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Degrees of Freedom Calculator

Select the statistical distribution type, enter the necessary inputs, and the calculator will compute the probability of a continuous random variable

Test type

Variance (N)

Variance (N₁)

Variance (N₂)

Sample Size (σ₁)

Sample Size (σ₂)

Number of rows in the table

Number of columns in the table

Number of groups (k)

Degrees of freedom within groups

Degrees of freedom between groups

Hypothesized mean (h)

Sample Standard Deviation (x)

Sample mean

Related

Use this degrees of freedom (DOF) calculator to determine the key variable for one-sample and two-sample t-tests, chi-square tests, and ANOVA analyses.

What Are Degrees of Freedom?

Degrees of freedom (DOF) represent the number of independent values in a dataset that can vary while calculating a statistical parameter.

How To Find Degrees of Freedom?

While an online degrees of freedom calculator provides quick results, it is helpful to understand the manual calculation using the formulas below:

Degrees of Freedom Formulas:

Here are the formulas for common statistical tests:

1-Sample t-Test:

For a one-sample t-test:

df = N - 1

Where:

N = Total number of observations in the dataset

df = Degrees of freedom

2-Sample t-Test:

For two-sample t-tests, the formula depends on whether variances are equal or unequal:

Equal Variances:

df = N₁ + N₂ - 2

N₁ = Number of observations in the first sample

N₂ = Number of observations in the second sample

Unequal Variances:

df = (σ₁²/N₁ + σ₂²/N₂)² / [(σ₁²)² / (N₁² * (N₁ - 1)) + (σ₂²)² / (N₂² * (N₂ - 1))]

Where:

σ² = Variance of each sample (can be calculated using a variance calculator)

ANOVA:

For ANOVA tests, the degrees of freedom are calculated as:

Between Groups: df = k - 1

Within Groups: df = N - k

Total Degrees of Freedom: df = N - 1

Chi-Square Test:

For a Chi-Square test:

df = (rows - 1) * (columns - 1)

For faster and precise calculations, you can also use an online degrees of freedom calculator.

Example Calculations

Example #1:

Find the degrees of freedom for a t-test with 23 observations.

Solution:

Given: N = 23

Using the formula for a one-sample t-test:

df = N - 1 = 23 - 1 = 22

Example #2:

Calculate degrees of freedom for a Chi-Square table showing marital status by education:

Status	Middle or Lower School (%)	High School (%)	Bachelor’s (%)	Master’s (%)	PhD (%)	Total (%)
Single	46	40	25	17	18	30
Married	31	40	54	67	64	50
Divorced	15	10	11	6	9	10
Widowed	8	10	11	11	9	10
Total	100	100	100	100	100	100