# t - Distribution (Student t - Distribution)

### From MM*Stat International

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t-Distribution is also known as the Student t-Distribution. If Z has a standard normal distribution N(0;1) and Y, the sum of squared standard normal random variables, has a -distribution with degrees of freedom , then we define
as the t-distribution with parameter (shortly written as t()), if Z and Y are independent. The parameter represents the degrees of freedom for the random variable Y. The random variable T has range and expected value and variance:
The following diagram plots the density function a t-distribution for different numbers of degrees of freedom .

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The Chi-square, t-, and F- distributions are distributions that are functions of Normal random variables that are particularly useful in statistics.
**On the t-Distribution.**
The density function of a t-distribution is a bell-shaped symmetricdistribution with expected value (as a standard Normal distribution). However, a t-distribution has heavier tails than a standard Normal distribution. In other words, the t-distribution will be more dispersed than a standard Normal distribution. The variance of the standard normal distribution is 1, but the variance of a t-distribution equals (for ).
As , the density function of the t-distribution converges to the standard normal distribution. For , a Normal distribution can produce a good approximation to a t-distribution.
The t-distribution is tabulated different values of .