Student's t-distribution

{\sqrt{\pi\nu}\, \Gamma{\left(\frac{\nu}{2}\right) \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2</math>

| cdf = <math>\begin{align}

& \frac{1}{2} + x \frac{\Gamma{\left(\frac{\nu + 1}{2}\right){\sqrt{\pi\nu}\, \Gamma{\left(\frac{\nu}{2}\right) \times \\

&\qquad {}_{2}F_1\!\left(\frac{1}{2}, \frac{\nu + 1}{2}; \frac{3}{2}; -\frac{x^2}{\nu}\right),

\end{align}</math>

where <math>{}_{2}F_1</math> is the hypergeometric function

| mean = <math>0</math> for <math>\nu > 1,</math> otherwise undefined

| median = <math>0</math>

| mode = <math>0</math>

| variance = <math>\frac{\nu}{\nu -2}</math> for <math>\nu > 2,</math> <math>\infty</math> for <math>1 < \nu \le 2,</math> otherwise undefined

| skewness = <math>0</math> for <math>\ \nu > 3\ ,</math> otherwise undefined

| kurtosis = <math>\frac{6}{\nu - 4}</math> for <math>\nu > 4,</math> <math>\infty</math> for <math>2 < \nu \le 4,</math> otherwise undefined

| entropy = <math>\begin{align}

& \frac{\nu + 1}{2} \left[\psi{\left(\frac{\nu + 1}{2}\right)} -

\psi{\left(\frac{\nu}{2}\right)}\right] \\

&\quad + \ln\left[\sqrt{\nu}\, \mathrm{B}{\left(\frac{\nu}{2}, \frac{1}{2}\right)}\right]~\text{(nats)},

\end{align}</math>

where <math>\psi</math> is the digamma function and <math>\mathrm{B}</math> is the beta function

| mgf = undefined

| char = <math>\frac{\big(\sqrt{\nu}\, |t|\big)^{\nu/2}\, K_{\nu/2}\big(\sqrt{\nu}\, |t|\big)}{\Gamma(\nu/2)\, 2^{\nu/2-1</math> for <math>\nu > 0</math>,

where <math>K_\nu</math> is the modified Bessel function of the second kind

| ES = <math>\mu + s\left(\frac{\big(\nu + [T^{-1}(1 - p)]^2\big) \times \tau\big(T^{-1}(1 - p)\big)}{(\nu - 1)(1 - p)}\right),</math>

where <math>T^{-1}</math> is the inverse standardized Student  CDF, and <math>\tau</math> is the standardized Student t PDF.

In probability theory and statistics, Student's  distribution (or simply the  distribution) <math>t_\nu </math> is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

However, <math>t_\nu</math> has heavier tails, and the amount of probability mass in the tails is controlled by the parameter <math>\nu</math>. For <math>\nu = 1</math> the Student's distribution <math>t_\nu</math> becomes the standard Cauchy distribution, which has very "fat" tails; whereas for <math>\nu \to \infty</math> it becomes the standard normal distribution <math>\mathcal{N}(0, 1),</math> which has very "thin" tails.

The name "Student" is a pseudonym used by William Sealy Gosset in his scientific paper publications during his work at the Guinness Brewery in Dublin, Ireland.

The Student's  distribution plays a role in a number of widely used statistical analyses, including Student's -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.

In the form of the location-scale  distribution <math>\operatorname{\ell st}(\mu, \tau^2, \nu)</math> it generalizes the normal distribution and also arises in the Bayesian analysis of data from a normal family as a compound distribution when marginalizing over the variance parameter.

Definitions

Probability density function

Student's  distribution has the probability density function (PDF) given by

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

where <math>\nu</math> is the number of degrees of freedom, and <math>\Gamma</math> is the gamma function. This may also be written as

<math display="block">f(t) = \frac{1}{\sqrt{\nu}\,\mathrm{B}{\left(\frac{1}{2}, \frac{\nu}{2}\right) \left(1 + \frac{t^2}{\nu}\right)^{-(\nu+1)/2},</math>

where <math>\mathrm{B}</math> is the beta function. In particular, for positive integer-valued degrees of freedom we have:

<math display="block">\begin{align}

\nu & \in \N \geq 2\\

k &= \begin{cases}

2, &\nu\text{ even}\\

\pi, &\nu\text{ odd}\\

\end{cases}\\

\frac{\Gamma{\left(\frac{\nu + 1}{2}\right){\sqrt{\pi\nu}\, \Gamma{\left(\frac{\nu}{2}\right) &= \frac{(\nu-1)!!}{k\sqrt{\nu}(\nu-2)!!}\\

\end{align}</math>

The probability density function is symmetric, and its overall shape 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  distribution approaches the normal distribution with mean 0 and variance 1. For this reason <math>{\nu}</math> is also known as the normality parameter.

