Compute Pearson Chi-Square test of independence (χ²), degrees of freedom, p-value, expected frequencies, and Cramer’s V for 2x2 contingency tables.
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The Chi Square Calculator performs Pearson’s Chi-Square Test of Independence on 2×2 contingency tables to determine whether there is a statistically significant association between two categorical variables.
E_{ij} = \frac{R_i \times C_j}{N}, \quad \chi^2 = \sum_{i,j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}, \quad V = \sqrt{\frac{\chi^2}{N \cdot \min(r-1, c-1)}}Where O is observed frequency, E is expected frequency based on independent marginal totals, N is grand sample size, and V is Cramer’s V effect size.
Scenario: Testing preference between two website designs across 100 visitors (30 vs 20, 15 vs 35)
Row totals: R₁ = 50, R₂ = 50. Column totals: C₁ = 45, C₂ = 55. N = 100.
Expected frequencies: E₁₁ = 22.5, E₁₂ = 27.5, E₂₁ = 22.5, E₂₂ = 27.5.
Sum of (O - E)² / E across all 4 cells: 2.5 + 2.05 + 2.5 + 2.05 = 9.09.
Degrees of freedom: (2 - 1) × (2 - 1) = 1.
A Chi-Square test of independence tells you whether an observed difference between categorical groups is likely due to real relationships or mere random sampling chance.
The standard scientific threshold is α = 0.05. If the calculated p-value is less than 0.05, you reject the null hypothesis of independence.
Standard statistical guidelines recommend that all expected cell frequencies should be 5 or greater. If expected counts are less than 5, Fisher’s Exact Test is recommended.
Cramer’s V measures the strength of association (effect size) between categorical variables on a scale from 0 (no relationship) to 1 (perfect correlation).
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