From Wikipedia, the free encyclopedia  View original article
Probability density function  
Cumulative distribution function  
Parameters  ν > 0 degrees of freedom (real) 

Support  x ∈ (−∞; +∞) 
CDF  where _{2}F_{1} is the hypergeometric function 
Mean  0 for ν > 1, otherwise undefined 
Median  0 
Mode  0 
Variance  for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined 
Skewness  0 for ν > 3, otherwise undefined 
Ex. kurtosis  for ν > 4, ∞ for 2 < ν ≤ 4, otherwise undefined 
Entropy  
MGF  undefined 
CF  for ν > 0

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (February 2010) 
Probability density function  
Cumulative distribution function  
Parameters  ν > 0 degrees of freedom (real) 

Support  x ∈ (−∞; +∞) 
CDF  where _{2}F_{1} is the hypergeometric function 
Mean  0 for ν > 1, otherwise undefined 
Median  0 
Mode  0 
Variance  for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined 
Skewness  0 for ν > 3, otherwise undefined 
Ex. kurtosis  for ν > 4, ∞ for 2 < ν ≤ 4, otherwise undefined 
Entropy  
MGF  undefined 
CF  for ν > 0

In probability and statistics, Student's tdistribution (or simply the tdistribution) is 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 population standard deviation is unknown. It plays a role in a number of widely used statistical analyses, including the Student's ttest for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's tdistribution also arises in the Bayesian analysis of data from a normal family.
If we take a sample of n = v+1 observations from a normal distribution (the black curve on the figure on the right of this page, representing a very large v), 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 (v=n1).
If we also compute the sample variance for these n observations, then the tdistribution (for n1) 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 , where n is the sample size. In this way, the tdistribution can be used to estimate how likely it is that the true mean lies in any given range.
The tdistribution is symmetric and bellshaped, 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 tdistribution is a special case of the generalised hyperbolic distribution.
In statistics, the tdistribution was first derived as a posterior distribution in 1876 by Helmert^{[2]}^{[3]}^{[4]} and Lüroth.^{[5]}^{[6]}^{[7]}
In the Englishlanguage literature it takes its name from William Sealy Gosset's 1908 paper in Biometrika under the pseudonym "Student".^{[8]}^{[9]} Gosset worked at the 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 ttest to test the quality of raw material.^{[10]}
Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". It became wellknown through the work of Ronald A. Fisher, who called the distribution "Student's distribution" and referred to the value as t.^{[11]}^{[12]}
Student's tdistribution has the probability density function given by
where ν is the number of degrees of freedom and Γ is the gamma function. This may also be written as
where B is the Beta function.
For ν even,
For ν odd,
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 tdistribution approaches the normal distribution with mean 0 and variance 1.
The following images show the density of the tdistribution for increasing values of ν. The normal distribution is shown as a blue line for comparison. Note that the tdistribution (red line) becomes closer to the normal distribution as ν increases.
The cumulative distribution function can be written in terms of I, the regularized incomplete beta function. For t > 0,^{[13]}
with
Other values would be obtained by symmetry. An alternative formula, valid for t^{2} < ν, is^{[13]}
where _{2}F_{1} is a particular case of the hypergeometric function.
Certain values of ν give an especially simple form.
Let x_{1}, ..., x_{n} be the numbers observed in a sample from a continuously distributed population with expected value μ. The sample mean and sample variance are given by:
The resulting tvalue is
The tdistribution with n − 1 degrees of freedom is the sampling distribution of the tvalue when the samples consist of independent identically distributed observations from a normally distributed population. Thus for inference purposes t is a useful "pivotal quantity" in the case when the mean and variance (μ, σ^{2}) are unknown population parameters, in the sense that the tvalue has then a probability distribution that depends on neither μ nor σ^{2}.
In Bayesian statistics, a (scaled, shifted) tdistribution arises as the marginal distribution of the unknown mean of a normal distribution, when the dependence on an unknown variance has been marginalised out:^{[14]}
where D stands for the data {x_{i}} and I represents any other information that may have been used to create the model. The distribution is thus the compounding of the conditional distribution of μ given the data and σ^{2} with the marginal distribution of σ^{2} given the data.
