Mean and variance

The t distribution is difficult to manipulate mathematically, so we only state the formulae for its mean and variance without proof.

Mean and variance

If \(T \sim \TDistn(k \text{ df})\), then

\[ E[T] \;=\; \begin{cases} 0 & \quad\text{if } k > 1 \\[0.2em] \text{undefined} & \quad\text{otherwise} \end{cases} \]

and

\[ \Var(T) \;=\; \begin{cases} \dfrac {k}{k-2} & \quad\text{if } k > 2 \\[0.5em] \infty & \quad\text{otherwise} \end{cases} \]

Shape of the distribution

The t distribution's probability density function, \(f(x)\), is a function of \(x^2\) and is therefore symmetric around zero. This supports our assertion that the distribution's mean is usually zero. However when its degrees of freedom are very small, the tails of the distribution become so extremely long that its variance becomes infinite and its mean becomes undefined.

The distribution was defined to be that of

\[ T \;\;=\;\; \frac{Z}{\sqrt{\diagfrac{Y}{k}}} \;\;=\;\; Z \times \frac{1}{\sqrt{\diagfrac{Y}{k}}} \]

where \(Z \sim \NormalDistn(0,1)\) and the second term is independent of it. Multiplying \(Z\) by another random quantity makes it more variable, so the spread of the T distribution is greater than that of the standard normal distribution.

However since \(Y\) has a \(\ChiSqrDistn(k \text{ df})\) distribution, its mean is \(E[Y] = k\) and its variance is \(\Var(Y) = 2k\). Therefore

\[ E\left[\diagfrac Y k\right] = 1 \spaced{and} \Var\left(\diagfrac Y k\right) = \diagfrac 2 k \]

As \(k \to \infty\), the variance of \(\diagfrac Y k\) therefore tends to zero and its value approaches its mean, 1.

As \(k \to \infty\), the t distribution therefore approaches the shape of a standard normal distribution.


Relationship to standard normal distribution

The diagram below shows the shape of the t distribution for various different values of the degrees of freedom. The red curve shows the shape of a standard normal distribution.

Drag the slider to see how the shape of the t distribution depends on the degrees of freedom. Note that

A standard normal distribution can be used as an approximation to a t distribution if the degrees of freedom are large (say 30 or more) but this approximation should be avoided for smaller degrees of freedom.