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Student's t-distribution: definition, properties, history, and applications

A continuous probability distribution used for inference with small samples; describes behavior of sample means, underpins t-tests, confidence intervals, and appears in regression and Bayesian analysis.

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Overview

The Student's t-distribution is a family of continuous probability distributions that model how sample-based estimates vary when population variance is unknown. It is most relevant when inference is based on small samples drawn from an approximately normal population. Each member of the family is determined by a parameter called the degrees of freedom.

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William Sealy Gosset
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Mathematical form and main properties

Analytically, the t-distribution with ν degrees of freedom has a bell-shaped, symmetric density similar to the normal distribution but with heavier tails: extreme values are more probable. The standard form for its probability density function uses the Gamma function and can be written as f(t) = Gamma((ν+1)/2) / (sqrt(νπ)·Gamma(ν/2)) · (1 + t²/ν)^{-(ν+1)/2}. Key properties include symmetry about zero, location at the sample mean when standardized, and finite moments only up to certain orders (for instance, variance exists only for ν>2). As ν increases, the t-distribution approaches the normal distribution; in the limit ν→∞ it becomes identical to the standard normal.

Degrees of freedom and relation to sampling

The shape of the t-distribution depends on the degrees of freedom, commonly ν = n−1 for a single sample of size n when estimating a mean. Smaller sample size (smaller ν) produces a distribution with heavier tails, reflecting greater uncertainty about the population variance. This adjustment for uncertainty is the reason the t family is preferred to the normal for many small-sample inferential tasks.

Origins and historical notes

The distribution was discovered in 1908 by William Sealy Gosset, who published under the pseudonym "Student." He introduced the distribution to address quality-control and brewing experiments, where samples were small and the population variance unknown. The name "Student's t" honors that original byline.

Primary uses and examples

The t-distribution underlies several standard statistical procedures. Most familiar is the Student's t-test, used to assess differences between means and to evaluate statistical significance for small samples. Relatedly, it is used to form confidence intervals for a mean or for the difference between two means when variances are estimated from data. In regression, t-statistics for coefficients rely on the t-distribution when residuals are approximately normal and variance is estimated; see applications in linear regression. The behavior of sample averages, and comparisons of sample means, are therefore commonly interpreted through this distribution.

Computation, interpretation, and notable facts

  • Tables of critical t-values have historically been used, but modern software computes tail probabilities and quantiles directly.
  • Because of heavier tails, t-based intervals are wider than normal-based intervals for the same confidence level when ν is small; this reflects added uncertainty from estimating variance.
  • The t-distribution also appears in Bayesian models as a robust alternative to the normal likelihood and as a marginal distribution when certain priors are used.

In practice, analysts choose t methods when sample sizes are modest and normality is plausible; for large samples the normal approximation becomes adequate. For non-normal data, alternative robust techniques or permutation methods may be more appropriate.

Further technical details, derivations, and tables can be found in standard statistical texts and online references. For quick reference links to definitions and tests see probability distribution, Student's t-test, and additional resources on estimation and regression at linear regression.

Densities of t-distributed random variables

Definition

A continuous random variable Xsatisfies the student tdistribution with n>0if it has probability density

{\displaystyle f_{n}(x)={\frac {\Gamma \left({\frac {n+1}{2}}\right)}{{\sqrt {n\pi }}~\Gamma \left({\frac {n}{2}}\right)}}\left(1+{\frac {x^{2}}{n}}\right)^{-{\frac {n+1}{2}}}\quad \mathrm {f{\ddot {u}}r} \quad -\infty <x<+\infty }

owns. Thereby is

\Gamma(x)=\int\limits_{0}^{+\infty}t^{x-1}e^{-t}\operatorname{d}t

the gamma function. For natural numbers napplies in particular (here n!the factorial of n)

{\displaystyle \Gamma (n+1)=n!,\quad \Gamma \left(n+{\tfrac {1}{2}}\right)={\frac {(2n)!}{n!\,4^{n}}}\,{\sqrt {\pi }}.}

