|\right)
                    \cdot \left(\sqrt{\nu}|t| \right)^{\nu/2}}
                    {\Gamma(\nu/2)2^{\nu/2-1}}</math> for ν > 0
* ''K''<sub>ν</sub>(''x''): [[Bessel function|Modified Bessel function of the second kind]]<ref>Hurst, Simon. ''[http://wwwmaths.anu.edu.au/research.reports/srr/95/044/ The Characteristic Function of the Student-t Distribution]'', Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95{{dead link|date=April 2013}}</ref>
}}
In [[probability]] and [[statistics]], '''Student's  ''t''-distribution''' (or simply the '''''t''-distribution''') is a family of continuous [[probability distribution]]s that arise when estimating the [[expected value|mean]] of a [[normal distribution|normally distributed]] [[Statistical population|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|Student's ''t''-test]] for assessing the [[statistical significance]] of the difference between two sample [[mean]]s, the construction of [[confidence interval]]s 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.


If we take a sample of ''n = ν+1'' observations from a normal distribution (the black curve on the figure on the right of this page, representing a very large ''ν''), compute the sample mean and plot it, and repeat this process infinitely many times (for the same ''n''), we get the probability density function for that ''n'', as shown in the image on the right.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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If we also compute the [[Sample variance#Population variance and sample variance|sample variance]] for these ''n'' observations, then the ''t''-distribution (for ''n-1'') can be defined as the distribution of the location of the true mean, relative to the sample mean and divided by the sample standard deviation, after multiplying by the normalizing term <math>\sqrt{n}</math>, where ''n'' is the sample size. In this way, the ''t''-distribution can be used to estimate how likely it is that the true mean lies in any given range.
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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]].
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==History and etymology==
<!--'''PNG'''  (currently default in production)
In statistics, the ''t''-distribution was first derived as a [[posterior distribution]] in 1876 by [[Friedrich Robert Helmert|Helmert]]<ref name=HFR1/><ref name=HFR2/><ref name=HFR3/> and [[Jacob Lüroth|Lüroth]].<ref name=L1876/><ref>{{Cite journal|first1=J.|last1=Pfanzagl| first2=O.|last2=Sheynin | title=A forerunner of the ''t''-distribution (Studies in the history of probability and statistics XLIV) | year=1996 | journal=Biometrika | volume=83 | issue=4 |pages=891–898 | doi=10.1093/biomet/83.4.891 <!-- abstract=The t-distribution first occurred in a paper by Lüroth (1876) on the classical theory of errors in connection with a Bayesian result --> |mr=1766040}}</ref><ref>{{Cite journal| doi=10.1007/BF00374700 | last=Sheynin | first=O. | year=1995 | title=Helmert's work in the theory of errors | journal=Arch. Hist. Ex. Sci. | volume=49 | pages=73–104}}</ref>
:<math forcemathmode="png">E=mc^2</math>


In the English-language literature it takes its name from [[William Sealy Gosset]]'s 1908 paper in [[Biometrika]] under the pseudonym "Student".<ref>{{Cite journal|author="Student" <nowiki>[</nowiki>[[William Sealy Gosset]]<nowiki>]</nowiki>|date=March 1908 |title=The probable error of a mean |journal=[[Biometrika]] |volume=6 |issue=1 |pages=1–25 |url=http://www.york.ac.uk/depts/maths/histstat/student.pdf |doi=10.1093/biomet/6.1.1}}</ref><ref>"Student" (William Sealy Gosset), original Biometrika paper as a [http://www.atmos.washington.edu/~robwood/teaching/451/student_in_biometrika_vol6_no1.pdf scan]</ref> Gosset worked at the [[St. James's Gate Brewery|Guinness Brewery]] in [[Dublin, Ireland]], and was interested in the problems of small samples, for example of the chemical properties of barley where sample sizes might be as low as 3. One version of the origin of the pseudonym 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 competitors to know that they were using the ''t''-test to test the quality of raw material.<ref>Mortimer, Robert G. (2005)  ''Mathematics for Physical Chemistry'', Academic Press. 3 edition. ISBN 0-12-508347-5 (page 326)</ref>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population".  It became well-known through the work of [[Ronald A. Fisher]], who called the distribution "Student's distribution" and referred to the value as ''t''.<ref name="Fisher 1925 90–104">{{Cite journal|last=Fisher |first=R. A. |authorlink=Ronald Fisher |year=1925 |title=Applications of "Student's" distribution |journal=Metron |volume=5 |pages=90–104 |url=http://www.sothis.ro/user/content/4ef6e90670749a86-student_distribution_1925.pdf}}</ref><ref>Walpole, Ronald; Myers, Raymond; Myers, Sharon; Ye, Keying. (2002) ''Probability and Statistics for Engineers and Scientists''. Pearson Education, 7th edition, pg. 237 ISBN 81-7758-404-9</ref>
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==Definition==
==Demos==


