How the Number of Degrees of Freedom Affect the Graph of a t-Distribution

By Alan Anderson

One of the interesting properties of the t-distribution is that the greater the degrees of freedom, the more closely the t-distribution resembles the standard normal distribution. As the degrees of freedom increases, the area in the tails of the t-distribution decreases while the area near the center increases. (The tails consist of the extreme values of the distribution, both negative and positive.) Eventually, when the degrees of freedom reaches 30 or more, the t-distribution and the standard normal distribution are extremely similar.

The following figures illustrate the relationship between the t-distribution with different degrees of freedom and the standard normal distribution. The first figure shows the standard normal and the t-distribution with two degrees of freedom (df). Notice how the t-distribution is significantly more spread out than the standard normal distribution.

The standard normal and t-distribution with two degrees of freedom.
The standard normal and t-distribution with two degrees of freedom.

The graph in the first figure shows that the t-distribution has more area in the tails and less area around the mean than the standard normal distribution. (The standard normal distribution curve is shown with square markers.) As a result, more extreme observations (positive and negative) are likely to occur under the t-distribution than under the standard normal distribution.

The standard normal and t-distribution with ten degrees of freedom.
The standard normal and t-distribution with ten degrees of freedom.

The second figure compares the standard normal distribution with the t-distribution with ten degrees of freedom. The two are much closer to each other here than in the first figure.

The standard normal and t-distribution with 30 degrees of freedom.
The standard normal and t-distribution with 30 degrees of freedom.

As you can see in the third figure, with 30 degrees of freedom, the t-distribution and the standard normal distribution are almost indistinguishable.