Probability density function  
Cumulative distribution function  
Parameters  $ \nu > 0\! $ deg. of freedom (real) 
Support  $ x \in (\infty; +\infty)\! $ 
$ \frac{\Gamma((\nu+1)/2)} {\sqrt{\nu\pi}\,\Gamma(\nu/2)\,(1+x^2/\nu)^{(\nu+1)/2}}\! $  
cdf  $ \frac{1}{2} + \frac{x \Gamma \left( (\nu+1)/2 \right) 2\, F_1 \left ( \frac{1}{2},(\nu+1)/2;\frac{3}{2};\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma (\nu/2)} $ 
Mean  $ 0 $ 
Median  $ 0 $ 
Mode  $ 0 $ 
Variance  $ \frac{\nu}{\nu2}\! $ for $ \nu>2 $ 
Skewness  $ 0 $ for $ \nu>3 $ 
Kurtosis  $ \frac{6}{\nu4}\! $ for $ \nu>4 $ 
Entropy  $ \begin{matrix} \frac{\nu+1}{2}\left[ \psi(\frac{1+\nu}{2})  \psi(\frac{\nu}{2}) \right] \\[0.5em] + \log{\left[\sqrt{\nu}B(\frac{\nu}{2},\frac{1}{2})\right]} \end{matrix} $

mgf  see text for raw/central moments 
Char. func. 
Although the tdistribution is attributed to William Sealy Gosset, it was actually first derived as a posterior distribution in 1876 by Helmert and Luroth. Before William Gosset published the derivation of tdistribution, it also appeared in a more general form as Pearson Type IV in Karl Pearson's 1895 paper. In English literature, the derivation of the tdistribution was first published in 1908 by William Sealy Gosset, while he worked at a Guinness brewery in Dublin. He published it in the paper Biometrika and as he was not allowed to publish under his own name, he penned it 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.
Occurrence and specification of Student's tdistributionEdit
Suppose X_{1}, ..., X_{n} are independent random variables that are normally distributed with expected value μ and variance σ^{2}. Let
 $ \overline{X}_n=(X_1+\cdots+X_n)/n $
be the sample mean, and
 $ {S_n}^2=\frac{1}{n1}\sum_{i=1}^n\left(X_i\overline{X}_n\right)^2 $
be the sample variance. It is readily shown that the quantity
 $ Z=\frac{\overline{X}_n\mu}{\sigma/\sqrt{n}} $
is normally distributed with mean 0 and variance 1. Gosset studied a related quantity,
 $ T=\frac{\overline{X}_n\mu}{S_n/\sqrt{n}} $
and showed that T has the probability density function
 $ f(t) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi\,}\,\Gamma(\nu/2)} (1+t^2/\nu)^{(\nu+1)/2} $
with
$ \nu $ equal to n − 1. The distribution of T is now called the tdistribution.
The parameter
$ \nu $ is conventionally called the number of degrees of freedom. The distribution depends on
$ \nu $ , but not
$ \mu $ or
$ \sigma $
 the lack of dependence on
$ \mu $ and
$ \sigma $ is what makes the tdistribution important in both theory and practice.
$ \Gamma $ is the Gamma function.
Confidence intervals derived from Student's tdistributionEdit
Suppose the number A is so chosen that
 $ \Pr(A < T < A)=0.9,\, $
when T has a tdistribution with n − 1 degrees of freedom (this is the same as
 $ \Pr(T < A) = 0.95,\, $
so A is the "95th percentage point" of this probability distribution). Then
 $ \Pr\left(A < {\overline{X}_n  \mu \over S_n/\sqrt{n}} < A\right)=0.9, $
and this is equivalent to
 $ \Pr\left(\overline{X}_n  A{S_n \over \sqrt{n}} < \mu < \overline{X}_n + A{S_n \over \sqrt{n}}\right) = 0.9. $
Therefore the interval whose endpoints are
 $ \overline{X}_n\pm A\frac{S_n}{\sqrt{n}} $
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 is normally distributed, the onesided (1 − a)upper confidence limit (UCL) of the mean, can be calculated using the following equation:
 $ \mathrm{UCL}_{1a} = \overline{X}_n+\frac{t_{a,n1} S}{\sqrt{n}}. $
The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words,
$ \overline{X}_n $ being the mean of the set of observations, the probability that the mean of the distribution is inferior to
$ \mathrm{UCL}_{1a} $ is equal to the confidence level
$ 1a. $
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.