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

Cumulative distribution function

The cumulative distribution function (CDF) can be written in terms of , the regularized

incomplete beta function. For

<math display="block">F(t) = \int_{-\infty}^t f(u) \, du ~=~ 1 - \frac{1}{2} I_{x(t)}{\left( \frac{\nu}{2},\, \frac{1}{2} \right)} ,</math>

where

Other values would be obtained by symmetry. An alternative formula, valid for <math> t^2 < \nu\, ,</math> is

<math display="block">\int_{-\infty}^t f(u) \, du = \frac{1}{2} + t\, \frac{\Gamma\!\left( \frac{\nu+1}{2} \right) }{ \sqrt{\pi \nu }\, \Gamma\!\left( \frac{ \nu }{\ 2\ }\right) } \; {}_{2}F_1\!\left( \frac{1}{2}, \frac{ \nu+1 }{2} ;\, \frac{3}{2} ;\, - \frac{t^2}{\nu} \right) ,</math>

where <math>{}_{2}F_1(\ ,\ ;\ ;\ ) </math> is a particular instance of the hypergeometric function.

For information on its inverse cumulative distribution function, see .

Special cases

Certain values of <math>\ \nu\ </math> give a simple form for Student's t-distribution.

{| class="wikitable"

! <math>\ \nu\ </math>

! PDF

! CDF

! notes

! 1

| <math>\frac{1}{ \pi (1 + t^2) } </math>

| <math>\frac{1}{2} + \frac{1}{\pi} \arctan(t) </math>

| See Cauchy distribution

! 2

| <math>\frac{1}{2\, \sqrt{2}\, \left(1+\frac{t^2}{2}\right)^{3/2 </math>

| <math>\frac{1}{2} + \frac{t}{ 2\sqrt{2} \, \sqrt{ 1 + \frac{t^2}{2} } } </math>

! 3

| <math>\frac{2}{ \pi \sqrt{3} \, \left( 1 + \frac{t^2}{3} \right)^2 } </math>

| <math>\frac{1}{2} + \frac{1}{\pi} \left[ \frac{\frac{t}{\sqrt{3}{ 1 + \frac{t^2}{3} } + \arctan\frac{t}{\sqrt{3 \right] </math>

! 4

| <math>\frac{3}{ 8 \left( 1 + \frac{t^2}{4} \right)^{5/2 </math>

| <math>\frac{1}{2} + \frac{3}{8} \left[ \frac{ t }{ \sqrt{ 1 + \frac{t^2}{4} \right] \left[ 1 - \frac{t^2}{12 \left(1 + \frac{t^2}{4} \right) } \right] </math>

! 5

| <math>\frac{8}{ 3\pi \sqrt{5} \, \left( 1 + \frac{t^2}{5} \right)^3 } </math>

| <math>\frac{1}{2} + \frac{1}{\pi} \left[ \frac{t}{ \sqrt{5 } \left(1 + \frac{t^2}{5}\right) } \left(1 + \frac{2}{ 3 \left(1 + \frac{t^2}{5}\right) }\right) + \arctan \frac{t}{\sqrt{5 \right] </math>

! <math>\ \infty\ </math>

| <math>\frac{1}{ \sqrt{2 \pi } }\, e^{-t^2/2}</math>

| <math>\frac{1}{2} \left[ 1 + \operatorname{erf}\left( \frac{t}{\sqrt{2 \right) \right] </math>

| See Normal distribution, Error function

Properties

Moments

For the raw moments of the  distribution are

<math display="block">\operatorname{\mathbb E}\left\{ T^k \right\} = \begin{cases}

\quad 0 & k \text{ odd }, \quad 0 < k < \nu\, , \\[2ex]

\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 <math>\ \nu\ </math> or higher do not exist.

The term for even, may be simplified using the properties of the gamma function to

<math display="block">\operatorname{\mathbb E}\left\{ T^k \right\} = \nu^{ \frac{k}{2} } \, \prod_{j=1}^{k/2} \frac{2j - 1}{\nu - 2j} \qquad k \text{ even}, \quad 0 < k < \nu ~.</math>

For a  distribution with <math>\nu </math> degrees of freedom, the expected value is <math> 0 </math> if <math>\nu > 1\, ,</math> and its variance is <math>\frac{\nu}{\nu-2} </math> if <math>\nu > 2 \,.</math> The skewness is 0 if <math>\nu > 3 </math> and the excess kurtosis is <math>\frac{6}{\nu - 4} </math> if <math>\nu > 4 \,.</math>

How the  distribution arises (characterization)