With n data points, if uninformative location and scale priors and can be taken for μ and σ^{2}, then Bayes' theorem gives
a Normal distribution and a scaled inverse chisquared distribution respectively, where ν = n − 1 and
The marginalisation integral thus becomes
This can be evaluated by substituting , where , giving
so
But the z integral is now a standard Gamma integral, which evaluates to a constant, leaving
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
The derivation above has been presented for the case of uninformative priors for μ and σ^{2}; but it will be apparent that any priors which lead to a Normal distribution being compounded with a scaled inverse chisquared distribution will lead to a t distribution with scaling and shifting for P(μD,I), although the scaling parameter corresponding to s^{2}/n above will then be influenced both by the prior information and the data, rather than just by the data as above.
Student's tdistribution with ν degrees of freedom can be defined as the distribution of the random variable T with ^{[13]}^{[15]}
where
A different distribution is defined as that of the random variable defined, for a given constant μ, by
This random variable has a noncentral tdistribution with noncentrality parameter μ. This distribution is important in studies of the power of Student's ttest.
Suppose X_{1}, ..., X_{n} are independent random variables that are normally distributed with expected value μ and variance σ^{2}. Let
be the sample mean, and
be an unbiased estimate of the variance from the sample. It can be shown that the random variable
has a chisquared distribution with v=n−1 degrees of freedom (by Cochran's theorem).^{[16]} It is readily shown that the quantity
is normally distributed with mean 0 and variance 1, since the sample mean is normally distributed with mean μ and variance σ^{2}/n. Moreover, it is possible to show that these two random variables (the normally distributed one Z and the chisquareddistributed one V) are independent. Consequently^{[clarification needed]} the pivotal quantity,
which differs from Z in that the exact standard deviation σ is replaced by the random variable S_{n}, has a Student's tdistribution as defined above. Notice that the unknown population variance σ^{2} 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.^{[11]}
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 tdistribution important in both theory and practice.
Student's tdistribution is the maximum entropy probability distribution for a random variate X for which is fixed.^{[17]}
The raw moments of the tdistribution are
Moments of order ν or higher do not exist.^{[18]}
The term for 0 < k < ν, k even, may be simplified using the properties of the gamma function to
For a tdistribution 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.
There are various approaches to constructing random samples from the Student's tdistribution. The matter depends on whether the samples are required on a standalone basis, or are to be constructed by application of a quantile function to uniform samples; e.g., in the multidimensional applications basis of copuladependency.^{[citation needed]} In the case of standalone sampling, an extension of the Box–Muller method and its polar form is easily deployed.^{[19]} It has the merit that it applies equally well to all real positive degrees of freedom, ν, while many other candidate methods fail if ν is close to zero.^{[19]}
The function A(tν) 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(tν) 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 ttests. For the statistic t, with ν degrees of freedom, A(tν) 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_{ν}(t) of the tdistribution:
where I_{x} is the regularized incomplete beta function (a, b).
For statistical hypothesis testing this function is used to construct the pvalue.
Student's t distribution can be generalized to a three parameter locationscale family, introducing a location parameter μ and a scale parameter σ, through the relation
The resulting nonstandardized Student's tdistribution has a density defined by^{[20]}
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 , where
Equivalently, the distribution can be written in terms of σ^{2}, the square of this scale parameter:
Other properties of this version of the distribution are:^{[20]}
This distribution results from 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 . 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 nonstandardized Student's tdistribution arises naturally in many Bayesian inference problems. See below.
Equivalently, this distribution results from compounding a Gaussian distribution with a scaledinversechisquared distribution with parameters ν and σ^{2}. The scaledinversechisquared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. ν = a/2, σ^{2} = b/a.
An alternative parameterization in terms of an inverse scaling parameter λ (analogous to the way precision is the reciprocal of variance), defined by the relation λ = σ^{−2}. Then the density is defined by^{[21]}
Other properties of this version of the distribution are:^{[21]}
This distribution results from compounding a Gaussian distribution with mean μ and unknown 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.