===Probability density function===
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
Student's '''''t''-distribution''' has the [[probability density function]] given by


:<math>f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\!</math>


where <math>\nu</math> is the number of ''[[degrees of freedom (statistics)|degrees of freedom]]'' and <math>\Gamma</math> is the [[gamma function]]. This may also be written as
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:<math>f(t) = \frac{1}{\sqrt{\nu}\, B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!,</math>
==Test pages ==


where ''B'' is the [[Beta function]].
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For <math>\nu</math> even,
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
: <math>\frac{\Gamma(\frac{\nu+1}{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\,}. </math>
==Bug reporting==
For <math>\nu</math> odd,
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
: <math>\frac{\Gamma(\frac{\nu+1}{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\,}.\!</math>
 
The probability density function is [[Symmetric distribution|symmetric]], and its overall shape resembles the bell shape of a [[normal distribution|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 <math>\nu</math>.  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 <math>\nu</math> increases.
 
{| align="center"
|+ Density of the ''t''-distribution (red) for 1, 2, 3, 5, 10, and 30 degrees of freedom compared to the standard normal distribution (blue).<br>Previous plots shown in green.
|-
| [[Image:T distribution 1df enhanced.svg|thumb|center|240x240px|alt=1df|1 degree of freedom]]
| [[Image:T distribution 2df enhanced.svg|thumb|center|240x240px|alt=2df|2 degrees of freedom]]
| [[Image:T distribution 3df enhanced.svg|thumb|center|240x240px|alt=3df|3 degrees of freedom]]
|-
| [[Image:T distribution 5df enhanced.svg|thumb|center|240x240px|alt=5df|5 degrees of freedom]]
| [[Image:T distribution 10df enhanced.svg|thumb|center|240x240px|alt=10df|10 degrees of freedom]]
| [[Image:T distribution 30df enhanced.svg|thumb|center|240x240px|alt=30df|30 degrees of freedom]]
|}
 
===Cumulative distribution function===
The [[cumulative distribution function]] can be written in terms of ''I'', the regularized
[[incomplete beta function]]. For ''t''&nbsp;>&nbsp;0,<ref name=JKB/>
 
:<math>F(t) = \int_{-\infty}^t f(u)\,du = 1- \tfrac{1}{2} I_{x(t)}\left(\tfrac{\nu}{2}, \tfrac{1}{2}\right),</math>
 
with
 
:<math>x(t) = \frac{\nu}{{t^2+\nu}}.</math>
 
Other values would be obtained by symmetry. An alternative formula, valid for ''t''<sup>2</sup> < ν, is<ref name=JKB/>
 
:<math>\int_{-\infty}^t f(u)\,du =\tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}(\nu+1) \right)} {\sqrt{\pi\nu}\,\Gamma \left(\tfrac{\nu}{2}\right)}  {}_2F_1 \left ( \tfrac{1}{2},\tfrac{1}{2}(\nu+1); \tfrac{3}{2};  -\tfrac{t^2}{\nu} \right)</math>
 
where <sub>2</sub>''F''<sub>1</sub> is a particular case of the [[hypergeometric function]].
 
===Special cases===
Certain values of ν give an especially simple form.
 
