Degrees of Freedom Calculator
Compute degrees of freedom for common tests, see the formula with your numbers plugged in, and get the one-tailed critical value at α.
Full tests: t-test, chi-square, ANOVA.
Related Calculators
What degrees of freedom mean
df counts how many pieces of information are free to vary after fitting the model or estimating parameters. For a sample variance with mean fixed, only n − 1 deviations can vary independently — the last is pinned by Σ(xᵢ − x̄) = 0.
Why Welch df is fractional
Welch treats the variance ratio as uncertain; the Satterthwaite formula blends the two group variances into a non-integer df between min(n₁−1, n₂−1) and n₁+n₂−2.
df_W = (Sx₁²/n₁ + Sx₂²/n₂)² / [(Sx₁²/n₁)²/(n₁−1) + (Sx₂²/n₂)²/(n₂−1)]
Pooled t: df = n₁ + n₂ − 2
χ² independence: df = (r − 1)(c − 1)
Quick reference
| Test | df |
|---|---|
| One-sample / paired t | n − 1 |
| Two-sample pooled t | n₁ + n₂ − 2 |
| Two-sample Welch | Satterthwaite (non-integer) |
| χ² goodness of fit | k − 1 − (# estimated parameters) |
| χ² independence | (r − 1)(c − 1) |
| One-way ANOVA | k − 1 and N − k |
| Simple regression t | n − 2 |
Frequently Asked Questions
Why subtract 1 for a t test?
One degree is spent estimating the sample mean before estimating spread, leaving n − 1 for the variance.
Can df be a decimal?
Yes for Welch’s t; textbook t tables use the nearest lower integer, but software uses the exact value.
What df goes with chi-square?
Categories minus 1 minus any parameters you estimated from the same data (goodness of fit) or (r−1)(c−1) for tables.
How do I get F critical values?
For one-way ANOVA the between-groups F uses df₁ = k−1 and df₂ = N−k; this page reports F at your α for that pair.
Regression says n − p − 1 — why?
You estimate p slopes plus an intercept, consuming p+1 degrees, leaving n − p − 1 error df.
Is correlation df also n − 2?
For testing H₀: ρ = 0 with one sample of pairs (x,y), the t test uses df = n − 2.
Two-way ANOVA error df?
N − ab when you have one observation per cell; each cell mean costs one degree.
Does this replace the full calculator?
No — it explains df and critical values; run the linked t, χ², or ANOVA tool for test statistics and p-values.
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