The noncentral tdistribution is a different way of generalizing the tdistribution to include a location parameter. Unlike the nonstandardized tdistributions, the noncentral distributions are not symmetric (the median is not the same as the mode).
The discrete Student's tdistribution is defined by its probability mass function at r being proportional to^{[22]}
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.^{[23]}
Student's tdistribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the tdistribution 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 tdistribution.
Confidence intervals and hypothesis tests are two statistical procedures in which the quantiles of the sampling distribution of a particular statistic (e.g. the standard score) are required. In any situation where this statistic is a linear function of the data, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's tdistribution. 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 tdistribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a databased 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.
A number of statistics can be shown to have tdistributions for samples of moderate size under null hypotheses that are of interest, so that the tdistribution 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]}.
Suppose the number A is so chosen that
when T has a tdistribution with n − 1 degrees of freedom. By symmetry, this is the same as saying that A satisfies
so A is the "95th percentile" of this probability distribution, or . Then
and this is equivalent to
Therefore the interval whose endpoints are
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 tdistribution 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 ttests: since the difference between the means of samples from two normal distributions is itself distributed normally, the tdistribution can be used to examine whether that difference can reasonably be supposed to be zero.
If the data are normally distributed, the onesided (1 − a)upper confidence limit (UCL) of the mean, can be calculated using the following equation:
The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL_{1−a} is equal to the confidence level 1 − a.
The tdistribution can be used to construct a prediction interval for an unobserved sample from a normal distribution with unknown mean and variance.
The Student's tdistribution, especially in its threeparameter (locationscale) 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 tdistribution. Equivalent constructions with the same results involve a conjugate scaledinversechisquared distribution over the variance, or a conjugate gamma distribution over the precision. If an improper prior proportional to σ^{−2} is placed over the variance, the tdistribution 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 normally distributed prior, or is unknown distributed according to an improper constant prior.
Related situations that also produce a tdistribution are:
The tdistribution is often used as an alternative to the normal distribution as a model for data.^{[24]} It is frequently the case that real data have heavier tails than the normal distribution allows for. The classical approach was to identify outliers and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in high dimensions), and the tdistribution 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 tdistribution 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.
Most statistical textbooks list t distribution tables. Nowadays, the better way to a fully precise critical t value or a cumulative probability is the statistical function implemented in spreadsheets (Office Excel, OpenOffice Calc, etc.), or an interactive calculating web page. The relevant spreadsheet functions are TDIST and TINV, while online calculating pages save troubles like positions of parameters or names of functions. For example, a Mediawiki page supported by R extension can easily give the interactive result of critical values or cumulative probability, even for noncentral tdistribution.
The following table lists a few selected values for tdistributions with ν degrees of freedom for a range of onesided or twosided 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 onesided (90% for twosided). 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,
and so
Note that the last row also gives critical points: a tdistribution with infinitely many degrees of freedom is a normal distribution. (See Related distributions above).
The first column is the number of degrees of freedom.