*ν = 1
 
:Distribution function:
 
::<math>F(x) = \tfrac{1}{2} + \tfrac{1}{\pi}\arctan(x).</math>
 
:Density function:
 
::<math>f(x) =  \frac{1}{\pi (1+x^2)}.</math>
 
:See [[Cauchy distribution]]
 
*ν = 2
:Distribution function:
 
::<math>F(x) = \tfrac{1}{2}+\frac{x}{2\sqrt{2+x^2}}.</math>
 
:Density function:
 
::<math>f(x) = \frac{1}{\left(2+x^2\right)^{\frac{3}{2}}}.</math>
 
*ν = 3
 
:Density function:
 
::<math>f(x) = \frac{6\sqrt{3}}{\pi\left(3+x^2\right)^2}.</math>
 
*ν = ∞
 
:Density function:
 
::<math>f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}.</math>
 
:See [[Normal distribution]]
 
==How the ''t''-distribution arises==
 
===Sampling distribution===
Let ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> be the numbers observed in a sample from a continuously distributed population with expected value μ. The sample mean and [[sample variance]] are given by:
 
<math>
\begin{align}
\bar{x} &= \frac{x_1+\cdots+x_n}{n} \\
s^2 &= \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2
\end{align}
</math>
 
The resulting ''t-value'' is
 
: <math> t = \frac{\bar{x} - \mu}{s/\sqrt{n}}. </math>
 
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 [[normal distribution|normally distributed]] population. Thus for inference purposes ''t'' is a useful "[[pivotal quantity]]" in the case when the mean and variance (μ, σ<sup>2</sup>) are unknown population parameters, in the sense that the ''t''-value has then a probability distribution that depends on neither μ nor σ<sup>2</sup>.
 
<span id="Bayesiantdistribution">
 
===Bayesian inference===
In Bayesian statistics, a (scaled, shifted) ''t''-distribution arises as the marginal distribution of the unknown mean of a normal distribution, when the dependence on an unknown variance has been marginalised out:<ref>A. Gelman ''et al'' (1995), ''Bayesian Data Analysis'', Chapman & Hall. ISBN 0-412-03991-5. p. 68</ref>
 
:<math>\begin{align}
p(\mu|D, I) = & \int p(\mu,\sigma^2|D, I) \; d \sigma^2 \\
= & \int p(\mu|D, \sigma^2, I) \; p(\sigma^2|D, I) \; d \sigma^2
\end{align}</math>
 
where ''D'' stands for the data {''x''<sub>i</sub>} and ''I'' represents any other information that may have been used to create the model. The distribution is thus the [[compound distribution|compounding]] of the conditional distribution of μ given the data and σ<sup>2</sup> with the marginal distribution of σ<sup>2</sup> given the data.<br>
<!--
:<math>p(\mu|D, \sigma^2, I) \propto \int p(D|\mu, \sigma^2, I) \; p(\mu|\sigma^2, I)</math>
and
:<math>\begin{align}p(\sigma^2|D, I) = & \int p(\mu, \sigma^2|D, I) \; d\mu \\
\propto & \int p(D |\sigma^2, \mu, I) \; p(\mu|\sigma^2, I) \; p(\sigma^2 | I)\end{align}</math>
-->
With ''n'' data points, if [[Jeffreys prior|uninformative]] location and scale priors <math>\scriptstyle{p(\mu|\sigma^2, I) = \mbox{const}}</math> and <math>\scriptstyle{p(\sigma^2|I)\; \propto \;1/\sigma^2}</math> can be taken for μ and σ<sup>2</sup>, then [[Bayes' theorem]] gives
 
:<math>\begin{align}
p(\mu|D, \sigma^2, I) \sim & N(\bar{x}, \sigma^2/n) \\
p(\sigma^2 | D, I) \sim & \operatorname{Scale-inv-}\chi^2(\nu, s^2)
\end{align}</math>
 
a Normal distribution and a [[scaled inverse chi-squared distribution]] respectively, where ν = ''n'' − 1 and
 
:<math>s^2 = \sum \frac{(x_i - \bar{x})^2}{n-1}</math>.
 
The marginalisation integral thus becomes
 
:<math>\begin{align}
p(\mu|D, I) &\propto \int_0^{\infty} \frac{1}{\sqrt{\sigma^2}} \exp \left(-\frac{1}{2\sigma^2} n(\mu - \bar{x})^2\right) \;\cdot\; \sigma^{-\nu-2}\exp(-\nu s^2/2 \sigma^2) \; d\sigma^2 \\
&\propto \int_0^{\infty} \sigma^{-\nu-3} \exp \left(-\frac{1}{2 \sigma^2} \left(n(\mu - \bar{x})^2 + \nu s^2\right) \right)  \; d\sigma^2
\end{align}</math>
 
