Degrees of Freedom Calculator

What Are Degrees of Freedom?

Degrees of freedom (df) count how many independent values can vary when you compute a statistic. Sample size alone is not enough: constraints such as estimating a mean or matching row and column totals reduce the free pieces of information.

Formulas by Test

• 1-sample t-test: $df = N - 1$
• 2-sample t-test (equal variances): $df = N_1 + N_2 - 2$
• Welch t-test (unequal variances): Welch–Satterthwaite approximation using both sample sizes and variances
• Chi-square independence: $df = (rows - 1)(columns - 1)$
• ANOVA: $df_{\text{between}} = k - 1$, $df_{\text{within}} = N - k$, $df_{\text{total}} = N - 1$

How to Use This Calculator

Choose the test type, enter the required sample sizes or table dimensions, and read df instantly. After you know df, use the Critical Value Calculator or Chi Square Test Calculator for critical values and p-values.

Example

For a one-sample t-test with $N = 30$, $df = 29$. For a $3 \times 2$ contingency table, $df = (3-1)(2-1) = 2$. For ANOVA with $k = 4$ groups and $N = 40$ observations, $df_{\text{between}} = 3$ and $df_{\text{within}} = 36$.

Related Tools

Explore related statistics tools:

• Critical Value Calculator
• ANOVA Calculator
• Chi Square Test Calculator