Degrees of Freedom Calculator
Degrees of freedom (df) determine the shape of the t-distribution, F-distribution or chi-square distribution used in a hypothesis test. Select your test type and enter your sample sizes to calculate df instantly.
Statistical test
df = n − 1
- 1
Sample size
n = 30 = 30 - 2
Degrees of freedom
n − 1 = 30 − 1 = 29
How does this calculator work?
Degrees of freedom tell a test how many independent pieces of information the data contribute. One-sample t: n−1. Pooled two-sample t: n₁+n₂−2. Welch's t: Satterthwaite formula (depends on both n and s). ANOVA: k−1 between, N−k within. Chi-square: (r−1)(c−1). More df = distribution closer to normal = smaller critical values.
Formula
How this is calculated
Degrees of freedom represent the number of independent values in a statistic that are free to vary once constraints (like knowing the mean) are imposed. A sample of n observations has n − 1 degrees of freedom when estimating variance, because the mean consumes one degree of freedom. This is why the one-sample t-test uses df = n − 1 and the pooled two-sample t-test uses df = n₁ + n₂ − 2 (both means are estimated).
When two samples have different variances, the Welch–Satterthwaite approximation computes df from the sample sizes and standard deviations: df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]. This gives a non-integer value that is usually rounded down; the result is always less than or equal to the pooled df and provides a more conservative (and often more accurate) test when variances differ.
For a one-way ANOVA: the between-groups df = k − 1 (k groups each consume one parameter, the group mean, minus one for the grand mean), and within-groups df = N − k (N total observations minus one per group). For a chi-square test of independence the df = (rows − 1)(columns − 1), reflecting the number of cell frequencies that are free once row and column totals are fixed. Degrees of freedom govern which row of a critical-value table you look up.
Frequently asked questions
With a small sample the sample variance is a noisy estimate of the population variance, so the t-distribution must have heavier tails to capture that uncertainty. As df increases (larger sample), the variance estimate improves and the t-distribution converges to the standard normal (z) distribution. At df = ∞ they are identical.
The Satterthwaite approximation involves a ratio of sums of variances, which generally yields a fractional result. In practice you round down to a whole number when consulting a printed t-table, but statistical software uses the exact fractional value with a continuous t-distribution.
Very small df (e.g. df = 1 or 2) means the t-distribution has extremely heavy tails, so you need a much larger test statistic to reach statistical significance. This reflects high uncertainty from a tiny sample. As a rule of thumb, df < 5 requires caution and large critical values.
Also known as
TG we-Calculate Editorial Team. (2026). Degrees of Freedom Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/degrees-of-freedom-calculator
TG we-Calculate Editorial Team. "Degrees of Freedom Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/degrees-of-freedom-calculator.
TG we-Calculate Editorial Team, "Degrees of Freedom Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/degrees-of-freedom-calculator
@misc{wecalculate_degrees_of_freedom_calculator, title = {Degrees of Freedom Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/degrees-of-freedom-calculator}}, year = {2026}, note = {TG we-Calculate} }
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