This can be evaluated by substituting <math>\scriptstyle{z = A / 2\sigma^2}</math>, where <math>\scriptstyle{A = n(\mu - \bar{x})^2 + \nu s^2}</math>, giving
:<math>dz = -\frac{A}{2 \sigma^4} d \sigma^2,</math>
so
 
:<math>p(\mu|D, I) \propto \; A^{-\frac{\nu + 1}{2}} \int_0^\infty z^{(\nu-1)/2} \exp(-z) dz</math>
 
But the ''z'' integral is now a standard [[Gamma integral]], which evaluates to a constant, leaving
 
:<math>\begin{align}p(\mu|D, I) \propto & \; A^{-\frac{\nu + 1}{2}} \\
\propto & \left( 1 + \frac{n(\mu - \bar{x})^2}{\nu s^2} \right)^{-\frac{\nu + 1}{2}} \end{align}</math>
 
This is a form of the ''t'' distribution with an explicit scaling and shifting that will be explored in more detail in a further section below.  It can be related to the standardised ''t'' distribution by the substitution
 
:<math>t = \frac{\mu - \bar{x}}{s / \sqrt{n}}</math>
 
The derivation above has been presented for the case of uninformative priors for μ and σ<sup>2</sup>; but it will be apparent that any priors which lead to a Normal distribution being compounded with a scaled inverse chi-squared distribution will lead to a ''t'' distribution with scaling and shifting for ''P''(μ|''D'',''I''), although the scaling parameter corresponding to ''s''<sup>2</sup>/''n'' above will then be influenced both by the prior information and the data, rather than just by the data as above.
 
</span>
 
==Characterization==
 
===As the distribution of a test statistic===
Student's ''t''-distribution with ν degrees of freedom can be defined as the distribution of the [[random variable]] ''T'' with <ref name=JKB>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995)
''Continuous Univariate Distributions, Volume 2,'' 2nd Edition. Wiley, ISBN 0-471-58494-0 (Chapter 28)</ref><ref name=Hogg>Hogg & Craig (1978, Sections 4.4 and 4.8.)</ref>
 
:<math> T=\frac{Z}{\sqrt{V/\nu}} = Z \sqrt{\frac{\nu}{V}} ,</math>
 
where
 
* ''Z'' is [[normal distribution|normally distributed]] with [[expected value]] 0 and variance 1;
* ''V'' has a [[chi-squared distribution]] with ν [[Degrees of freedom (statistics)|degrees of freedom]];
* ''Z'' and ''V'' are [[statistical independence|independent]].
 
A different distribution is defined as that of the random variable defined, for a given constant&nbsp;μ, by
:<math>(Z+\mu)\sqrt{\frac{\nu}{V}}.</math>
This random variable has a [[noncentral t-distribution|noncentral ''t''-distribution]] with [[noncentrality parameter]] μ. This distribution is important in studies of the [[statistical power|power]] of Student's ''t''-test.
 
====Derivation====
Suppose ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[statistical independence|independent]] random variables that are normally distributed with expected value μ and [[variance]] σ<sup>2</sup>. Let
 
:<math>\overline{X}_n = \frac{1}{n}(X_1+\cdots+X_n)</math>
 
be the sample mean, and
 
:<math>S_n^{\;2} = \frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2</math>
 
be an unbiased estimate of the variance from the sample.  It can be shown that the random variable
 
: <math>V = (n-1)\frac{S_n^2}{\sigma^2} </math>
 
has a [[chi-squared distribution]] with ''v=n−1'' degrees of freedom (by [[Cochran's theorem]]).<ref>{{cite journal|last=Cochran|first=W. G.|authorlink=William Gemmell Cochran|title=The distribution of quadratic forms in a normal system, with applications to the analysis of covariance|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|date=April 1934|volume=30|issue=2|pages=178–191|doi=10.1017/S0305004100016595|bibcode = 1934PCPS...30..178C }}</ref>  It is readily shown that the quantity
 
:<math>Z = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{\sigma}</math>
 
is normally distributed with mean 0 and variance 1, since the sample mean <math>\overline{X}_n</math> is normally distributed with mean μ and variance σ<sup>2</sup>/''n''.  Moreover, it is possible to show that these two random variables (the normally distributed one '''''Z''''' and the chi-squared-distributed one '''''V''''') are independent.  Consequently{{clarify|date=November 2012}} the [[pivotal quantity]],
 
