Understanding Degrees of Freedom Through Example

May 15, 2018

Defining Degrees of Freedom

Degrees of freedom refer to the number of values in a study that are free to vary. 

You’ve probably heard the term “degrees of freedom” thrown around while discussing the various forms of hypothesis testing that exist, such as a chi-square

This is because in order to understand the importance of a chi-square statistic, and whether or not a null hypothesis is valid, you must calculate degrees of freedom.

Related: An Introduction to the Chi-Square Test and When to Use It

Understanding Degrees of Freedom Through Example

Many people find the concept of degrees of freedom confusing at first, but the idea is often made more complicated than it needs to be. 

To better understand degrees of freedom, consider the following high-level example. 

In order to graduate on time, a university student that works part-time must receive credits from 12 courses in the field of her major during her senior year. 

Given the complexities of her schedule, and the concentration of her major, there are only 12 possible courses offered by the university that the student can take. 

In this case there are 11 degrees of freedom, because the university student is able to enroll in 11 of the classes that fit her schedule and support the concentration of her major. The 12th class is the only possible class left for the student to choose if she wants to graduate on time. 

More on Degrees of Freedom and Chi-Square Tests

Now let’s take a closer look at degrees of freedom in the context of chi-square tests.

If you need to brush up on your chi-square knowledge, be sure to read the following articles:

There are two different types of chi-square tests: the test of independence and the goodness-of-fit test.

The test of independence asks a question pertaining to relationships, such as, “Is there a relationship between age and annual income?”

The goodness-of-fit test asks a question such as, “If a quarter is flipped 200 times, will it land on heads and tails 100 times each?”

While carrying out these tests, degrees of freedom are evaluated to measure if a certain null hypothesis can be rejected due to the total number of variables and samples that the study consists of.

For example, let’s reconsider the example of our student selecting courses in order to graduate on time. 

If this example represented the experiences of 40 or 50 students in a sample, then researchers could observe and analyze data, but it might not be significant enough for them to be confident in their findings. 

Instead, if a sample of 1,000 students was developed, and the same results were witnessed statistically, the findings would be much more statistically valid.


Degrees of freedom can be a challenging concept to wrap your mind around if you are in the beginning stages of learning about market research and statistical best practices.

However, we hope the examples explained in this article have distilled the concept into one that is more digestible. Once you develop a thorough baseline understanding of degrees of freedom, you will then have no problem incorporating this metric into your research.

Struggling with the concept of degrees of freedom? Need some questions answered? We’ve got you covered! Just drop us a line in the comments below!

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