Quick answer
Degrees of freedom depend on the test: n − 1 for a one-sample test, n₁ + n₂ − 2 for pooled and the Welch–Satterthwaite formula for unequal variances. A one-sample test with n = 25 has 24 degrees of freedom.
Choose the test and enter the sample sizes (and standard deviations for Welch). The calculator returns the degrees of freedom.
Related tools: P-Value Calculator: P-value from a z, t, chi-square or F statistic. T Statistic Calculator: t statistic from summary statistics. Critical Value Calculator: Critical values for z, t, chi-square and F distributions.
How to use the Degrees of Freedom Calculator
- Choose test: One sample (n − 1), Two samples, pooled (n₁ + n₂ − 2) and Two samples, Welch.
- Enter n₁ (a whole number, 2 or more).
- Optionally, enter n₂ (two-sample) (a whole number, 2 or more).
- Optionally, enter s₁ (Welch) (above 0).
- Optionally, enter s₂ (Welch) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Test | Choose one: One sample (n − 1); Two samples, pooled (n₁ + n₂ − 2); Two samples, Welch | Starts at One sample (n − 1) |
| n₁ | A whole number | ≥ 2 |
| n₂ (two-sample) (optional) | A whole number | ≥ 2 |
| s₁ (Welch) (optional) | A number | > 0 |
| s₂ (Welch) (optional) | A number | > 0 |
Formula and method
Formula
one-sample: n − 1; pooled: n₁ + n₂ − 2; Welch–Satterthwaite: (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]
IEEE-754 double precision; shown to at most 6 decimal places.
Welch's df is generally fractional.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Test = One sample (n − 1); n₁ = 25.
| Result | Value |
|---|---|
| Degrees of freedom | 24 |
Example 2
Inputs: Test = Two samples, pooled (n₁ + n₂ − 2); n₁ = 10; n₂ (two-sample) = 12.
| Result | Value |
|---|---|
| Degrees of freedom | 20 |
Example 3
Inputs: Test = Two samples, Welch; n₁ = 10; n₂ (two-sample) = 12; s₁ (Welch) = 2; s₂ (Welch) = 3.
| Result | Value |
|---|---|
| Degrees of freedom | 19.190546 |
The first line is the headline result (degrees of freedom). IEEE-754 double precision; shown to at most 6 decimal places.
Why n − 1
Once the sample mean is fixed, only n − 1 values can vary freely; the last is determined. That is why the sample variance divides by n − 1.
Welch
With unequal variances the Welch formula (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)] gives a non-integer df. n₂ is required for two samples, and s₁ and s₂ for Welch.
How the result changes with n₁
Holding every other input at the first worked example, this table changes only n₁. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| n₁ | Degrees of freedom |
|---|---|
| 13 | 12 |
| 25 | 24 |
| 50 | 49 |
| 100 | 99 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- n₂ (two-sample): left empty shows "Enter n₂." beside that field and no result.
- s₂ (Welch): left empty shows "Enter s₂." beside that field and no result.
Choosing between the Degrees of Freedom Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [P-Value Calculator](/statistics/p-value/) | test statistic, statistic value and test type | p-value |
| [T Statistic Calculator](/statistics/t-statistic/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [Critical Value Calculator](/statistics/critical-value/) | distribution, significance level and tails | Critical value |
| [One-Sample T-Test Calculator](/statistics/t-test-calculator/) | sample mean, hypothesised mean and sample standard deviation | t statistic |
| [One-Sample Z-Test Calculator](/statistics/z-test/) | sample mean, hypothesised mean and population σ | z statistic |
Background: hypothesis testing and inference
A hypothesis test compares a test statistic with what the null hypothesis predicts. The decision rule is to reject the null hypothesis when the p-value is at or below the significance level α, which you should choose before seeing the data (0.05 is conventional, not special). For a two-tailed z test the p-value is 2 × (1 − Φ(|z|)); for a one-sample t test the statistic is (x̄ − μ₀) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Failing to reject the null hypothesis does not prove it true, and a statistically significant result can still be too small to matter in practice.
Assumptions and limitations
- n₂ required for two samples
- s₁ and s₂ required for Welch
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for n₂ (two-sample) and s₂ (Welch) to confirm this.
Frequently asked questions
What is the pooled df for 10 and 12?
A pooled test with samples of 10 and 12 has 10 + 12 − 2 = 20 degrees of freedom.
Why can df be fractional?
The Welch formula returns real numbers.
How do df affect the t distribution?
Fewer df give heavier tails.
What does the Degrees of Freedom Calculator calculate?
Degrees of freedom for one- and two-sample t-tests. It returns Degrees of freedom.
What formula does the Degrees of Freedom Calculator use?
one-sample: n − 1; pooled: n₁ + n₂ − 2; Welch–Satterthwaite: (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]. Welch's df is generally fractional.
What do I need to enter in the degrees of Freedom Calculator?
You enter Test and n₁. Optional: n₂ (two-sample), s₁ (Welch) and s₂ (Welch).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. n₂ required for two samples.
What are the limits of this calculator?
n₂ required for two samples. s₁ and s₂ required for Welch.
What does the “Test” option change?
It sets how the degrees of Freedom is worked out. The choices are One sample (n − 1), Two samples, pooled (n₁ + n₂ − 2) and Two samples, Welch.
What is an example of a degrees of Freedom result?
With test One sample (n − 1) and n₁ 25, the degrees of freedom is 24.
Related calculators
- P-Value Calculator — P-value from a z, t, chi-square or F statistic.
- T Statistic Calculator — t statistic from summary statistics.
- Critical Value Calculator — Critical values for z, t, chi-square and F distributions.
- One-Sample T-Test Calculator — One-sample t-test from summary statistics.
- One-Sample Z-Test Calculator — Z-test for a mean with known population standard deviation.
- One-Way ANOVA Calculator — One-way analysis of variance across groups.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.