:<math>T \equiv \frac{Z}{\sqrt{V/v}} = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{S_n},</math>
 
which differs from ''Z'' in that the exact standard deviation σ is replaced by the random variable ''S''<sub>''n''</sub>, has a Student's ''t''-distribution as defined above. Notice that the unknown population variance σ<sup>2</sup> does not appear in ''T'', since it was in both the numerator and the denominator, so it canceled. Gosset intuitively obtained the [[probability density function]] stated above, with ν equal to ''n'' − 1, and Fisher proved it in 1925.<ref name="Fisher 1925 90–104"/>
 
The distribution of the test statistic, ''T'', depends on ν, 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 <math>E(\ln(\nu+X^2))</math> is fixed.<ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |journal=Journal of Econometrics |volume= |issue= |pages=219–230 |publisher=Elsevier |doi= |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |accessdate=2011-06-02 }}</ref>
 
==Properties==
 
===Moments===
The [[raw moment]]s of the ''t''-distribution are
 
:<math>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+1}{2}\right)\Gamma\left(\frac{\nu-k}{2}\right)\nu^{\frac{k}{2}}\right] & k \text{ even}, \quad 0<k< \nu\\
\end{cases}</math>
 
Moments of order ν or higher do not exist.<ref>See, for example, page 56 of Casella and Berger, ''Statistical Inference'', 1990 Duxbury.</ref>
 
The term for 0 < ''k'' < ν, ''k'' even, may be simplified using the properties of the [[gamma function]] to
 
:<math>E(T^k)= \nu^{\frac{k}{2}} \, \prod_{i=1}^{\frac{k}{2}} \frac{2i-1}{\nu - 2i} \qquad k\text{ even},\quad 0<k<\nu. </math>
 
For a ''t''-distribution with ν degrees of freedom, the [[expected value]] is 0, and its [[variance]] is ν/(ν − 2) if ν > 2. The [[skewness]] is 0 if ν > 3 and the [[excess kurtosis]] is 6/(ν − 4) if ν > 4.
 
===Relation to F distribution===
*<math>Y \sim \mathrm{F}(\nu_1 = 1, \nu_2 = \nu)</math> has an [[F-distribution|''F''-distribution]] if ''Y'' = ''X''<sup>2</sup> and ''X'' ~ t(ν) has a Student's ''t''-distribution.
 
===Monte Carlo sampling===
There are various approaches to constructing random samples from the Student's ''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 (statistics)|copula-dependency]].{{citation needed|date=July 2011}} In the case of stand-alone sampling, an extension of the [[Box–Muller method]] and its [[Box–Muller transform#Polar form|polar form]] is easily deployed.<ref name=Bailey>{{Cite journal
| last1 = Bailey | first1 = R. W.
| title = Polar Generation of Random Variates with the ''t''-Distribution
| journal = Mathematics of Computation
| volume = 62
| issue = 206
| pages = 779–781
| doi = 10.2307/2153537
| year = 1994
| pmid = 
| pmc =
}}</ref> It has the merit that it applies equally well to all real positive [[degrees of freedom (statistics)|degrees of freedom]], ν, while many other candidate methods fail if ν is close to zero.<ref name=Bailey/>
 
===Integral of Student's probability density function and ''p''-value===
The function ''A''(''t''|ν) is the integral of Student's probability density function, ''f''(''t'') between −''t'' and ''t'', for ''t'' ≥ 0.  It thus gives the probability that a value of ''t'' less than that calculated from observed data would occur by chance.  Therefore, the function ''A''(''t''|ν) 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 test|''t''-tests]].  For the statistic ''t'', with ν degrees of freedom, ''A''(''t''|ν) 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 can be easily calculated from the [[cumulative distribution function]] ''F''<sub>ν</sub>(''t'') of the ''t''-distribution:
 
:<math>A(t|\nu) = F_\nu(t) - F_\nu(-t) = 1 - I_{\frac{\nu}{\nu +t^2}}\left(\frac{\nu}{2},\frac{1}{2}\right),</math>
 
where ''I''<sub>''x''</sub> is the regularized [[Beta function#Incomplete beta function|incomplete beta function]] (''a'',&nbsp;''b'').
 
For statistical hypothesis testing this function is used to construct the [[p-value|''p''-value]].
 