As the distribution of a test statistic

Student's t-distribution with <math>\nu</math> degrees of freedom can be defined as the distribution of the random variable T with

where

Z is a standard normal with expected value 0 and variance 1;
V has a chi-squared distribution () with <math>\nu</math> 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

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, ..., Xn are independent realizations of the normally-distributed, random variable X, which has an expected value μ and variance σ2. Let

<math display="block">\overline{X}_n = \frac{1}{n}(X_1+\cdots+X_n)</math>

be the sample mean, and

<math display="block">s^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 display="block">V = (n-1)\frac{s^2}{\sigma^2} </math>

has a chi-squared distribution with <math>\nu = n - 1</math> degrees of freedom (by Cochran's theorem). It is readily shown that the quantity

<math display="block">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 σ2/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 the pivotal quantity

<math display="block">T \equiv \frac{Z}{\sqrt{V/\nu = \left(\overline{X}_n - \mu\right) \frac{\sqrt{n{s},</math>

which differs from Z in that the exact standard deviation σ is replaced by the sample standard error s, 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 denominator, so it canceled. Gosset intuitively obtained the probability density function stated above, with <math>\nu</math> equal to n − 1, and Fisher proved it in 1925.

Maximum entropy distribution

Student's  distribution is the maximum entropy probability distribution for a random variate X having a certain value of <math display="inline">\operatorname{\mathbb E}\left\{ \ln(\nu+X^2) \right\} </math>.

This follows immediately from the observation that the pdf can be written in exponential family form with <math>\nu + X^2</math> as sufficient statistic.

Integral of Student's probability density function and -value

The function is the integral of Student's probability density function, between   and , for It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function 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 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  tests. For the statistic , with degrees of freedom, is the probability that 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 It can be easily calculated from the cumulative distribution function of the  distribution:

<math display="block"> A( t \mid \nu) = F_\nu(t) - F_\nu(-t) = 1 - I_{ \frac{\nu}{\nu +t^2} }\!\left(\frac{\nu}{2}, \frac{1}{2}\right),</math>

where is the regularized incomplete beta function.

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

In general

The noncentral  distribution generalizes the  distribution to include a noncentrality parameter. Unlike the nonstandardized  distributions, the noncentral distributions are not symmetric (the median is not the same as the mode).
The discrete Student's  distribution is defined by its probability mass function at r being proportional to: <math display="block"> \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 distributions for continuous distributions.
One can generate Student samples by taking the ratio of variables from the normal distribution and the square-root of the . If we use instead of the normal distribution, e.g., the Irwin–Hall distribution, we obtain over-all a symmetric 4 parameter distribution, which includes the normal, the uniform, the triangular, the Student  and the Cauchy distribution. This is also more flexible than some other symmetric generalizations of the normal distribution.
 distribution is an instance of ratio distributions.
The square of a random variable distributed is distributed as Snedecor's F distribution .

Location-scale -distribution

Location-scale transformation

Student's  distribution generalizes to the three parameter location-scale  distribution <math>\operatorname{\ell st}(\mu,\ \tau^2,\ \nu)\ </math> by introducing a location parameter <math>\ \mu\ </math> and a scale parameter <math>\ \tau ~.</math> With

and location-scale family transformation

we get

<math display="block"> X \sim \operatorname{\ell st}(\mu,\ \tau^2,\ \nu) ~.</math>

The resulting distribution is also called the non-standardized Student's  distribution.

Density and first two moments

The location-scale distribution has a density defined by:

<math display="block">p(x\mid \nu,\mu,\tau) = \frac{\Gamma{\left(\frac{\nu + 1}{2} \right){\Gamma{\left( \frac{\nu}{2}\right)} \tau \sqrt{\pi \nu \left(1 + \frac{1}{\nu} \left(\frac{x-\mu}{\tau} \right)^2 \right)^{-(\nu+1)/2}</math>

Equivalently, the density can be written in terms of <math>\tau^2</math>:

<math display="block">p(x \mid \nu, \mu, \tau^2) = \frac{\Gamma{\left(\frac{\nu + 1}{2}\right){\Gamma{\left(\frac{\nu}{2}\right)} \sqrt{\pi \nu \tau^2 \left(1 + \frac{1}{\nu} \frac{(x - \mu)^2}{\tau^2} \right)^{-(\nu+1)/2}</math>

Other properties of this version of the distribution are: The classical approach was to identify outliers (e.g., using Grubbs's test) and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the  distribution is a natural choice of model for such data and provides a parametric approach to robust statistics.

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.

Student's  process

For practical regression and prediction needs, Student's  processes were introduced, that are generalisations of the Student  distributions for functions. A Student's  process is constructed from the Student  distributions like a Gaussian process is constructed from the Gaussian distributions. For a Gaussian process, all sets of values have a multidimensional Gaussian distribution. Analogously, <math>X(t)</math> is a Student  process on an interval <math>I=[a,b]</math> if the correspondent values of the process <math>\ X(t_1),\ \ldots\ , X(t_n)\ </math> (<math>t_i \in I</math>) have a joint multivariate Student  distribution. These processes are used for regression, prediction, Bayesian optimization and related problems. For multivariate regression and multi-output prediction, the multivariate Student  processes are introduced and used.

Table of selected values

The following table lists values for  distributions with degrees of freedom for a range of one-sided or two-sided critical regions. The first column is , the percentages along the top are confidence levels <math>\ \alpha\ ,</math> and the numbers in the body of the table are the <math>t_{\alpha,n-1}</math> factors described in the section on confidence intervals.

The last row with infinite gives critical points for a normal distribution since a  distribution with infinitely many degrees of freedom is a normal distribution. (See Related distributions above).

{| 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%

|1.000

|1.376

|1.963

|3.078

|6.314

|12.706

|31.821

|63.657

|127.321

|318.309

|636.619

|0.816

|1.061

|1.386

|1.886

|2.920

|4.303

|6.965

|9.925

|14.089

|22.327

|31.599

|0.765

|0.978

|1.250

|1.638

|2.353

|3.182

|4.541

|5.841

|7.453

|10.215

|12.924

|0.741

|0.941

|1.190

|1.533

|2.132

|2.776

|3.747

|4.604

|5.598

|7.173

|8.610

|0.727

|0.920

|1.156

|1.476

|2.015

|2.571

|3.365

|4.032

|4.773

|5.893

|6.869

|0.718

|0.906

|1.134

|1.440

|1.943

|2.447

|3.143

|3.707

|4.317

|5.208

|5.959

|0.711

|0.896

|1.119

|1.415

|1.895

|2.365

|2.998

|3.499

|4.029

|4.785

|5.408

|0.706

|0.889

|1.108

|1.397

|1.860

|2.306

|2.896

|3.355

|3.833

|4.501

|5.041

|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

!∞

|0.674

|0.842

|1.036

|1.282

|1.645

|1.960

|2.326

|2.576

|2.807

|3.090

|3.291

! 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%

; Calculating the confidence interval :

Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom, the one-sided  value from the table is 1.372 . Then with confidence interval calculated from

<math display="block">\overline{X}_n \pm t_{\alpha,\nu} \, \frac{S_n}{\sqrt{n\, ,</math>

we determine that with 90% confidence we have a true mean lying below

In other words, 90% of the times that an upper threshold is calculated by this method from particular samples, this upper threshold exceeds the true mean.

And with 90% confidence we have a true mean lying above

In other words, 90% of the times that a lower threshold is calculated by this method from particular samples, this lower threshold lies below the true mean.

So that at 80% confidence (calculated from 100% − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval

<math display="block">\left(10 - 1.372 \, \frac{\sqrt{2{\sqrt{11, \, 10 + 1.372 \, \frac{\sqrt{2{\sqrt{11 \right) = ( 9.414,\, 10.585) \,.</math>

Saying that 80% of the times that upper and lower thresholds are calculated by this method from a given sample, the true mean is both below the upper threshold and above the lower threshold is not the same 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.

Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the  distribution and its inverse without tables.

Computational methods

Monte Carlo sampling

There are various approaches to constructing random samples from the Student's  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 form is easily deployed. It has the merit that it applies equally well to all real positive degrees of freedom, , while many other candidate methods fail if is close to zero. and Lüroth. As such, Student's t-distribution is an example of Stigler's Law of Eponymy. The  distribution also appeared in a more general form as Pearson type IV distribution in Karl Pearson's 1895 paper.

In the English-language literature, the distribution takes its name from William Sealy Gosset's 1908 paper in Biometrika under the pseudonym "Student" during his work at the Guinness Brewery in Dublin, Ireland. One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Another version is that Guinness did not want their competitors to know that they were using the  test to determine the quality of raw material.

Gosset worked at Guinness and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3. 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 Fisher, who called the distribution "Student's distribution" and represented the test value with the letter .

Notes

References

External links

Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution")
First Students on page 112.
Student's t-Distribution,

Definitions

Probability density function

Cumulative distribution function

Special cases

Properties

Moments

How the &nbsp;distribution arises (characterization)

As the distribution of a test statistic

Derivation

Maximum entropy distribution

Integral of Student's probability density function and -value

Related distributions

In general

Location-scale -distribution

Location-scale transformation

Density and first two moments

Student's &nbsp;process

Table of selected values

Computational methods

Monte Carlo sampling

See also

Notes

References

External links

How the distribution arises (characterization)

Student's process