Chi-Square Test Calculator
The chi-square test compares observed frequencies to expected frequencies across categories to determine whether an observed distribution is significantly different from what you expected. Enter observed and expected counts (comma-separated, same number of values) for each category. The calculator computes the chi-square statistic, degrees of freedom, and a reference critical value at the 5% significance level. You can then compare your statistic to the critical value or use a chi-square table for your exact degrees of freedom.
Chi-square formula
χ² = ∑ (O - E)² / E
Where O is the observed count in each cell/category and E is the expected count. Degrees of freedom for a goodness-of-fit test = number of categories minus 1. The statistic follows a chi-square distribution under the null hypothesis of no difference.
Interpreting the result
- If the chi-square statistic exceeds the critical value at your chosen alpha level and degrees of freedom, reject the null hypothesis (the observed distribution differs significantly from expected).
- A larger chi-square value indicates greater disagreement between observed and expected frequencies.
- This calculator provides the critical value at alpha = 0.05. For other significance levels, consult a chi-square distribution table.
- The chi-square test is one-tailed (we only care about values being too large, not too small).
Frequently asked questions
What is the chi-square test?
The chi-square (chi-squared) test is a statistical test used to determine whether observed frequencies in data differ significantly from expected frequencies. It is commonly used for goodness-of-fit tests (does data follow a specified distribution?) and tests of independence (are two categorical variables related?).
What is the formula for the chi-square statistic?
Chi-squared = sum of (O minus E) squared divided by E, where O is the observed frequency in each category and E is the expected frequency. Summing over all categories gives the test statistic. Larger values indicate greater discrepancy between observed and expected.
What are degrees of freedom?
For a goodness-of-fit test, degrees of freedom equal the number of categories minus 1 (minus additional estimated parameters). For a test of independence in a two-way table with r rows and c columns, df = (r-1)(c-1).
What critical value do I compare against?
Compare the chi-square statistic to the critical value from the chi-square distribution at your chosen significance level (usually 0.05) and your degrees of freedom. For example, at alpha = 0.05 and df = 3, the critical value is 7.815. If chi-squared exceeds this, the result is significant.
What are the assumptions of the chi-square test?
The chi-square test requires: observations are independent; each cell's expected frequency is at least 5 (commonly recommended); and data are counts, not proportions or percentages. When expected frequencies are small, use Fisher's exact test instead.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Chi-Square Goodness of Fit Test.
- NIST/SEMATECH e-Handbook: Engineering Statistics Handbook.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.