Degrees of Freedom Calculator

Degree of Freedom Calculator

Degrees of Freedom calculator is an online tool that helps to calculate the degrees of freedom for different statistical tests such as:

• ANOVA
• Chi-Square
• 1-Sample
• 2-Sample t-test (samples with equal variances)
• 2-Sample t-test with unequal variances (Welch’s t-test)

This degree of freedom calculator calculates df value quickly and provides a step-by-step solution for every selection. Our df calculator also gives a feature to copy and downloads the result for easy saving or sharing. 

What are Degrees of Freedom?

Degrees of freedom (df) is a statistical concept that shows the finding of independent values that vary in an analysis without violating any constraints. It depends on the sample size of any statistics. 

If the sample size is small then the degree of freedom is a small number while if the sample size is large then the df is large. Calculating degrees of freedom plays an important role in the reliability and accuracy of statistical estimates. It is used in different statistical tests to find the critical values and p-values.  

Degree of Freedom Formula

There are different formulas for the calculation of df and each formula depends on the type of statistical test that you’re performing. Below we explain these formulas that are used to calculate degree of freedom:

Formula for One Sample T-test

In one sample t test the below formula is used to calculate degrees of freedom value:

df = N-1

Where: “N” is the sample size of the data set.

Formula for Chi-Square:

In chi-square test degrees of freedom depend on the number of rows and columns in the contingency table. Its formula can be written as:

df = (r-1) (c-1)

Where: “r” is the number of rows and “c” is the number of columns in the table.  

DF-Formula for ANOVA:

In ANOVA test the degrees of freedom are split into two types: between-group and within-group. But total degree of freedom is equal to the sum of between & within groups of dof.

Degree of Freedom between Groups: df_between = k−1

Degree of Freedom within Groups: df_within = N−k

Total degrees of freedom: df_total = N−1

Where: “k” is the number of groups and “N” is the total number of observations. 

Formula for Two-sample T-tests (with Equal Variances)

In two sample tests having data with equal variances then the below formula is used to calculate df:

df = N₁+ N₂ −2

Where: N₁ & N₂ are number of data for 1^st & 2^nd sample data set, respectively.

Formula for Two-sample T-tests (with Unequal Variances)

If two samples have unequal variances then the below formula is used for df calculation. This formula is also known as Welch’s formula.

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

Where: “σ” is the variance and “N” is the sample size of data.

All the above formulas are used to find degrees of freedom manually for their respective test. However, for instant df calculation use our above df calculator.

How to Find Degrees of Freedom?

For quick and accurate df calculation it’s best to use our degrees of freedom calculator. But for manual calculations see the below example and understand how to calculate degree of freedom.

In calculation, we explain detailed steps that help to understand the use of a degree of freedom formula for different statistical tests and calculation of degrees of freedom of given data. 

Example 1:

Find the degrees of freedom for the given sample: {12, 32, 9, 23, 43}.

Solution:

The given data is in one sample form, then a 1-sample degree of freedom formula is used to find the DF-value. 

N = 5 

Step 1: Put the value of “N” and solve with the help of a 1-sample t-test formula.

Df = N-1

Df = 5 -1

Df = 4

So, the Df of the given data is 4.

Example 2:

Suppose, conducting a ANOVA test to compare the mean scores of students from 4 different schools. You have the following information:

• Number of groups (schools): 4
• Total sample size (N): 20 (5 students from each school)

Then find degrees of freedom for a given data set by ANOVA Test.

Solution

Step 1: According to the ANOVA test, we calculate the degrees of freedom between the groups:

df_between = k−1

and Put, k = 4, in the above equation

df_between = 4−1

df_between = 3 

Step 2: Now, calculate the degrees of freedom within groups:

df_within = N−k

Put the N = 20 and k = 4 in above formula:

df_within = 20−4 

df_within = 16 

Step 3: Find the total degrees of freedom by the sum of degrees of freedom between & within groups.

df_total = 20−1 

df_total= 19 

Results: 

Thus,                      df_between = 3:          df_within = 16:          df_total = 19

To verify the answers of above data use our degrees of freedom calculator.

Frequently Asked Questions

What is the purpose of degrees of freedom?

The degree of freedom shows the number of independent values involved in a calculation and freedom to vary. It helps to manage the amount of information used in the estimation process. It is also used in various statistical tests such as t-tests, F-tests, chi-square tests, and regression analysis.

How to find the degrees of freedom?

For automate finding use our degree of freedom calculator. However, if do it manually then use one of the statistical formulas of degrees of freedom according to given data and statistical test.

How to Find degrees of freedom for T-test?

To calculate the degrees of freedom for a t-test, you can use df-formula: Degrees of Freedom = Sample size (n) – Number of Parameters. Also, use df calculator.

How to find degrees of freedom chi-square?

The degrees of freedom for a chi-square test can be calculated using the formula df = (r-1)(c-1). Some important results about degrees of freedom of chi-square tests: 

• If r = 1 and c > 1, then df = c - 1.
• If r > 1 and c = 1, then df = r - 1.
• If r = 1 and c = 1, then N/A is returned.