Further theoryEdit
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
 $ \frac{Z}{\sqrt{V/\nu\ }} $
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,
 $ \int_{\infty}^t f(u)\,du = \left\{ \begin{matrix} 1  \frac{1}{2} I_x(\nu/2,1/2) & \mbox{if}\quad t > 0, \\ \\ \frac{1}{2} I_x(\nu/2,1/2) & \mbox{otherwise}, \end{matrix}\right. $
with
 $ x = \frac{1}{1+t^2/\nu}. $
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 valuesEdit
The following table lists a few selected values for distributions with r degrees of freedom for the 90%, 95%, 97.5%, and 99.5% confidence intervals. These are "onesided", i.e., where we see "90%", "4 degrees of freedom", and "1.533",
 it means Pr(T < 1.533) = 0.9;
 it does not mean Pr(−1.533 < T < 1.533) = 0.9.
Consequently, by the symmetry of this distribution, we have
 Pr(−1.533 < T) = 0.9,
and consequently
 Pr(−1.533 < T < 1.533) = 0.8.
r  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  1.37218  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 
$ \infty $  0.674  0.842  1.036  1.282  1.645  1.960  2.326  2.576  2.807  3.090  3.291 
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:
 $ \overline{X}_n\pm A\frac{S_n}{\sqrt{n}} $
We can determine that at 90% confidence, we have a true mean lying below:
 $ 10+1.37218 \frac{\sqrt{2}}{\sqrt{11}}=10.58510 $
And, still at 90% confidence, we have a true mean lying over:
 $ 101.37218 \frac{\sqrt{2}}{\sqrt{11}}=9.41490 $
So that at 80% confidence, we have a true mean lying between
 $ 10\pm1.37218 \frac{\sqrt{2}}{\sqrt{11}}=[9.41490,10.58510] $
Special cases Edit
Certain values of
$ \nu $ give an especially simple form.
Edit
$ \nu=1 $Distribution function
 $ F(x) = \frac{1}{2} + \frac{1}{\pi}\tan^{1}(x) $
Edit
$ \nu=2 $Distribution function
 $ F(x) = \frac{1}{2}\left[1+\frac{x}{\sqrt{2+x^2}}\right] $
Density function
 $ f(x) = \frac{1}{\left(2+x^2\right)^{3/2}} $
Related distributionsEdit
 $ Y \sim \mathrm{F}(\nu_1 = 1, \nu_2 = \nu) $is a Fdistribution if$ Y = X^2 \, $ and$ X \sim \mathrm{t}(\nu) $ is a Student's tdistribution.
 $ Y \sim N(0,1) $is a normal distribution as$ Y = \lim_{\nu \to \infty} X $ where$ X \sim \mathrm{t}(\nu) $.
 $ X \sim \mathrm{Cauchy}(0,1) $is a Cauchy distribution if$ X \sim \mathrm{t}(\nu = 1) $.
See alsoEdit
ReferencesEdit
 "Student" (W.S. Gosset) (1908) The probable error of a mean. Biometrika 6(1):125.
 M. Abramowitz and I. A. Stegun, eds. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. (See Section 26.7.)
 R.V. Hogg and A.T. Craig (1978) Introduction to Mathematical Statistics. New York: Macmillan.
 Helmert FR (1875). "Über die Berechnung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler". Z. Math. U. Physik. 20: 300–3.
 Lüroth J (1876). "Vergleichung von zwei Werten des wahrscheinlichen Fehlers". Astron. Nachr. 87 (14): 209–20. Bibcode:1876AN.....87..209L. doi:10.1002/asna.18760871402.
 T Table  History of T Table, Etymology, onetail T Table, twotail T Table and Tstatistic
 Pearson, K. (18950101). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 186: 343–414 (374). doi:10.1098/rsta.1895.0010. ISSN 1364503X.
 Wendl MC (2016). "Pseudonymous fame". Science. 351 (6280): 1406. doi:10.1126/science.351.6280.1406. PMID 27013722.
External links Edit
 VassarStats Density plot, critical values, etc., calculated for a userspecified number of d.f.
 Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term "Student's distribution")
 Distribution Calculator Calculates probabilities and critical values for normal, t, chi2 and Fdistribution
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