A Test of Independence

Statistics • Chi Square Tests

Written by STEM Calculators Team Published December 25, 2025 Updated February 24, 2026

Task

Enter a contingency table of observed counts. The calculator computes expected counts, the χ² statistic, df, and the p-value (right-tailed). If you choose the test option, it also compares to a chosen significance level α.

Rows (R)

Number of row categories (exclude totals).

Columns (C)

Number of column categories (exclude totals).

Significance level α

The test of independence uses a right-tail χ² rejection region.

Labels

You can paste a labeled CSV below to auto-fill the table.

Observed counts (editable table)

Enter nonnegative counts. Totals are computed automatically.

Key formulas used

\[ \begin{aligned} df &= (R-1)(C-1) \\ E_{ij} &= \frac{(\text{row total}_i)(\text{column total}_j)}{n} \\ \chi^2 &= \sum \frac{(O_{ij}-E_{ij})^2}{E_{ij}} \end{aligned} \]

The p-value is computed as the right-tail area: \(\;p = P(\chi^2_{df} \ge \chi^2_{\text{obs}})\).

How to use

1) Build the table, 2) enter observed counts, 3) click Calculate. Use batch mode if you prefer pasting CSV.

Ready

χ² model visualization (right tail)

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Blue shading shows the p-value region (right tail). Red shading shows the rejection region at level α.

Outputs

The results below include: expected counts, χ², df, and (if testing) the p-value and critical value.

Tip

Use Copy tables to paste into documents/spreadsheets.

Enter observed counts and click Calculate.

Batch mode: paste CSV (or paste from a CSV file)

Paste a contingency table as CSV. Supported formats:
A) Matrix only (numbers): each row is a category, each column is a category.
B) With labels: first row contains column labels and first column contains row labels.
Example (with labels): ,In Favor,Against,No Opinion Men,93,70,12 Women,87,32,6

CSV input

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Chi-square Test of Independence

This calculator analyzes a contingency table (two categorical variables) using the chi-square distribution. It can compute expected counts and perform the test of independence (right-tailed).

How to use the calculator

Choose a task:
- Compute expected frequencies only, or
- Chi-square test of independence.
Set the number of rows (R) and columns (C), then build the table.
Enter the observed counts in each cell (nonnegative integers).
For the test, select a significance level α and click Calculate.
Optional: paste a table as CSV in the batch section and apply it to the editable table.

What the calculator computes

Degrees of freedom

\[ df = (R-1)(C-1) \]

Expected counts

\[ E_{ij}=\frac{(\text{row total}_i)\cdot(\text{column total}_j)}{n} \]

Chi-square test statistic

\[ \chi^2=\sum_{i=1}^{R}\sum_{j=1}^{C}\frac{(O_{ij}-E_{ij})^2}{E_{ij}} \]

Right-tail p-value

\[ p=P\!\left(\chi^2_{df}\ge \chi^2_{\text{obs}}\right) \]

Hypotheses and decision rule

\[ \begin{aligned} H_0&:\ \text{The two categorical variables are independent.}\\ H_1&:\ \text{The two categorical variables are dependent.} \end{aligned} \]

The chi-square test of independence is right-tailed. At significance level α: reject \(H_0\) if the p-value is \(\le \alpha\), equivalently if \(\chi^2_{\text{obs}} \ge \chi^2_{1-\alpha,df}\).

\[ \text{Reject }H_0\ \text{if}\ \chi^2_{\text{obs}}\ge \chi^2_{1-\alpha,df} \]

Validity guideline

A common guideline for using the chi-square approximation is that all expected counts should be at least 5. If some expected counts are smaller, consider combining categories or increasing the sample size.

Frequently Asked Questions

What does a chi-square test of independence determine?

It tests whether two categorical variables are independent by comparing the observed contingency table to the expected table under independence. A small p-value suggests an association between the variables.

How are expected counts computed in a contingency table?

For each cell, the expected count is Eij = (row total i x column total j) / n, where n is the grand total. This is the expected frequency if the variables are independent.

What degrees of freedom are used for the chi-square independence test?

The test uses df = (R - 1)(C - 1), where R is the number of row categories and C is the number of column categories (excluding totals).

Why is the chi-square test of independence right-tailed?

The chi-square statistic is a sum of squared standardized differences, so it is always nonnegative. Larger values indicate larger departures from independence, so evidence against H0 is in the right tail.

What should I do if some expected counts are less than 5?

A common guideline is that expected counts should be at least 5 in every cell for the chi-square approximation to be reliable. If some expected counts are small, consider combining categories or increasing the sample size.