Degrees of Freedom Calculator

This Degree of Freedom Calculator is used to calculate the degree of freedom for many statistical tests such as (one-sample and two-sample t-tests, chi-square tests, and ANOVA). Read further to find out how to calculate the degrees of freedom for different tests using the degree of freedom formulas.

What are Degrees of Freedom?

Degrees of Freedom represent the maximum number of independent values that are free to vary in a dataset. This is generally calculated by subtracting one from the sample size.  It is important for validating statistical tests such as chi-square tests, ANOVA tests, t-tests, and F-tests.

The number of degrees of freedom for a statistic varies based on the sample size:

• If the sample size (n) is small, then the degrees of freedom will also be small.

• If the sample size (n) is large, then the degrees of freedom will also be large.

Note: The concept of degrees of freedom is connected to sample size, but not the same.  The degrees of freedom are always fewer than the sample size.

Formulas to Calculate Degrees of Freedom

The calculation of degrees of freedom depends on the type of statistical test that you are conducting. Here are some common formulas for the degree of freedom:

ANOVA Test 

• Degrees of Freedom Within Groups

DFwithin ​= N−k

• Degrees of Freedom Between Groups

DFBetween ​= k -1

• Total Degrees of Freedom 

DFtotal = N – 1

where:

• N is the total number of observations across all groups.

• K is the number of groups.

Chi-Square Test

DF = (r−1) (c−1)

• r is the number of rows. 

• c is the number of columns in the contingency table.

Single Sample t-test

DF = N−1

Where N is the sample size.

2-sample t-test (samples with equal variances)

DF = N1 + N2 -2

• ​N1 = Number of values from the 1st sample.​ 

• N2 = Number of values from the 2nd sample.

2-sample t-test with unequal variances (Welch’s t-test)

DF ≈ [(σ12/N1 + σ22/N2)2 / [(σ12/N1)2 / (N1 - 1) + (σ22/N2)2 / (N2 - 1)]

• σ = Variance 

• N = Sample Size

How to Calculate Degrees of Freedom?

In this section, we’ll solve some examples and understand how to find the degree of freedom for different statistical tests.

Example 1:

Calculate the degree of freedom for the provided sample: 15, 46, 67, 23, 45

Solution:

Given: n = 5 

Subtract 1 from the sample size to get the degree of freedom.

DF = N -1 

DF = 5 -1 = 4 

So, the pdf of the given sample is 4.

Example 2:

Evaluate the degree of freedom for the provided sample data:

• Observation 1: 1, 7, 5, 12, 17 

• Observation 2: 14, 15, 21, 29

Solution:

Given: n1 = 5, n2 = 4

There are two sequences so we need to apply a 2-sample t-test 

DF = N1 + N2 -2

DF = 5 + 4 -2 = 7

So, the df of given sequences is 7.

Frequently Asked Questions

1. How to calculate degrees of freedom for t-tests?

To calculate the degree of freedom for the t-test, you need to follow the below 3 simple steps:

• Calculate the size (N)of your sample.
• Subtract 1 from the sample size.
• The result represents the number of degrees of freedom.

2. How to calculate degrees of freedom for Chi-Square?

To find the degrees of freedom for chi-square by using the chi-square test, follow the below simple steps:

• Find the total number of rows in the chi-square table, then subtract one.
• Find the number of columns and subtract one.
• Multiply the both numbers you get from Step 1 and Step 2.

3. Why do we use degrees of freedom?

We use degrees of freedom because they indicate the number of independents that can vary in an analysis without breaking any constraints. Furthermore, it is an important idea in many contexts throughout statistics including hypothesis tests, probability distributions, and linear regression.

4. Can the number of degrees of freedom be negative?

No, the number of degrees of freedom cannot be negative. According to the Gibbs & Duhem equation, the degrees of freedom can be zero.