8.2.1 - t Distribution

The height of the t distribution is determined by the number of degrees of freedom (df). For a one sample mean test, \(df=n-1\).

The first plot below compares the standard normal distribution (i.e., z distribution) to a t distribution. The solid blue line is the standard normal distribution and the dashed red line is a t distribution with 2 degrees of freedom. Here, the tails of the t distribution are higher than the tails of the normal distribution.

A plot showing the z distribution compared to a t distribution with df=2

If you think about the area under the curve, the higher tails mean that more area will fall in the tails. For example, as seen in the following two plots, \(P(z>2.00)=0.0227501\) while \(P(t_{df=2}>2.00)=0.0917517\).

Standard normal (i.e., z) distribution showing the area above z=2

Probability distribution plot showing the area greater than t=2 on a distribution with 2 degrees of freedom

The next plot compares the standard normal distribution to a t distribution with 10 degrees of freedom. Notice that the two distributions are becoming more similar as the sample size increases.

Plot comparing the z distribution to a t distribution with 10 degrees of freedom

The next plot compares the standard normal distribution to a t distribution with 30 degrees of freedom. 

Plot comparing the standard normal distribution to a t distribution with 30 degrees of freedom

In the final graph, the standard normal distribution is compared to a t distribution with 500 degrees of freedom. Here, the two distributions are nearly identical. As the degrees of freedom approach infinity, the t distribution approaches (i.e., becomes more similar to) the standard normal distribution.

Plot comparing the standard normal distribution to a t distribution with 500 degrees of freedom