# Student's t-distribution

Student's ''t''-distributiont-distributiont''-distributiont distributionStudent t-distributionStudent's ''tStudent's tconfidence intervalsStudentStudent distribution
In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.wikipedia
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### William Sealy Gosset

StudentGossetGosset, William Sealy
It was developed by William Sealy Gosset under the pseudonym Student.
Gosset published under the pen name Student and developed most famously Student's t-distribution – originally called Student's "z" – and "Student's test of statistical significance".

### Bayesian inference

BayesianBayesian analysisBayesian method
The Student's t-distribution also arises in the Bayesian analysis of data from a normal family.
For example, confidence intervals and prediction intervals in frequentist statistics when constructed from a normal distribution with unknown mean and variance are constructed using a Student's t-distribution.

### Confidence interval

confidence intervalsconfidence levelconfidence
The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.
In the theoretical example below, the parameter σ is also unknown, which calls for using the Student's t-distribution.

### Guinness Brewery

St. James's Gate BreweryGuinnessGuinness plc
Gosset worked at the Guinness Brewery in Dublin, Ireland, 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.
The brewery hired the statistician William Sealy Gosset in 1899, who achieved lasting fame under the pseudonym "Student" for techniques developed for Guinness, particularly Student's t-distribution and the even more commonly known Student's t-test.

### Degrees of freedom (statistics)

degrees of freedomdegree of freedomEffective degrees of freedom
where \nu is the number of degrees of freedom and \Gamma is the gamma function.
While Gosset did not actually use the term 'degrees of freedom', he explained the concept in the course of developing what became known as Student's t-distribution.

### Ronald Fisher

R.A. FisherR. A. FisherFisher
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 t.
Fisher's 1924 article On a distribution yielding the error functions of several well known statistics presented Pearson's chi-squared test and William Gosset's Student's t-distribution in the same framework as the Gaussian distribution and is where he developed Fisher's z-distribution a new statistical method, commonly used decades later as the F distribution.

### Pearson distribution

Pearson Type III distributionPearson's system of continuous curvesPearson
The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper.
Pearson's 1895 paper introduced the type IV distribution, which contains Student's t-distribution as a special case, predating William Sealy Gosset's subsequent use by several years.

### Statistics

statisticalstatistical analysisstatistician
In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.
Widely used pivots include the z-score, the chi square statistic and Student's t-value.

### Noncentral t-distribution

noncentral ''t''-distributionnon-central t-distributionNoncentral ''t''-distribution in power analysis
:This random variable has a noncentral t-distribution with noncentrality parameter μ.
As with other probability distributions with noncentrality parameters, the noncentral t-distribution generalizes a probability distribution – Student's t-distribution – using a noncentrality parameter.

### Generalised hyperbolic distribution

Generalized hyperbolic distribution
The Student's t-distribution is a special case of the generalised hyperbolic distribution.
As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the Laplace distribution, the hyperbolic distribution, the normal-inverse Gaussian distribution and the variance-gamma distribution.

### Chi-squared distribution

chi-squaredchi-square distributionchi square distribution
has a chi-squared distribution with \nu = n - 1 degrees of freedom (by Cochran's theorem).
It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.

### Student's t-test

t-testt''-testStudent's ''t''-test
The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.
The t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis.

### Noncentrality parameter

:This random variable has a noncentral t-distribution with noncentrality parameter μ.
For example, the Student's t-distribution is the central family of distributions for the noncentral t-distribution family.

### Normal distribution

normally distributedGaussian distributionnormal
In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. The t-distribution with n - 1 degrees of freedom is the sampling distribution of the t-value when the samples consist of independent identically distributed observations from a normally distributed population.
However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).

### Jacob Lüroth

LürothLüroth, Jacob
In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth.
Following up on Carl Friedrich Gauss' work on statistics, Lüroth discovered the t-distribution usually credited to William Sealy Gosset.

### Sampling distribution

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

### Pivotal quantity

pivotal quantitiespivotal method
Thus for inference purposes t is a useful "pivotal quantity" in the case when the mean and variance are unknown population parameters, in the sense that the t-value has then a probability distribution that depends on neither \mu nor \sigma^2.
Using x=\mu the function g(\mu,X) becomes a pivotal quantity, which is also distributed by the Student's t-distribution with \nu = n-1 degrees of freedom.

### Kurtosis

excess kurtosisleptokurticplatykurtic
The skewness is 0 if \nu > 3 and the excess kurtosis is if \nu > 4.
Examples of leptokurtic distributions include the Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution.

### Ratio distribution

ratio distributionsComplex normal ratio distributionGaussian ratio distribution
the t-distribution arises from a Gaussian random variable divided by an independent chi-distributed random variable,

### P-value

p''-valuepp''-values
For statistical hypothesis testing this function is used to construct the p-value.
Some such tests are z-test for normal distribution, t-test for Student's t-distribution, f-test for f-distribution.

### Compound probability distribution

compound distributionmixturecompounding
The distribution is thus the compounding of the conditional distribution of \mu given the data and \sigma^2 with the marginal distribution of \sigma^2 given the data.

### Power (statistics)

statistical powerpowerpowerful
This distribution is important in studies of the power of Student's t-test.
The test statistic under the null hypothesis follows a Student t-distribution.

### Cochran's theorem

Cochran's QCochran’s theorem
has a chi-squared distribution with \nu = n - 1 degrees of freedom (by Cochran's theorem).
where F 1,n − 1 is the F-distribution with 1 and n − 1 degrees of freedom (see also Student's t-distribution).

### Folded-t and half-t distributions

folded non-standardized t distributionHalf-''t'' distributionhalf-Student's t
In statistics, the folded-t and half-t distributions are derived from Student's t-distribution by taking the absolute values of variates.

### Posterior predictive distribution

prior predictive distributionpredictive uncertainty quantification
For example, the Student's t-distribution can be defined as the prior predictive distribution of a normal distribution with known mean μ but unknown variance σ x 2, with a conjugate prior scaled-inverse-chi-squared distribution placed on σ x 2, with hyperparameters ν and σ 2 .