Chi-Square Test Calculator

The chi-square test (χ² test) is one of the most widely used tools in statistics. It answers a fundamental question: is the pattern you observe in your data the result of genuine structure, or could it simply be random variation? Unlike tests that compare means, the chi-square test works entirely with counts — how many items fall into each category.

Two versions of the test are most common. The goodness-of-fit test takes a single categorical variable and compares your observed frequency distribution to an expected one. Did the die land on each face the expected number of times? Do blood type frequencies in your sample match the known population distribution? The test for independence takes two categorical variables — typically arranged in a contingency table — and asks whether they are associated. Does smoking status depend on disease status? Is product preference related to customer age group?

Both tests compute the same core statistic: χ² = Σ((O − E)² / E), where O is the observed count and E is the expected count in each cell. A large χ² value indicates the observed data deviate substantially from what independence or the specified distribution would predict.

The goodness-of-fit test compares one observed frequency distribution to a theoretical one. You have k categories, each with an observed count O_i and an expected count E_i. The test statistic is:

Example — Dice fairness test: Roll a die 60 times. If the die is fair, you expect 10 rolls on each face. Your actual results are [8, 12, 9, 11, 7, 13]. Compute each cell's contribution: (8−10)²/10 + (12−10)²/10 + (9−10)²/10 + (11−10)²/10 + (7−10)²/10 + (13−10)²/10 = 0.4 + 0.4 + 0.1 + 0.1 + 0.9 + 0.9 = 2.8. With df = 5, the right-tail p-value is approximately 0.73 — the die appears fair.

If the user does not know the expected distribution, it is valid to assume equal proportions across all categories. This tests the null hypothesis that every category is equally likely. Geneticists use this for Mendelian ratios; marketers use it for survey response distributions; educators use it for grade distributions across classes.

The test for independence uses a two-way contingency table — a grid where each row represents one categorical variable and each column represents another. The cells contain the observed counts at each combination of categories.

Example — Smoking and disease: In a study of 100 people, 50 are smokers and 50 are non-smokers. 30 smokers have disease, 20 do not; 10 non-smokers have disease, 40 do not. The expected counts under independence are E[smoker, disease] = 50×40/100 = 20, and so on. The resulting χ²(1) ≈ 16.67 with p < 0.001 — a highly significant association between smoking and disease.

Common applications include survey analysis (gender × satisfaction level), A/B testing (device type × click-through), biology (treatment group × outcome), and social science research. The only requirement is that the data consist of counts in mutually exclusive categories, not means or percentages.

The chi-square test returns three key numbers: the χ² statistic, degrees of freedom, and p-value. A large χ² relative to the degrees of freedom produces a small p-value, which is evidence against the null hypothesis.

The p-value is always a right-tail probability — the probability of observing a χ² at least as large as yours if the null hypothesis were true. Because deviations from the null are always positive (squared), chi-square tests are inherently one-directional (right-tailed only).

Conventional thresholds: p < 0.05 is "significant at the 5% level"; p < 0.01 is "significant at the 1% level"; p < 0.001 is "highly significant." However, statistical significance does not imply practical importance. Use Cramér's V or Cohen's w to gauge effect size:

The chi-square test rests on four key assumptions. Violating them does not always make the test useless, but it does reduce its reliability.

• 1
  Random sampling: The data should come from a random or representative sample. Results from a self-selected sample (e.g., online opt-in surveys) may not generalise to the broader population.
• 2
  Independence of observations: Each observation must belong to exactly one cell. If one person can appear in multiple categories, the chi-square approximation breaks down.
• 3
  Expected cell count ≥ 5: This is the most commonly violated assumption. When expected frequencies are small, the chi-square distribution is a poor approximation to the true sampling distribution. The calculator warns you when any expected cell falls below 5.
• 4
  Counts, not proportions: Enter the raw counts of observations, not percentages or proportions. If you only have relative frequencies, you must know the total sample size to back-calculate actual counts.

When expected cell counts are small and you have a 2×2 table, consider Fisher's exact test instead. For larger sparse tables, consider combining adjacent categories before running the chi-square test.