When should I use goodness of fit vs. test of independence?

Use goodness of fit when you have one categorical variable and want to test if its distribution matches a theoretical/expected distribution. Use the test of independence when you have two categorical variables and want to test if they are associated. Goodness of fit has one set of observed counts; independence has a contingency table.

Can I use proportions instead of frequencies for expected values?

Yes. If you provide expected proportions (summing to 1), multiply each proportion by the total observed count to get expected frequencies. The calculator supports both input formats. For example, expected proportions of [0.25, 0.25, 0.25, 0.25] with N = 100 gives expected frequencies of [25, 25, 25, 25].

What happens if expected frequency is very small?

If any expected frequency is below 5, the chi-square approximation becomes unreliable. Consider combining adjacent categories to increase expected counts, or use an exact test. Some references allow expected counts as low as 1 if no more than 20% of cells have E < 5.

All Tools

Chi-Square Calculator

Perform chi-square tests of independence and goodness of fit with step-by-step solutions, expected frequencies, and Python code.

Test of Independence

Rows:2

Cols:2

Quick Examples

Enter your data and click Calculate to see results.

Quick Reference:

Independence: Test if two categorical variables are associated
Goodness of Fit: Test if observed data matches expected distribution

Significance Levels:

p < 0.05: Significant (reject H₀)
p < 0.01: Very significant
p < 0.001: Highly significant

Frequently Asked Questions

What are the assumptions of the chi-square test of independence?

The test requires: (1) observations are independent, (2) data are categorical (nominal or ordinal), (3) expected frequencies should be at least 5 in each cell (some texts allow 80% of cells with E >= 5). For small samples or when assumptions are violated, use Fisher's exact test.

What does Cramér's V tell me?

Cramér's V is an effect size measure ranging from 0 (no association) to 1 (perfect association). For a 2x2 table, V equals the absolute value of the phi coefficient. Guidelines: V = 0.1 is small, V = 0.3 is medium, V = 0.5 is large. It helps assess practical significance beyond the p-value.

What is the difference between chi-square test of independence and homogeneity?

Mathematically, the calculations are identical. The difference is in study design: the test of independence uses one sample classified on two variables, while the test of homogeneity compares distributions across two or more populations. The interpretation differs even though the formula is the same.

How do I handle small expected frequencies?

When expected frequencies are below 5, the chi-square approximation may be poor. Options: (1) combine categories to increase expected counts, (2) use Fisher's exact test for 2x2 tables, (3) use a Monte Carlo simulation for larger tables, or (4) report results with a caveat about the approximation.

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Frequently Asked Questions