Degrees of Freedom Calculator

This Degrees of Freedom Calculator is used to determine 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 is 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: DF_within = N - k
• Degrees of Freedom Between Groups: DF_between = k - 1
• Total Degrees of Freedom: DF_total = 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 ≈ [(σ²₁ / N₁ + σ²₂ / N₂)² / [(σ²₁ / N₁)² / (N₁ - 1) + (σ²₂ / N₂)² / (N₂ - 1)]
• σ = Variance
• N = Sample Size

How to Find Degrees of Freedom?

In this section, we’ll solve some examples and understand how to calculate df 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 degrees of freedom 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 degrees of freedom of the given sequences is 7.