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)

It provides the result with a complete step-by-step solution for every test selection. This df calculator allows users to copy and download the result as PDF 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 there is a small number of degrees of freedom. On the other hand, if the sample size is large then degrees of freedom in large numbers or many pieces of independent values.

Degrees of freedom play an important role in calculating the reliability and accuracy of statistical estimates. It is used in different statistical tests to find the critical values and p-values. These values help in hypothesis testing like T-test, F-test, and Chi-square test. 

Df is also used in Regression analysis and ANOVA (Analysis of Variance) to measure the number of independent values, which are used to find the significant differences between two or more groups. 

Degree of Freedom Formula

The numerical value of the degrees of freedom is determined using different formulas according to the different statistical tests or given data set values. These formulas are discussed below:

DF-Formula for ANOVA

In ANOVA, the degrees of freedom are split into two types: between-group and within-group. But total degrees of freedom is equal to the sum of between & within-groups of degrees of freedom. The degrees of freedom formula for ANOVA is given below: 

Between groups: df_between = k−1

Within groups: df_within = N−k

Total degrees of freedom: df_total = N−1

Where:

• k is a number of groups or cell means.
• N is the total number of observations taken in the test. 

Formula for Chi-Square Test

In the chi-square test, the degrees of freedom depend on the number of rows and columns in the contingency table. The chi-square degrees of freedom value can calculated with below formula:

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

where 

• r is the number of rows in a table. 
• c is the number of columns table.  

Formula for One Sample T-test

If one sample data is given then the below degrees of freedom formula is used to calculate the df-value:

df = N-1

Where:

• Df = Degree of Freedom
• N = Sample Size

For Two-sample T-tests with Equal Variances

The degrees of freedom value for two sample data having equal variances are calculated by using the below formula:

df = N₁+ N₂ −2

Where:

• N₁ = Number of data values of the first sample data set.
• N₂ = Number of data values of a second sample data set.

For Two-sample T-tests with Unequal Variances (Welch’s t-test)

 Degrees of freedom for unequal variances can be calculated with the below formula:

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

Where:

• σ = Variance
• N = Total Number of Sample

All the above formulas are difficult to remember for students or analysts, face difficulty in finding their value. To remove this difficulty use our above online Degrees of Freedom calculator. It provides the answer instantly by just putting the values in the data fields.

How to Find Degrees of Freedom?

In this part, we’ll calculate the degrees of freedom using the right formulas of the degrees of freedom for given data. Also, explain the calculation with detailed steps that help to understand the use of a degree of freedom formula for different statistical tests or 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 main purpose of df is to represent  the number of independent values involved in a calculation and freedom to vary. 

How to find the degrees of freedom?

The most common manual method used to calculate degrees of freedom is “Df = N-1”. To find the quick Df-value, use the above Degrees of Freedom calculator.

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.