Chi-Squared Test Calculator
Calculate the chi-squared statistic to test whether observed data fits an expected distribution, or whether two categorical variables are independent.
Chi-Squared Test Guide
Chi-Squared Formula and Steps
χ² = Σ (O − E)² / E. Where O = observed frequency, E = expected frequency. Steps: state the null hypothesis (H₀: no significant difference / variables are independent). Calculate expected frequencies. Calculate (O−E)²/E for each category. Sum all values to get χ². Find degrees of freedom (df = number of categories − 1 for goodness-of-fit). Compare to critical value at p = 0.05 (5% significance level). If χ² > critical value: reject H₀ (result is statistically significant). If χ² < critical value
Degrees of Freedom and Critical Values
For goodness-of-fit: df = number of categories − 1. Critical values at p = 0.05: df 1: 3.841, df 2: 5.991, df 3: 7.815, df 4: 9.488, df 5: 11.070. Example: genetics cross expecting 3:1 ratio (2 categories) → df = 1 → critical value = 3.841. If calculated χ² = 2.1: 2.1 < 3.841, do not reject H₀ — data consistent with expected ratio. If χ² = 5.2: 5.2 > 3.841, reject H₀ — data significantly differs from expected 3:1 ratio.
Chi-Squared in Biology (Genetics)
The chi-squared test is used to test whether observed genetic ratios match expected Mendelian ratios. Example: cross expected to give 3:1 phenotype ratio. Observed: 72 dominant, 28 recessive (n=100). Expected: 75 dominant, 25 recessive. χ² = (72−75)²/75 + (28−25)²/25 = 9/75 + 9/25 = 0.12 + 0.36 = 0.48. df = 1, critical value = 3.841. 0.48 < 3.841 — do not reject H₀. The observed results are consistent with the expected 3:1 Mendelian ratio. Always state what the null hypothesis is before calculat
Assumptions and Limitations
Chi-squared assumptions: data must be frequencies (counts), not proportions or percentages. Expected frequency in each category must be ≥ 5 (small expected frequencies inflate χ² artificially). Observations must be independent. The test does not work for continuous data (use t-test or ANOVA instead). For a 2×2 contingency table with small samples (any expected count < 5), use Fisher's exact test instead. Chi-squared only detects whether differences exist — it does not indicate the direction or m
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