Alternatively, the tdistribution with ndegrees of freedom can also be defined as the distribution of size

{\displaystyle t_{n}\equiv {\frac {Z}{\sqrt {\chi _{n}^{2}/n}}}},

where Zis a standard normally distributed random variable and χ \chi_n^2is a chi-squared distributed random variable with ndegrees of freedom, independent of Z

Distribution

The distribution function can be expressed in closed form as

{\displaystyle F_{n}(t)=I\left({\frac {t+{\sqrt {t^{2}+n}}}{2{\sqrt {t^{2}+n}}}},{\frac {n}{2}},{\frac {n}{2}}\right)}

or as

{\displaystyle F_{n}(t)={\frac {1}{2}}\left(1+{\frac {t}{|t|}}I\left({\frac {t^{2}}{t^{2}+n}},{\frac {1}{2}},{\frac {n}{2}}\right)\right)}

with

{\displaystyle I(z,a,b)={\frac {1}{B(a,b)}}\int _{0}^{z}t^{a-1}(1-t)^{b-1}\mathrm {d} t,}

where Brepresents the beta function.

F_n(t)calculates the probability that a random variable f_n(x)distributed according to treceives Xa value less than or equal to

Properties

Let be Xa t-distributed random variable with ndegrees of freedom and density f_n(x).

Turning Points

The density has inflection points at

x=\pm\,\sqrt{\frac{n}{n+2}}.

Median

The median is

\tilde{x}=0.

Mode

The mode results in

{\displaystyle x_{D}=0.}

Symmetry

The Student's tdistribution is symmetric about 0.

Expected value

For the expected value we get for n>1

\operatorname{E}(X)=0.

The expected value for does not exist. n=1

Variance

The variance for results in

\operatorname{Var}(X)=\frac{n}{n-2}.

Skew

The skewness is for n>3

\operatorname{v}(X)=0.

Cambers

For kurtosis kurtosis β \beta _{2}and excess kurtosis γ \gamma_2we obtain for n>4

\operatorname{\beta_2}(X)=\frac{\mu_4}{\mu_2^2}=\frac{3n-6}{n-4},\qquad \operatorname{\gamma_2}(X)=\frac{\mu_4}{\mu_2^2}-3=\frac{6}{n-4}.

Moments

For the k-th moments m_k=\operatorname{E}(X^k)and the k-th central moments μ \mu_k=\operatorname{E}([X-\operatorname{E}(X)]^k)holds:

m_{k}=\mu _{k}=0,{\text{ falls }}n>k{\text{ und }}k{\text{ ungerade}}

m_{k}=\mu _{k}=n^{{k/2}}\cdot {\frac {1\cdot 3\cdot 5\cdot 7\dotsm (k-1)}{(n-2)\cdot (n-4)\cdot (n-6)\dotsm (n-k)}},{\text{ falls }}n>k{\text{ und }}k{\text{ gerade}}

Relationship to beta distribution

The integral

{\displaystyle \int _{0}^{z}t^{a-1}(1-t)^{b-1}\mathrm {d} t}

is the incomplete beta function

{\displaystyle B(z;a,b),}

where

{\displaystyle B(a,b)=B(1;a,b)}establishes the connection to the complete beta function. Then for t>0

{\displaystyle F_{n}(t)={\tfrac {1}{2}}+{\tfrac {1}{2}}I(z,{\tfrac {1}{2}},{\tfrac {n}{2}})={\tfrac {1}{2}}+{\tfrac {1}{2}}{\frac {B(z_{t};{\tfrac {1}{2}},{\tfrac {n}{2}})}{B(1;{\tfrac {1}{2}},{\tfrac {n}{2}})}}}

with

{\displaystyle z_{t}={\frac {t^{2}}{t^{2}+n}}.}

If t goes to infinity, z_{t}to 1. In the limiting case, the numerator and denominator of the above fraction are the same, i.e., we get:

{\displaystyle F_{n}(t)={\tfrac {1}{2}}+{\tfrac {1}{2}}I(z_{t},{\tfrac {1}{2}},{\tfrac {n}{2}})\rightarrow {\tfrac {1}{2}}+{\tfrac {1}{2}}=1}

Non-central t-distribution

The size

{\displaystyle {\frac {Z+\delta }{\sqrt {\chi _{n}^{2}/n}}}}

with {\displaystyle Z\sim {\mathcal {N}}(0,1)}and δ \delta as non-centrality parameter follows the so-called non-central t-distribution. This distribution is mainly used to determine the β-error in hypothesis tests with t-distributed test variable. Its probability density is:

{\displaystyle f(x)={\frac {n^{n/2}n!e^{-\delta ^{2}/2}}{2^{n}\Gamma \left(n/2\right)(x^{2}+n)^{(n+1)/2}}}\left({\frac {{\sqrt {2}}\delta x}{\sqrt {x^{2}+n}}}{\frac {_{1}{\mathcal {F}}_{1}\left(n/2+1,3/2,{\frac {(\delta x)^{2}}{2(x^{2}+n)}}\right)}{\Gamma \left((n+1)/2\right)}}+{\frac {_{1}{\mathcal {F}}_{1}\left((n+1)/2,1/2,{\frac {(\delta x)^{2}}{2(x^{2}+n)}}\right)}{\Gamma \left(n/2+1\right)}}\right)}

The parenthesis with the sum of hypergeometric functions can be written a bit simpler still, resulting in a shorter alternative expression for the density:

{\displaystyle f(x)={\frac {2^{n}n^{n/2+1}\Gamma \left((n+1)/2\right)}{\pi (x^{2}+n)^{(n+1)/2}}}e^{-\delta ^{2}/2}H_{-n-1}\left(-{\frac {\delta x}{{\sqrt {2}}{\sqrt {x^{2}+n}}}}\right),}

where {\displaystyle H_{-n-1}\left(z\right)}represents a Hermitian polynomial with negative index with {\displaystyle H_{-n-1}\left(0\right)={\frac {\sqrt {\pi }}{2^{n+1}\Gamma \left(n/2+1\right)}}}.

The expected value for

{\displaystyle {\frac {\delta {\sqrt {n}}\Gamma \left((n-1)/2\right)}{{\sqrt {2}}\Gamma \left(n/2\right)}}}

and the variance (for

{\displaystyle {\frac {(1+\delta ^{2})n}{n-2}}-{\frac {\delta ^{2}n\Gamma \left((n-1)/2\right)^{2}}{2\Gamma \left(n/2\right)^{2}}}.}

With δ \delta =0we obtain the characteristics of the central tdistribution.

Some densities of non-central tdistributions

Relationship with other distributions

Relationship to Cauchy distribution

For n=1and with Γ \Gamma(1/2)=\sqrt{\pi}the Cauchy distribution results as a special case from the Student's tdistribution.

Relationship to Chi-Square Distribution and Standard Normal Distribution

The tdistribution describes the distribution of an expression

{\displaystyle t_{n}\equiv {\frac {{\mathcal {N}}(0,1)}{\sqrt {\frac {\chi _{n}^{2}}{n}}}},}

where denotes {\mathcal {N}}(0,1)a standard normally distributed and χ \chi_n^2denotes a chi-squared distributed random variable with ndegrees of freedom. The numerator variable must be independent of the denominator variable. The density function of the tdistribution is then symmetric with respect to its expected value {\displaystyle 0}. The values of the distribution function are usually available in tabular form.

Distribution with heavy edges

The distribution belongs to the distributions with heavy edges.

Approximation by the normal distribution

As the number of degrees of freedom increases, the distribution values of the tdistribution can be approximated using the normal distribution. As a rule of thumb, from 30 degrees of freedom, the tdistribution function can be approximated by the normal distribution.

Use in mathematical statistics

Various estimators are t-distributed.

If the independent random variables X_1, X_2, \dotsc, X_nare identically normally distributed with expected value μ \mu and standard deviation σ \sigma , it can be proved that the sample mean is

{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}

and the sample variance

{\displaystyle S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}

are stochastically independent.

Because the random variable {\displaystyle {\tfrac {{\overline {X}}-\mu }{\sigma /{\sqrt {n}}}}}has a standard normal distribution and (n-1)\, S^2/\sigma^2follows a chi-squared distribution with n-1degrees of freedom, it follows that the quantity

{\displaystyle t_{n-1}={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}\cdot {\frac {\sigma }{\sigma }}={\frac {{\overline {X}}-\mu }{\sigma /{\sqrt {n}}}}\cdot {\frac {\sigma }{S}}={\frac {{\overline {X}}-\mu }{\sigma /{\sqrt {n}}}}/\left({\frac {S}{\sigma }}\right)={\frac {{\overline {X}}-\mu }{\sigma /{\sqrt {n}}}}/{\sqrt {\chi _{n-1}^{2}/(n-1)}}}

by definition is t-distributed with n-1degrees of freedom.

So the distance of the measured mean from the mean of the population is distributed as t_{n-1} S/\sqrt{n}. This is then used to calculate the 95 % confidence interval for the mean μ \mu to be

{\displaystyle {\overline {x}}-t\cdot S/{\sqrt {n}}\leq \mu \leq {\overline {x}}+t\cdot S/{\sqrt {n}},}

where tis determined by F_{n-1}(t)=0{,}975This interval is somewhat larger for n < \inftyslightly larger than the one that would have resulted with known σ \sigma from the distribution function of the normal distribution at the same confidence level {\displaystyle \left(\mu \in \left[{\overline {x}}\pm 1{,}96\cdot {\tfrac {\sigma }{\sqrt {n}}}\right]\right)}.

Density derivation

The probability density of the tdistribution can be derived from the joint density of the two independent random variables Zand χ \chi^2_nwhich are standard normal and chi-squared distributed, respectively:

{\displaystyle f_{Z,\chi _{n}^{2}}(z,y)={\frac {e^{-{\frac {1}{2}}z^{2}}}{\sqrt {2\pi }}}\cdot {\frac {y^{{\frac {n}{2}}-1}e^{-{\frac {1}{2}}y}}{2^{\frac {n}{2}}\Gamma ({\frac {n}{2}})}}}

With the transformation

{\displaystyle t=z/{\sqrt {y/n}},v=y}

get the joint density of T=Z/\sqrt{\chi^2_n/n}and χ \chi^2_n, where -\infty <t<\infty <

The Jacobian determinant of this transformation is:

\det\frac{\partial(z,y)}{\partial(t,v)}=\begin{vmatrix} \sqrt{\frac{v}{n}}&0\\ \Diamond&1 \end{vmatrix}=\sqrt{\frac{v}{n}}

The value \Diamondis unimportant because it is multiplied by 0 when calculating the determinant. The new density function is thus written

f_{T,\chi^2_n}(t,v)=\frac{e^{-\frac 12 v \frac{t^2}n}}{\sqrt{2\pi}} \cdot \frac{1}{2^\frac n2 \Gamma(\frac n2)}v^{\frac n2-1}e^{-\frac 12v}\cdot\sqrt{\frac{v}{n}}.

We are now looking for the marginal distribution as an f_n(t)integral over the variable of no interest v:

{\displaystyle f_{n}(t)=\int \limits _{0}^{\infty }f_{T,\chi _{n}^{2}}(t,v)\,dv={\frac {1}{{\sqrt {n\pi }}\,2^{(n+1)/2}\Gamma (n/2)}}\int \limits _{0}^{\infty }v^{(n-1)/2}e^{-v(1+t^{2}/n)/2}\,dv={\frac {\Gamma \left({\frac {n+1}{2}}\right)}{{\sqrt {n\pi }}\Gamma \left({\frac {n}{2}}\right)}}\left(1+{\frac {t^{2}}{n}}\right)^{-{\frac {n+1}{2}}}}

Selected quantiles of the t-distribution

Tabulated are tvalues for various degrees of freedom nand common probabilities P(0.75 to 0.999), whereof holds:

P_{{{\text{einseitig}}}}=F_{n}(t)=P(T_{n}\leq t)

Due to the mirror symmetry of the density, one only needs to adjust the probability scale for the case of the interval bounded symmetrically on both sides. Thereby the probabilities decrease for the same tbecause the integration interval is - treduced by cutting away the range from -\infty to

P_{{{\text{zweiseitig}}}}=F_{n}(t)-F_{n}(-t)=P(-t<T_{n}\leq t)=2P_{{{\text{einseitig}}}}-1

If in a sample Nobservations are made and mparameters are estimated from the sample , then n=N-mthe number of degrees of freedom.

To the number of degrees of freedom nin the first column and the significance level α \alpha (represented as 1-\alpha second row), in each cell of the following table the value of the (one-sided) quantile t_{n,\alpha}, corresponding to DIN 1319-3, is given. This satisfies the following equations for the density of f_{n}the t_ndistribution:

One-sided: \int_{-\infty}^{t_{n,\alpha}}f_n(x)\,\mathrm{d}x=1-\alpha

Two-sided: \int_{-t_{n,\alpha/2}}^{t_{n,\alpha/2}}f_n(x)\,\mathrm{d}x=1-\alpha

So, for example, with n=4and α \alpha = 0{,}05find the tvalues of 2.776 (two-sided) or 2.132 (one-sided).

The quantile function of the tdistribution x_pis the solution of the equation p=F(x_p|m,\,n)and thus in principle can be calculated via the inverse function. Concretely applies here

{\displaystyle x_{p}={\frac {{\sqrt {n}}\left(2I^{-1}(p,{\frac {n}{2}},{\frac {n}{2}})-1\right)}{2{\sqrt {\left(1-I^{-1}(p,{\frac {n}{2}},{\frac {n}{2}})\right)\cdot I^{-1}(p,{\frac {n}{2}},{\frac {n}{2}})}}}}}

with I^{{-1}}as inverse of the regularized incomplete beta function. This value x_pis entered in the quantile table under the coordinates p and n.

For few values n(1,2,4) the quantile function simplifies:

{\displaystyle n=1:x_{p}=\operatorname {tan} (\pi (p-1/2))}

{\displaystyle n=2:x_{p}=(p-1/2){\sqrt {\frac {2}{p(1-p)}}}}

{\displaystyle n=4:x_{p}={\sqrt {{\frac {2\cos \left({\frac {1}{3}}\arccos \left(2{\sqrt {p(1-p)}}\,\right)\right)}{\sqrt {p(1-p)}}}-4}}}

