Chi-Square Test of Independence

Enter observed frequencies in a contingency table (up to 4x4) to test whether two categorical variables are independent. The calculator shows expected frequencies, chi-square statistic, p-value, and Cramer's V effect size.

Observed vs Expected

Contingency Table

Rows

Columns

	Col 1	Col 2
Row 1
Row 2

Significance Level (alpha)

Presets

Results

Chi-Square Statistic

11.43

Degrees of Freedom

1

P-value

0.0007232

Cramer's V (effect size)

0.378 (medium)

Decision (alpha = 0.05): Reject the null hypothesis. The variables are not independent.

Expected Frequencies

23	18
23	18

Step-by-Step Calculation

1. Expected Frequencies

E_{ij} = \frac{R_i \times C_j}{N}

E_{11} = \frac{40 \times 45}{80} = 22.5

\text{(computed for all 2 \times 2 cells)}

2. Cell Contributions

\frac{(O_{ij} - E_{ij})^2}{E_{ij}}

\frac{(30 - 22.5)^2}{22.5} = 2.5, \quad \frac{(10 - 17.5)^2}{17.5} = 3.214, \quad \frac{(15 - 22.5)^2}{22.5} = 2.5, \quad \frac{(25 - 17.5)^2}{17.5} = 3.214

3. Chi-Square Statistic

\chi^2 = \sum_{i,j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}

\chi^2 = \text{sum of all cell contributions}

\chi^2 = 11.43

4. Degrees of Freedom

df = (r - 1)(c - 1)

df = (2 - 1)(2 - 1)

df = 1

5. P-value and Decision

p = P(\chi^2_{1} > 11.43)

p = 0.0007232

p < \alpha = 0.05 \implies \text{Reject } H_0

6. Cramer's V

V = \sqrt{\frac{\chi^2}{N \cdot \min(r-1, c-1)}}

V = \sqrt{\frac{11.43}{80 \cdot 1}}

V = 0.378

Reference Guide

The Chi-Square Test

The chi-square test of independence checks whether two categorical variables are related or independent. It compares observed counts to the counts you would expect if the variables were independent.

\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}

Expected Frequencies

Under the null hypothesis (no association), the expected count for each cell is calculated from the row and column totals.

E_{ij} = \frac{R_i \times C_j}{N}

As a rule of thumb, all expected frequencies should be at least 5 for the test to be reliable.

Cramer's V

Cramer's V measures the strength of association between the two variables. It ranges from 0 (no association) to 1 (perfect association).

V = \sqrt{\frac{\chi^2}{N \cdot \min(r-1, c-1)}}

Guidelines: below 0.1 is negligible, 0.1 to 0.3 is small, 0.3 to 0.5 is medium, and above 0.5 is large.

When to Use This Test

Use this test when you have count data for two categorical variables and want to know if they are related. Common examples include survey responses by demographic group, A/B test results, and medical study outcomes.

The test requires independent observations and sufficiently large expected counts (typically 5 or more per cell).