Student's tdistribution
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Probability density function 

Cumulative distribution function 

Parameters  degrees of freedom (real) 

Support  
CDF  where is the hypergeometric function 
Mean  , otherwise undefined 
Median  
Mode  
Variance  , otherwise undefined 
Skewness  
Ex. kurtosis  
Entropy 

MGF  (Not defined) 
The Student's tdistribution (or also tdistribution), in probability and statistics, is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. It is the basis of the popular Student's ttests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means. The Student's tdistribution is a special case of the generalised hyperbolic distribution.
The derivation of the tdistribution was first published in 1908 by William Sealy Gosset, while he worked at a Guinness Brewery in Dublin. He was prohibited from publishing under his own name, so the paper was written under the pseudonym Student. The ttest and the associated theory became wellknown through the work of R.A. Fisher, who called the distribution "Student's distribution".
Student's distribution arises when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data. Textbook problems treating the standard deviation as if it were known are 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.
Why use the Student's tdistribution
Confidence intervals and hypothesis tests rely on Student's tdistribution to cope with uncertainty resulting from estimating the standard deviation from a sample, whereas if the population standard deviation were known, a normal distribution would be used.
How Student's tdistribution comes about
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 the sample variance. 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 and standard error .
Gosset studied a related pivotal quantity,
which differs from Z in that the exact standard deviation is replaced by the random variable . Technically, has a distribution by Cochran's theorem. Gosset's work showed that T has the probability density function
with ν equal to n − 1 and where Γ is the Gamma function.
This may also be written as
where B is the Beta function.
The distribution of T is now called the tdistribution. The parameter ν is called the number of degrees of freedom. The distribution 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.
The moments of the tdistribution are
It should be noted that the term for 0 < k < ν, k even, may be simplified using the properties of the Gamma function to
Confidence intervals derived from Student's tdistribution
Suppose the number A is so chosen that
when T has a tdistribution with n − 1 degrees of freedom. This is the same as
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 90percent 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 is equal to the confidence level
A number of other 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 in other situations as well as when examining the differences between means. For example, the distribution of Spearman's rank correlation coefficient, rho, in the null case (zero correlation) is well approximated by the t distribution for sample sizes above about 20.
See prediction interval for another example of the use of this distribution.
Integral of Student's probability density function and pvalue
The function is the integral of Student's probability density function, ƒ(t) between −t and t. 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 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 degrees of freedom, 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 is defined for real t by the following formula:
where B is the Beta function. For t > 0, there is a relation to the regularized incomplete beta function I_{x}(a, b) as follows:
The probability that a value of the t statistic greater than or equal to that observed would happen by chance, if the two sets of data were drawn from the same population, is given by
Further theory
Gosset's result can be stated more generally. (See, for example, Hogg and Craig, Sections 4.4 and 4.8.) Let Z have a normal distribution with mean 0 and variance 1. Let V have a chisquare distribution with ν degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio
has a tdistribution with ν degrees of freedom.
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 and the kurtosis is 6/(ν − 4) if ν > 4.
The cumulative distribution function is given by an incomplete beta function,
with
The tdistribution is related to the Fdistribution as follows: the square of a value of t with ν degrees of freedom is distributed as F with 1 and ν degrees of freedom.
The overall shape of the probability density function of the tdistribution 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. For ν = 30 the tdistribution is almost the same as the normal distribution.
Table of selected values
The following table lists a few selected values for tdistributions with ν degrees of freedom for a range of onesided confidence intervals. 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%. The value of that entry is "2.132". Then the probability that T is less than 2.132 is 95% or ; the entry does not mean (as it might with other distributions) that .
In fact, by the symmetry of the distribution,
 Pr(T < −2.132) = 1 − Pr(T > −2.132) = 1 − 0.95 = 0.05,
and so
 Pr(−2.132 < T < 2.132) = 1 − 2(0.05) = 0.9.
Note that the last row also gives critical points: a tdistribution with infinitelymany degrees of freedom is a normal distribution. (See below: Related distributions).
75%  80%  85%  90%  95%  97.5%  99%  99.5%  99.75%  99.9%  99.95%  

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 
See also ttable.
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, the true mean lies below this upper threshold.) 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, the true mean lies above this lower threshold.) So that at 80% confidence, we have a true mean lying between
(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.
Special cases
Certain values of ν give an especially simple form.
ν = 1
Distribution function:
Density function:
See Cauchy distribution
ν = 2
Distribution function:
Density function:
Robust parametric modelling
The tdistribution is often used as an alternative to the normal distribution as a model for data. 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 modelling 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.
Related distributions
 has a tdistribution if has a scaled inverseχ^{2} distribution and has a normal distribution.
 has an Fdistribution if and has a Student's tdistribution.
 has a normal distribution as where .
 has a Cauchy distribution if .