=={{anchor|Three-parameter version|Non-standardized}}Non-standardized Student's ''t''-distribution==
 
===In terms of scaling parameter σ, or σ<sup>2</sup>===
Student's t distribution can be generalized to a three parameter [[location-scale family]], introducing a [[location parameter]] μ and a [[scale parameter]] σ, through the relation
:<math>X = \mu + \sigma T</math>
The resulting '''non-standardized Student's ''t''-distribution''' has a density defined by<ref name="Jackman" >{{Cite book |last= Jackman |first= Simon |title= Bayesian Analysis for the Social Sciences |publisher= Wiley |year=2009 |page=507}}</ref>
 
:<math>p(x|\nu,\mu,\sigma) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\pi\nu}\sigma} \left(1+\frac{1}{\nu}\left(\frac{x-\mu}{\sigma}\right)^2\right)^{-\frac{\nu+1}{2}} </math>
 
Here, σ does ''not'' correspond to a [[standard deviation]]: it is not the standard deviation of the scaled ''t'' distribution, which may not even exist; nor is it the standard deviation of the underlying [[normal distribution]], which is unknown. σ simply sets the overall scaling of the distribution.  In the Bayesian derivation of the marginal distribution of an unknown Normal mean μ above, σ as used here corresponds to the quantity <math>\scriptstyle{s/\sqrt{n}}</math>, where
 
:<math>s^2 = \sum \frac{(x_i - \bar{x})^2}{n-1}</math>.
 
Equivalently, the distribution can be written in terms of σ<sup>2</sup>, the square of this scale parameter:
 
:<math>p(x|\nu,\mu,\sigma^2) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\pi\nu\sigma^2}} \left(1+\frac{1}{\nu}\frac{(x-\mu)^2}{\sigma^2}\right)^{-\frac{\nu+1}{2}} </math>
 
Other properties of this version of the distribution are:<ref name="Jackman" />
 
:<math>\begin{align}
\operatorname{E}(X) &= \mu \quad \quad \quad \text{for }\,\nu > 1 ,\\
\text{var}(X) &= \sigma^2\frac{\nu}{\nu-2}\, \quad \text{for }\,\nu > 2 ,\\
\text{mode}(X) &= \mu.
\end{align} </math>
 
This distribution results from [[compound distribution|compounding]] a [[Gaussian distribution]] ([[normal distribution]]) with [[mean]] μ and unknown [[variance]], with an [[inverse gamma distribution]] placed over the variance with parameters ''a'' = ν/2 and <math>b = \nu\sigma^2/2</math>. In other words, the [[random variable]] ''X'' is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is [[marginalized out]] (integrated out). The reason for the usefulness of this characterization is that the inverse gamma distribution is the [[conjugate prior]] distribution of the variance of a Gaussian distribution. As a result, the non-standardized Student's ''t''-distribution arises naturally in many [[Bayesian inference]] problems.  See below.
 
Equivalently, this distribution results from compounding a Gaussian distribution with a [[scaled-inverse-chi-squared distribution]] with parameters ν and σ<sup>2</sup>. The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. ν = ''a''/2, σ<sup>2</sup> = ''b''/''a''.
 
===In terms of inverse scaling parameter λ===
An alternative [[parameterization]] in terms of an inverse scaling parameter λ (analogous to the way [[precision (statistics)|precision]] is the reciprocal of variance), defined by the relation λ = σ<sup>−2</sup>.  Then the density is defined by<ref name="Bishop2006">{{Cite book |last= Bishop |first= C.M. |title= Pattern recognition and machine learning |publisher= [[Springer Science+Business Media|Springer]] |year=2006}}</ref>
 
:<math>p(x|\nu,\mu,\lambda) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left(1+\frac{\lambda(x-\mu)^2}{\nu}\right)^{-\frac{\nu+1}{2}}.</math>
 
Other properties of this version of the distribution are:<ref name="Bishop2006" />
 
:<math> \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} </math>
 
This distribution results from [[compound distribution|compounding]] a [[Gaussian distribution]] with [[mean]] μ and unknown [[precision (statistics)|precision]] (the reciprocal of the [[variance]]), with a [[gamma distribution]] placed over the precision with parameters ''a'' = ν/2 and ''b'' = ν/(2λ).  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.
 
