Chi-Square Goodness-of-Fit Test

Chi-square goodness-of-fit test comparing observed counts to an expected distribution.

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About this tool

Enter observed counts and, optionally, expected counts. Leave expected blank to test against equal frequencies. The tool reports χ², df, p and Cohen's w.

How to use

1
Step 1
Enter the observed counts.
2
Step 2
Enter expected counts, or leave blank for uniform.
3
Step 3
Add extra estimated parameters if any and press Calculate.

How it works

The test compares observed counts with those a hypothesis predicts: Σ(O − E)² ÷ E. df is categories − 1 − estimated parameters. A low-count warning appears when expected counts are too small for the chi-square approximation to be trustworthy.

χ² = Σ (O − E)² ÷ E      df = k − 1 − (parameters estimated)

Worked example

Is the die fair?

Input
Observed 5, 8, 9, 8, 10, 20 (n = 60)
Output
χ²(5) = 13.40, p = .020, w = 0.47
Walkthrough
Expected 10 each. (25 + 4 + 1 + 4 + 0 + 100) ÷ 10 = 13.4. The right-tail p for 13.4 on 5 df is.0199.

Limitations

  • Counts, not percentages.
  • Expected counts under about 5 make the p-value unreliable.

Questions

How do I test a die for fairness?
Enter the six counts and leave expected blank; the expected count is n ÷ 6.
What does 18, 22, 20, 19, 21 give?
χ²(4) = 0.50, p =.974: very consistent with equal frequencies.
How many degrees of freedom?
Categories minus one, minus any parameters estimated from the data.

Features

  • Uniform or custom expected
  • Cohen's w
  • Low-count warning

Related resources

Guides and videos from around the web—opens on YouTube or the publisher’s site.

  • Student's t-distribution

    Khan Academy

    Why t-tables differ from z for small samples.

    Watch on YouTube

  • Hypothesis testing and p-values

    StatQuest with Josh Starmer

    Intuition for t/z/χ² decision rules.

    Watch on YouTube

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