# Student's t-distribution

Template:Probability distribution In probability and statistics, the t-distribution or Student's distribution 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 t-tests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means.

The derivation of the t-distribution was first published in 1908 by William Sealey Gosset, while he worked at a Guinness brewery in Dublin. He was not allowed to publish under his own name, so the paper was written under the pseudonym Student. The t-test and the associated theory became well-known 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 data-based 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.

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## Related distributions

• [itex]Y \sim \mathrm{F}(\nu_1 = 1, \nu_2 = \nu)[itex] is a F-distribution if [itex]Y = X^2[itex] and [itex]X \sim \mathrm{t}(\nu)[itex] is a Student's t-distribution.
• [itex]Y \sim N(0,1)[itex] is a normal distribution as [itex]Y = \lim_{\nu \to \infty} X[itex] where [itex]X \sim \mathrm{t}(\nu)[itex].
• [itex]X \sim \mathrm{Cauchy}(0,1)[itex] is a Cauchy distribution if [itex]X \sim \mathrm{t}(\nu = 1)[itex].

## How Student's t-distribution arises

Suppose X1, ..., Xn are independent random variables that are normally distributed with expected value μ and variance σ2. Let

[itex]\overline{X}_n=(X_1+\cdots+X_n)/n[itex]

be the "sample mean", and

[itex]S_n^2=\frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2[itex]

be the "sample variance". It is readily shown that the estimate of the mean is represented by Z where

[itex]Z=\frac{\overline{X}_n-\mu}{\sigma/\sqrt{n}}[itex]

is normally distributed with mean 0 and variance 1. Gosset studied a related quantity,

[itex]T=\frac{\overline{X}_n-\mu}{S_n/\sqrt{n}}[itex]

and showed that T has the probability density function

[itex]f(t) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi\,}\,\Gamma(\nu/2)} (1+t^2/\nu)^{-(\nu+1)/2}[itex]

with ν equal to n − 1. The distribution of T is now called the t-distribution. The parameter ν is conventionally 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 t-distribution important in both theory and practice.

## How Student's t-distribution is used

The interval whose endpoints are

[itex]\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}[itex]

where A is an appropriate percentage-point of the t-distribution, is a 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 t-distribution 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 t-tests: since the difference between the means of samples from two normal distributions is itself distributed normally, the t-distribution can be used to examine whether that difference can reasonably be supposed to be zero.

If the data is normally distributed, the one-sided (1 − a)-upper confidence limit (UCL) of the mean, can be calculated using the following equation:

[itex]UCL _{1-a} = \overline{X}+{t_{a,n-1}}{S}/{\sqrt{n}}[itex]

The resulting UCL will be the the greatest average value that will occur for a given confidence interval and population size.

A number of other statistics can be shown to have t-distributions for samples of moderate size under null hypotheses that are of interest, so that the t-distribution 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 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 chi-square distribution with ν degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio

[itex] \frac{Z}{\sqrt{V/\nu\ }} [itex]

has a t-distribution with ν degrees of freedom.

For a t-distribution 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,

[itex]\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.[itex]

with

[itex]x = \frac{1}{1+t^2/\nu}.[itex]

The t-distribution is related to the F-distribution 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 t-distribution 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 t-distribution approaches the normal distribution with mean 0 and variance 1.

The following images show the density of the t-distribution for increasing values of ν. The normal distribution is shown as a blue line for comparison. Note that the t-distribution (red line) becomes closer to the normal distribution as ν increases. For ν = 30 the t-distribution is almost the same as the normal distribution.

 Missing imageT_distribution_1df.png Missing imageT_distribution_30df.png

## References

• "Student" (W.S. Gosset) (1908) The probable error of a mean. Biometrika 6(1):1--25.
• 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.

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