==Related distributions==
 
===Noncentral ''t''-distribution===
The [[noncentral t-distribution|noncentral ''t''-distribution]] is a different way of generalizing the ''t''-distribution to include a location parameter.  Unlike the nonstandardized ''t''-distributions, the noncentral distributions are not symmetric (the median is not the same as the mode).
 
===Discrete Student's ''t''-distribution===
The '''discrete Student's ''t''-distribution''' is defined by its [[probability mass function]] at ''r'' being proportional to<ref>Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. ISBN 0-85264-137-0 (Table 5.1)</ref>
:<math> \prod_{j=1}^k \frac{1}{(r+j+a)^2+b^2}  \quad \quad r=\ldots, -1, 0, 1, \ldots  .</math>
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 distribution]]s for continuous distributions.<ref>Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. ISBN 0-85264-137-0 (Chapter 5)</ref>
 
==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 and residuals in statistics|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 interval]]s and [[hypothesis test]]s are two statistical procedures in which the [[quantile]]s 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 hypothesis|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 .{{citation needed|date=November 2010}}
 
====Confidence intervals====
Suppose the number ''A'' is so chosen that
 
:<math>\Pr(-A < T < A)=0.9,</math>
 
when ''T'' has a ''t''-distribution with ''n'' − 1 degrees of freedom.  By symmetry, this is the same as saying that ''A'' satisfies
 
:<math>\Pr(T < A) = 0.95,</math>
 
so ''A'' is the "95th percentile" of this probability distribution, or <math> A=t_{(0.05,n-1)}</math>. Then
 
:<math>\Pr \left (-A < \frac{\overline{X}_n - \mu}{\frac{S_n}{\sqrt{n}}} < A \right)=0.9,</math>
 
and this is equivalent to
 
:<math>\Pr\left(\overline{X}_n - A \frac{S_n}{\sqrt{n}} < \mu < \overline{X}_n + A\frac{S_n}{\sqrt{n}}\right) = 0.9.</math>
 
Therefore the interval whose endpoints are
 
:<math>\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}</math>
 
is a 90% [[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-test|Student's ''t''-test]]s: 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:
 
:<math>\mathrm{UCL}_{1-a} = \overline{X}_n + t_{a,n-1}\frac{S_n}{\sqrt{n}}.</math>
 
The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, <math>\overline{X}_n</math> being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL<sub>1−''a''</sub> 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.
 
===In Bayesian statistics===
The Student's ''t''-distribution, especially in its three-parameter (location-scale) version, arises frequently in [[Bayesian statistics]] as a result of its connection with the [[normal distribution]].  Whenever the [[variance]] of a normally distributed [[random variable]] is unknown and a [[conjugate prior]] placed over it that follows an [[inverse gamma distribution]], the resulting [[marginal distribution]] of the variable will follow a Student's ''t''-distribution.  Equivalent constructions with the same results involve a conjugate [[scaled-inverse-chi-squared distribution]] over the variance, or a conjugate [[gamma distribution]] over the [[precision (statistics)|precision]].  If an [[improper prior]] proportional to σ<sup>−2</sup> is placed over the variance, the ''t''-distribution also arises. This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to a [[conjugate prior|conjugate]] normally distributed prior, or is unknown distributed according to an improper constant prior.
 
Related situations that also produce a ''t''-distribution are:
*The [[marginal distribution|marginal]] [[posterior distribution]] of the unknown mean of a normally distributed variable, with unknown prior mean and variance following the above model.
*The [[prior predictive distribution]] and [[posterior predictive distribution]] of a new normally distributed data point when a series of [[independent identically distributed]] normally distributed data points have been observed, with prior mean and variance as in the above model.
 
===Robust parametric modeling===
The ''t''-distribution is often used as an alternative to the normal distribution as a model for data.<ref>{{Cite journal| last=Lange | first=Kenneth L. | coauthors=Little, Roderick J.A.; Taylor, Jeremy M.G. | journal=JASA | title=Robust statistical modeling using the ''t''-distribution | year=1989 | volume=84 | issue=408 | pages=881–896 | jstor=2290063}}</ref> 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 [[curse of dimensionality|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 (programming language)|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 ν 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 ν, 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''&nbsp;<&nbsp;−2.132)&nbsp;=&nbsp;1&nbsp;−&nbsp;Pr(''T'' > −2.132) = 1 − 0.95 = 0.05,
 
and so
 
: Pr(−2.132&nbsp;<&nbsp;''T''&nbsp;<&nbsp;2.132) = 1&nbsp;−&nbsp;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|Related distributions]] above).
 
