What are my degrees of freedom?
Pick your test and type the sample sizes. The degrees of freedom calculator gives the df with the formula written out, including Welch’s unequal-variance value and both ANOVA degrees of freedom.
- Degrees of freedom
- 9
A one-sample t test with n = 10 has 9 degrees of freedom.
- Formula
- n − 1 = 10 − 1 = 9
- In words
- A one-sample t test with n = 10 has 9 degrees of freedom.
Degrees of freedom: 9. A one-sample t test with n = 10 has 9 degrees of freedom.
How to calculate
Finds the degrees of freedom for a one-sample, paired, pooled or Welch t test, a chi-square goodness-of-fit or independence test, or a one-way ANOVA.
Example with the default inputs (Test One-sample t test, Sample size (n) 10): A one-sample t test with n = 10 has 9 degrees of freedom.
Method: t: n − 1; pooled: n₁ + n₂ − 2; Welch: (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]; goodness of fit: k − (parameters + 1); table: (r − 1)(c − 1); ANOVA: k − 1 and N − k.
- Welch’s degrees of freedom are not rounded; tables and some software round them down.
- For goodness of fit, k counts the non-empty categories, and the parameters are those estimated from the data.
- For a paired t test, n is the number of pairs.
Worked examples
Each example is checked against the calculator on every build.
- Test One-sample t test, Sample size (n) 10 gives Degrees of freedom 9.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (t with N − 1 degrees of freedom), https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (retrieved 2026-10-02)
- Test Two-sample t test, equal variances, Sample 1 size (n₁) 249, Sample 2 size (n₂) 79 gives Degrees of freedom 326.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.3 Two-Sample t-Test for Equal Means (ν = N₁ + N₂ − 2 for equal variances; the Welch–Satterthwaite ν otherwise; N₁ = 249, N₂ = 79 gives ν = 326), https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm (retrieved 2026-10-02)
- Test Welch’s t test, unequal variances, Sample 1 size (n₁) 249, Sample 2 size (n₂) 79, Sample 1 standard deviation 6.4147, Sample 2 standard deviation 6.10771 gives Degrees of freedom 136.87499, Pooled df (equal variances) 326.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.3 Two-Sample t-Test for Equal Means (ν = N₁ + N₂ − 2 for equal variances; the Welch–Satterthwaite ν otherwise; N₁ = 249, N₂ = 79 gives ν = 326), https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm (retrieved 2026-10-02)
- Test Chi-square goodness of fit, Categories (k) 6, Estimated parameters 2 gives Degrees of freedom 3.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.15 Chi-Square Goodness-of-Fit Test (k − c degrees of freedom, k non-empty cells, c = estimated parameters + 1), https://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm (retrieved 2026-10-02)
- Test Chi-square test of independence, Rows (r) 3, Columns (c) 4 gives Degrees of freedom 6.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.4.5 How can we compare the results of classifying according to several categories? ((r − 1)(c − 1) for r rows and c columns), https://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm (retrieved 2026-10-02)
- Test One-way ANOVA, Groups (k) 3, Total values (N) 15 gives Degrees of freedom 2, df within groups 12, df total 14.Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.4.3.4 One-way ANOVA calculations (k − 1 for treatments, N − k for error), https://www.itl.nist.gov/div898/handbook/prc/section4/prc434.htm (retrieved 2026-10-02)
How it works
| Test | Inputs | Degrees of freedom |
|---|---|---|
| One-sample t | sample size n | n − 1 |
| Paired t | number of pairs n | n − 1 |
| Two-sample t, equal variances | n₁, n₂ | n₁ + n₂ − 2 |
| Welch’s t, unequal variances | n₁, n₂, s₁, s₂ | (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)] |
| Chi-square goodness of fit | categories k, estimated parameters p | k − (p + 1) |
| Chi-square independence | rows r, columns c | (r − 1)(c − 1) |
| One-way ANOVA | groups k, total values N | k − 1 between, N − k within (N − 1 total) |
For Welch’s formula, s₁ and s₂ are first divided by the larger of the two, which leaves ν unchanged and keeps the squares in range. Welch’s ν is not rounded; the page also shows the pooled n₁ + n₂ − 2.
Rules
- Sample sizes, categories and groups are whole numbers from 2 to 1,000,000,000, and rows and columns from 2 to 1,000,000 (so (r − 1)(c − 1) stays exact); the total N is at least 3, and the number of estimated parameters is 0 or more.
- Standard deviations are greater than 0 and at most 10³⁰⁰.
- Goodness of fit needs k − (p + 1) ≥ 1, and ANOVA needs N > k; otherwise there is no answer.
Output format. Whole-number degrees of freedom with thousands separators. Welch’s ν to 10 significant digits, and to 6 in the formula and the sentence, which also gives ν rounded down.
Worked examples by hand
One-sample t, n = 10: 10 − 1 = 9.
NIST’s car data, pooled: 249 + 79 − 2 = 326.
The same data with Welch’s formula (s₁ = 6.4147, s₂ = 6.10771): s₁²/n₁ = 41.148 ÷ 249 = 0.165255 and s₂²/n₂ = 37.304 ÷ 79 = 0.472204. ν = (0.637459)² ÷ (0.165255² ÷ 248 + 0.472204² ÷ 78) = 0.406353 ÷ 0.002968793 = 136.87.
Goodness of fit, 6 categories, 2 estimated parameters: 6 − (2 + 1) = 3.
A 3 × 4 table: (3 − 1)(4 − 1) = 6.
ANOVA, 3 groups, 15 values: 2 and 12 (14 in total).
Other questions people ask
What are degrees of freedom?
The number of values that are free to vary once the estimates a statistic uses are fixed. A sample of n values has n − 1 degrees of freedom around its mean, because the deviations from the mean must add to 0.
How do I find the degrees of freedom for a t test?
One-sample or paired: n − 1 (n pairs for a paired test). Two samples with equal variances: n₁ + n₂ − 2. Two samples with unequal variances: Welch’s formula, which usually gives a decimal between the smaller n − 1 and n₁ + n₂ − 2.
What is the Welch–Satterthwaite formula?
ν = (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]. NIST gives it for the two-sample t test when the variances are not assumed equal. For NIST’s car data (249 and 79 cars, s = 6.4147 and 6.10771) it is 136.87, against 326 when pooled.
How many degrees of freedom does a chi-square test have?
Goodness of fit: k − c, where k is the number of non-empty categories and c is the number of estimated parameters plus 1, so k − 1 when nothing is estimated. Independence in an r × c table: (r − 1)(c − 1).
What are the degrees of freedom in a one-way ANOVA?
Two numbers: k − 1 between the groups and N − k within them, for k groups and N values. The F statistic uses both, written F(k − 1, N − k). With 3 groups of 5, that is F(2, 12).
Should I round Welch’s degrees of freedom?
Software uses the decimal value with the t distribution directly. Printed t tables list whole degrees of freedom, so with a table round down, which gives a slightly larger, more cautious critical value.