One Sided  75%  80%  85%  90%  95%  97.5%  99%  99.5%  99.75%  99.9%  99.95% 

Two Sided  50%  60%  70%  80%  90%  95%  98%  99%  99.5%  99.8%  99.9% 
1  1.000  1.376  1.963  3.078  6.314  12.71  31.82  63.66  127.3  318.3  636.6 
2  0.816  1.061  1.386  1.886  2.920  4.303  6.965  9.925  14.09  22.33  31.60 
3  0.765  0.978  1.250  1.638  2.353  3.182  4.541  5.841  7.453  10.21  12.92 
4  0.741  0.941  1.190  1.533  2.132  2.776  3.747  4.604  5.598  7.173  8.610 
5  0.727  0.920  1.156  1.476  2.015  2.571  3.365  4.032  4.773  5.893  6.869 
6  0.718  0.906  1.134  1.440  1.943  2.447  3.143  3.707  4.317  5.208  5.959 
7  0.711  0.896  1.119  1.415  1.895  2.365  2.998  3.499  4.029  4.785  5.408 
8  0.706  0.889  1.108  1.397  1.860  2.306  2.896  3.355  3.833  4.501  5.041 
9  0.703  0.883  1.100  1.383  1.833  2.262  2.821  3.250  3.690  4.297  4.781 
10  0.700  0.879  1.093  1.372  1.812  2.228  2.764  3.169  3.581  4.144  4.587 
11  0.697  0.876  1.088  1.363  1.796  2.201  2.718  3.106  3.497  4.025  4.437 
12  0.695  0.873  1.083  1.356  1.782  2.179  2.681  3.055  3.428  3.930  4.318 
13  0.694  0.870  1.079  1.350  1.771  2.160  2.650  3.012  3.372  3.852  4.221 
14  0.692  0.868  1.076  1.345  1.761  2.145  2.624  2.977  3.326  3.787  4.140 
15  0.691  0.866  1.074  1.341  1.753  2.131  2.602  2.947  3.286  3.733  4.073 
16  0.690  0.865  1.071  1.337  1.746  2.120  2.583  2.921  3.252  3.686  4.015 
17  0.689  0.863  1.069  1.333  1.740  2.110  2.567  2.898  3.222  3.646  3.965 
18  0.688  0.862  1.067  1.330  1.734  2.101  2.552  2.878  3.197  3.610  3.922 
19  0.688  0.861  1.066  1.328  1.729  2.093  2.539  2.861  3.174  3.579  3.883 
20  0.687  0.860  1.064  1.325  1.725  2.086  2.528  2.845  3.153  3.552  3.850 
21  0.686  0.859  1.063  1.323  1.721  2.080  2.518  2.831  3.135  3.527  3.819 
22  0.686  0.858  1.061  1.321  1.717  2.074  2.508  2.819  3.119  3.505  3.792 
23  0.685  0.858  1.060  1.319  1.714  2.069  2.500  2.807  3.104  3.485  3.767 
24  0.685  0.857  1.059  1.318  1.711  2.064  2.492  2.797  3.091  3.467  3.745 
25  0.684  0.856  1.058  1.316  1.708  2.060  2.485  2.787  3.078  3.450  3.725 
26  0.684  0.856  1.058  1.315  1.706  2.056  2.479  2.779  3.067  3.435  3.707 
27  0.684  0.855  1.057  1.314  1.703  2.052  2.473  2.771  3.057  3.421  3.690 
28  0.683  0.855  1.056  1.313  1.701  2.048  2.467  2.763  3.047  3.408  3.674 
29  0.683  0.854  1.055  1.311  1.699  2.045  2.462  2.756  3.038  3.396  3.659 
30  0.683  0.854  1.055  1.310  1.697  2.042  2.457  2.750  3.030  3.385  3.646 
40  0.681  0.851  1.050  1.303  1.684  2.021  2.423  2.704  2.971  3.307  3.551 
50  0.679  0.849  1.047  1.299  1.676  2.009  2.403  2.678  2.937  3.261  3.496 
60  0.679  0.848  1.045  1.296  1.671  2.000  2.390  2.660  2.915  3.232  3.460 
80  0.678  0.846  1.043  1.292  1.664  1.990  2.374  2.639  2.887  3.195  3.416 
100  0.677  0.845  1.042  1.290  1.660  1.984  2.364  2.626  2.871  3.174  3.390 
120  0.677  0.845  1.041  1.289  1.658  1.980  2.358  2.617  2.860  3.160  3.373 
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_{α, ν}. 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_{α, ν}. (Calculated as for a onetailed or onesided test, as opposed to a twotailed 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
We can determine that at 90% confidence, we have a true mean lying below
(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
(In other words, on average, 90% of the times that a lower threshold is calculated by this method, this lower threshold lies below the true mean.) So that at 80% confidence (calculated from 1 − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval
This is generally expressed in interval notation, e.g., for this case, at 80% confidence the true mean is within the interval [9.41490, 10.58510].
(In other words, on average, 80% of the times that upper and lower thresholds are calculated by this method, the true mean is both below the upper threshold and above the lower threshold. This is not the same thing as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method—see confidence interval and prosecutor's fallacy.)
For information on the inverse cumulative distribution function see quantile function.