The first column is the number of degrees of freedom.
 
{| class="wikitable"
|-
! ''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
|-
!'''<math>\infty</math>'''
|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 ν 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''<sub>α, ν</sub>.  If a quantity ''T'' is distributed as a Student's t distribution with ν degrees of freedom, then there is a probability 1 − α that ''T'' will be less than ''t''<sub>α, ν</sub>. (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
 
:<math>\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}.</math>
 
We can determine that at 90% confidence, we have a true mean lying below
 
:<math>10+1.37218 \frac{\sqrt{2}}{\sqrt{11}}=10.58510.</math>
 
(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
 
:<math>10-1.37218 \frac{\sqrt{2}}{\sqrt{11}}=9.41490.</math>
 
(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&nbsp;−&nbsp;2&nbsp;×&nbsp;(1&nbsp;−&nbsp;90%) = 80%), we have a true mean lying within the interval
 
:<math>\left(10-1.37218 \frac{\sqrt{2}}{\sqrt{11}}, 10+1.37218 \frac{\sqrt{2}}{\sqrt{11}}\right) = \left(9.41490, 10.58510\right). </math>
 
This is generally expressed in interval notation, e.g., for this case, at 80% confidence the true mean is within the interval [9.41490,&nbsp;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]]''.
 
==See also==
{{Portal|Statistics}}
{{Colbegin}}
* [[Chi-squared distribution]]
* [[F-distribution|''F''-distribution]]
* [[Gamma distribution]]
* [[Hotelling's T-squared distribution|Hotelling's ''T''-squared distribution]]
* [[Multivariate Student distribution]]
* [[t-statistic|''t''-statistic]]
* [[Wilks' lambda distribution]]
* [[Wishart distribution]]
{{Colend}}
 
==Notes==
{{Reflist|30em|refs=
 
<ref name=HFR1>Helmert, F. R. (1875). "Über die Bestimmung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler". ''Z. Math. Phys.'', 20, 300–3.</ref>
<ref name=HFR2>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.</ref>
<ref name=HFR3>Helmert, F. R. (1876b). "Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit", ''Astron. Nachr.'', 88, 113–32.</ref>
<ref name=L1876>{{Cite journal| doi=10.1002/asna.18760871402 | last=Lüroth | first=J | year=1876 | title=Vergleichung von zwei Werten des wahrscheinlichen Fehlers | journal=Astron. Nachr. | volume=87 | pages=209–20| issue=14 | bibcode=1876AN.....87..209L}}</ref>
}}
 
==References==
*{{Cite journal
| last1 = Senn | first1 = S.
| last2 = Richardson | first2 = W.
| title = The first ''t''-test
| journal = [[Statistics in Medicine (journal)|Statistics in Medicine]]
| volume = 13
| issue = 8
| pages = 785–803
| year = 1994
| pmid = 8047737 |doi=10.1002/sim.4780130802
}}
* [[Robert V. Hogg|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
*{{Cite book
  | last = Gelman
  | first = Andrew
  | coauthors = John B. Carlin, Hal S. Stern, Donald B. Rubin
  | title = Bayesian Data Analysis (Second Edition)
  | publisher = CRC/Chapman & Hall
  | year = 2003
  | isbn = 1-58488-388-X
  | url = http://www.stat.columbia.edu/~gelman/book/
}}
 
==External links==
*{{springer|title=Student distribution|id=p/s090710}}
*[http://calculus-calculator.com/statistics/students-t-distribution-calculator.html Calculator for the pdf, cdf and critical values of the Student's t-distribution]
*[http://jeff560.tripod.com/s.html Earliest Known Uses of Some of the Words of Mathematics (S)] ''(Remarks on the history of the term "Student's distribution")''
 
{{ProbDistributions|continuous-infinite}}
{{Common univariate probability distributions|state=collapsed}}
{{Statistics|state=collapsed}}
 
{{DEFAULTSORT:Student's T-Distribution}}
[[Category:Continuous distributions]]
[[Category:Special functions]]
[[Category:Normal distribution]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Infinitely divisible probability distributions]]
[[Category:Probability distributions]]

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