What is the null hypothesis?

The null hypothesis is that the observed frequencies come from the specified expected distribution. A small p-value gives evidence to reject this and conclude the data do not fit the expected pattern.

What p-value is considered significant?

The conventional threshold is 0.05. If p < 0.05 you reject the null hypothesis. Some fields use 0.01 or 0.10 depending on the desired stringency.

When is the test not reliable?

The test assumes all expected cell counts are at least 5. If any expected count is below 5, the chi-square approximation becomes unreliable. Combine adjacent categories or collect more data.

What are degrees of freedom for this test?

Degrees of freedom = number of categories minus 1. If you estimated parameters from the data (e.g., mean and variance for a normal distribution), subtract those estimated parameters as well.

Can I enter proportions instead of expected counts?

Yes. Switch the "Expected type" toggle to Proportions. Enter values like 0.25, 0.25, 0.25, 0.25 (they will be normalized to the observed total automatically). Proportions do not need to sum to exactly 1 before normalization.

How is this different from the chi-square test of independence?

The goodness of fit test compares one categorical variable's distribution to a theoretical model. The independence test compares two categorical variables in a contingency table to see if they are related.

Chi-Square Goodness of Fit

Enter observed and expected category counts to compute the chi-square statistic, degrees of freedom, and p-value for a goodness-of-fit test.

Category Frequencies

Number of Categories

Expected Values Are

Label

Observed

Expected

Significance Level (α)

Enter frequencies and click Run Test to see results.

Summary

The chi-square goodness of fit test checks whether a set of observed categorical frequencies matches a hypothesized distribution. You supply the observed count for each category and either raw expected counts or proportions, and the tool calculates the chi-square statistic and the corresponding p-value. A small p-value (typically below 0.05) indicates the data do not follow the expected distribution.

How it works

Choose the number of categories (2 to 10).
Enter a label, the observed count, and the expected count or proportion for each category.
Select whether your expected values are counts or proportions; the tool normalizes proportions to the grand total automatically.
Click "Run Test" — the tool calculates the chi-square statistic as the sum of (observed - expected)^2 / expected across all categories.
Degrees of freedom equal the number of categories minus 1 (minus any estimated parameters if applicable).
The p-value is derived from the chi-square distribution; compare it to your chosen significance level.

Use cases

Test whether a six-sided die is fair by comparing roll frequencies to a uniform distribution.
Check whether customer complaint categories match historical proportions.
Validate whether website traffic follows a predicted day-of-week pattern.
Test whether genetic phenotype ratios fit a Mendelian 3:1 ratio.
Evaluate whether survey response distributions match prior-year benchmarks.
Determine if observed color preferences match a theorized distribution.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu