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Degrees of Freedom for t, Chi-Square, ANOVA & Welch

Calculate degrees of freedom for one-sample, pooled two-sample, Welch, contingency-table and one-way ANOVA designs.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Two independent samples of 10 and 12 observations have 20 pooled-test degrees of freedom. Welch degrees of freedom instead depend on both sample sizes and sample variances and may be fractional.

One sample: n − 1; pooled: n₁ + n₂ − 2; table: (rows − 1)(columns − 1); ANOVA: between k − 1, within N − k

Worked example

Enter these known values and leave the other values blank.

One-sample observation count
10
Degrees of freedom
9

Assumptions and limitations

  • Choose the test design before entering sample sizes. Counts must be whole numbers; the reviewed inference domains require at least two observations per t-test sample and at least two table rows and columns.
  • ANOVA uses total observations across all groups, not observations per group. There must be at least two groups and more observations than groups.
  • Welch mode retains positive sample variances and positive standard deviations. Its degrees of freedom are not rounded to a whole number for subsequent calculations.
  • Degrees of freedom alone do not verify test assumptions or provide a test statistic, p-value or conclusion. The selected design must match how the observations were collected.

Common questions

Why is Welch degrees of freedom not an integer?

It is an effective degrees-of-freedom approximation based on the two variance contributions. It is not simply a count of observations minus a fixed number.

References

Bookify requires usable positive degrees-of-freedom test designs and positive sample standard deviations, while retaining fractional Welch results.

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