How to Calculate Degrees of Freedom in Statistical Models

By Ari Reid
Close-up of statistics.

The degrees of freedom in a statistical calculation represent how many values involved in your calculation have the freedom to vary. Appropriately calculated degrees of freedom help ensure the statistical validity of chi-square tests, F tests, and t tests. You can think of degrees of freedom as a sort of checks-and-balances measure, where each piece of information that you are estimating has an associated "cost" of one degree of freedom.

Meaning of Degrees of Freedom

Statistics is designed to define and measure the strength of the relationship between a researcher's actual observations and the parameters that researcher wishes to establish. The degrees of freedom are dependent upon the sample size, or observations, and the parameters to be estimated. The degrees of freedom equal the number of observations minus the number of parameters, so you gain degrees of freedom with a larger sample size. The converse is also true: as you increase the number of parameters to be estimated, you lose degrees of freedom.

Single Parameter With Multiple Observations

If you are trying to fill in one missing piece of information, or estimating a single parameter, and you have three observations in your sample, you know that your degrees of freedom will equal your sample size: three minus the number of parameters you are estimating -- one -- gives you two degrees of freedom. For example, if you have three observations for the measurement of big-toe length that all add up to 15, and you know that the first and second observations are four and six, respectively, then you know that the third measurement must be five. This third measurement does not have the freedom to vary, while the first two do. Therefore, there are two degrees of freedom in this measurement.

Single Parameter, Multiple Observations From Two Groups

Calculating degrees of freedom for big-toe lengths when you have multiple big-toe measurements from two groups, say three from men and three from women, can be a little different. This is the type of situation a t-test may be used for -- when you want to know if there are differences in the mean big-toe lengths of these groups. To calculate the degrees of freedom, you add the total number of observations from men and women. In this example, you have six observations, from which you will subtract the number of parameters. Because you are working with the means of two different groups here, you have two parameters; thus your degrees of freedom is six minus two, or four.

More Than Two Groups

Calculating the degrees of freedom in more complex analyses, such as ANOVA or multiple regressions, depends upon several assumptions associated with those types of models. Chi-square degrees of freedom are equal to the product of the number of rows minus one times the number of columns minus one. Each degree of freedom calculation is dependent upon the statistical test that it is being applied to, and while the calculation is typically quite straightforward, it can be beneficial to make note cards or a quick reference sheet to keep them all straight.

About the Author

Ari Reid has a bachelor's degree in biology (behavior) and a master's in wildlife ecology. When Reid is not training to run marathons, she is operating a non-profit animal rescue organization. Reid has been writing web content for science, health and fitness blogs since 2008.