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Statistics & Data Science · pattern F, Dataset

Chi-square goodness of fit, category by category.

Enter observed and expected counts for up to four categories for the chi-square statistic, its degrees of freedom, and a comparison against the critical value you supply.

Inputs

Observed, category 1A count.Expected, category 1Under the null hypothesis.Observed, category 2A count.Expected, category 2Under the null hypothesis.Observed, category 3Zero if unused.Expected, category 3Zero if unused.Observed, category 4Zero if unused.Expected, category 4Zero if unused.Critical χ² at your αFrom tables for your degrees of freedom.

The formula used

χ² = Σ (O − E)² ÷ E

Counts, never percentages. Chi-square operates on frequencies, and converting to proportions first invalidates the test entirely.

Chi-square statistic

3.4400

Categories used: 4
Degrees of freedom: 3
Largest contribution: 1.5000
Total observed: 150
Total expected: 150.0
Decision: do not reject the null
Any expected count below 5?: no

The approximation requires expected counts of at least five in every category. Where one is smaller, combine categories or use an exact test instead.

Every term is a contribution you should look at.

The statistic is a sum, so the individual terms tell you which category drives the result — and that is usually the interesting part of the answer. A significant chi-square with one dominant term is a statement about that category, not about the distribution as a whole. Presenting the per-category contributions is what turns a test statistic into a finding.

Questions about chi square

What are the degrees of freedom?: Categories minus one for a goodness-of-fit test. For a contingency table it is (rows − 1)(columns − 1), which is a different calculation.
What if an expected count is below five?: Combine adjacent categories until every expected count reaches five, or use Fisher's exact test. Proceeding regardless makes the p-value unreliable.
Can chi-square show the direction of an effect?: No. It is always non-negative and only says the observed counts differ from expected. Inspect the individual contributions and their signs to describe how.

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