Table of some t-quantiles

→ Main article: Quantile table

NumberDegrees of freedom
n

P for two-sided confidence interval

0,5

0,75

0,8

0,9

0,95

0,98

0,99

0,998

P for one-sided confidence interval

0,75

0,875

0,90

0,95

0,975

0,99

0,995

0,999

1

1,000

2,414

3,078

6,314

12,706

31,821

63,657

318,309

2

0,816

1,604

1,886

2,920

4,303

6,965

9,925

22,327

3

0,765

1,423

1,638

2,353

3,182

4,541

5,841

10,215

4

0,741

1,344

1,533

2,132

2,776

3,747

4,604

7,173

5

0,727

1,301

1,476

2,015

2,571

3,365

4,032

5,893

6

0,718

1,273

1,440

1,943

2,447

3,143

3,707

5,208

7

0,711

1,254

1,415

1,895

2,365

2,998

3,499

4,785

8

0,706

1,240

1,397

1,860

2,306

2,896

3,355

4,501

9

0,703

1,230

1,383

1,833

2,262

2,821

3,250

4,297

10

0,700

1,221

1,372

1,812

2,228

2,764

3,169

4,144

11

0,697

1,214

1,363

1,796

2,201

2,718

3,106

4,025

12

0,695

1,209

1,356

1,782

2,179

2,681

3,055

3,930

13

0,694

1,204

1,350

1,771

2,160

2,650

3,012

3,852

14

0,692

1,200

1,345

1,761

2,145

2,624

2,977

3,787

15

0,691

1,197

1,341

1,753

2,131

2,602

2,947

3,733

16

0,690

1,194

1,337

1,746

2,120

2,583

2,921

3,686

17

0,689

1,191

1,333

1,740

2,110

2,567

2,898

3,646

18

0,688

1,189

1,330

1,734

2,101

2,552

2,878

3,610

19

0,688

1,187

1,328

1,729

2,093

2,539

2,861

3,579

20

0,687

1,185

1,325

1,725

2,086

2,528

2,845

3,552

21

0,686

1,183

1,323

1,721

2,080

2,518

2,831

3,527

22

0,686

1,182

1,321

1,717

2,074

2,508

2,819

3,505

23

0,685

1,180

1,319

1,714

2,069

2,500

2,807

3,485

24

0,685

1,179

1,318

1,711

2,064

2,492

2,797

3,467

25

0,684

1,178

1,316

1,708

2,060

2,485

2,787

3,450

26

0,684

1,177

1,315

1,706

2,056

2,479

2,779

3,435

27

0,684

1,176

1,314

1,703

2,052

2,473

2,771

3,421

28

0,683

1,175

1,313

1,701

2,048

2,467

2,763

3,408

29

0,683

1,174

1,311

1,699

2,045

2,462

2,756

3,396

30

0,683

1,173

1,310

1,697

2,042

2,457

2,750

3,385

40

0,681

1,167

1,303

1,684

2,021

2,423

2,704

3,307

50

0,679

1,164

1,299

1,676

2,009

2,403

2,678

3,261

60

0,679

1,162

1,296

1,671

2,000

2,390

2,660

3,232

70

0,678

1,160

1,294

1,667

1,994

2,381

2,648

3,211

80

0,678

1,159

1,292

1,664

1,990

2,374

2,639

3,195

90

0,677

1,158

1,291

1,662

1,987

2,368

2,632

3,183

100

0,677

1,157

1,290

1,660

1,984

2,364

2,626

3,174

200

0,676

1,154

1,286

1,653

1,972

2,345

2,601

3,131

300

0,675

1,153

1,284

1,650

1,968

2,339

2,592

3,118

400

0,675

1,152

1,284

1,649

1,966

2,336

2,588

3,111

500

0,675

1,152

1,283

1,648

1,965

2,334

2,586

3,107

\infty

0,674

1,150

1,282

1,645

1,960

2,326

2,576

3,090



Questions and answers

Q: What is Student's t-distribution?

A: Student's t-distribution is a probability distribution which was developed by William Sealy Gosset in 1908. It describes samples drawn from a full population, and the larger the sample size, the more it resembles a normal distribution.

Q: Who developed Student's t-distribution?

A: William Sealy Gosset developed Student's t-distribution in 1908. He used the pseudonym "Student" when he published the paper describing it.

Q: What are some of the uses of Student's t-distribution?

A: The Student's t-distribution plays a role in many widely used statistical analyses, including the Student's t-test for assessing the statistical significance of differences between two sample means, constructing confidence intervals for differences between two population means, and linear regression analysis. It also arises in Bayesian analysis of data from a normal family.

Q: How does sample size affect the shape of a t-distribution?

A: The larger the sample size, the more closely it will resemble a normal distribution. For each different sample size there is an associated unique t-distribution that describes it.

Q: Is there any relation between Student’s T Distribution and Normal Distribution?

A: Yes - while normal distributions describe full populations, student’s T Distributions describe samples drawn from those populations; as such they share similarities but differ depending on their respective sizes. As mentioned above, larger samples tend to look more like normal distributions than smaller ones do.

Q: Is there any other name for this type of distribution?

A: No - this type of distribution is known as "Student's T Distribution," named after its developer William Sealy Gosset who used his pseudonym "Student" when publishing his paper about it.

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AlegsaOnline.com Student's t-distribution: definition, properties, history, and applications

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