Calculator Form
Example Data Table
| Session | Buy | Browse | Leave |
|---|---|---|---|
| Morning | 35 | 28 | 17 |
| Afternoon | 30 | 40 | 20 |
| Evening | 25 | 22 | 33 |
This sample compares session timing with customer behavior. It helps demonstrate whether the two categorical variables appear independent.
Formula Used
Expected frequency: E = (Row Total × Column Total) / Grand Total
Chi-square statistic: χ² = Σ ((O - E)² / E)
Degrees of freedom: (rows - 1) × (columns - 1)
P-value: upper tail probability from the chi-square distribution using the calculated χ² and degrees of freedom.
Cramer's V: √(χ² / (n × min(r - 1, c - 1)))
O is the observed count. E is the expected count. Lower p-values suggest stronger evidence against independence.
How to Use This Calculator
- Select the number of rows and columns in your contingency table.
- Add clear row labels and column labels.
- Enter each observed frequency in the table.
- Choose your significance level. The default is 0.05.
- Use Yates correction only for a 2×2 table when needed.
- Click Calculate to view the statistic, p-value, expected counts, and decision.
- Download the result as CSV or PDF after calculation.
About the Chi Square Test of Independence
Why this test matters
The chi square test of independence checks whether two categorical variables are related. It is common in statistics, research, quality studies, surveys, education, healthcare, and market analysis. The method compares observed counts with expected counts under an independence assumption.
What the calculator returns
This calculator produces the chi square statistic, degrees of freedom, p-value, expected frequencies, row totals, column totals, grand total, and Cramer's V. These outputs help you judge significance and effect size in one place. The expected table is useful for checking assumptions.
How to read the result
A small p-value suggests that the variables are not independent. In practical terms, category membership in one variable may be associated with category membership in the other. A large p-value means your data does not provide enough evidence to claim an association.
Assumptions to review
Each observation should belong to one cell only. The sample should represent the population reasonably well. Expected counts should usually not be too small. Many analysts review how many expected frequencies fall below five before interpreting the test strongly.
When to use it
Use this tool for contingency tables such as device type by conversion outcome, shift by defect class, age group by preference, or treatment group by response category. It works best when the data are counts rather than percentages, averages, or continuous measurements.
Why expected counts matter
The chi square formula measures the gap between observed and expected values. Large gaps increase the test statistic. Very small gaps reduce it. If your expected counts are low, results may be less reliable, so review the assumption summary before reporting conclusions.
FAQs
1. What does this calculator test?
It tests whether two categorical variables are statistically independent. It compares observed frequencies with expected frequencies that would appear if no association existed.
2. Can I use percentages instead of counts?
No. The test requires raw frequency counts in each cell. If you only have percentages, convert them back to counts first.
3. What is an expected frequency?
An expected frequency is the count predicted for a cell if the row and column variables are independent. It comes from row totals, column totals, and the grand total.
4. What does a low p-value mean?
A low p-value means the observed table would be unlikely under independence. That gives evidence that the two categorical variables may be associated.
5. When should I use Yates correction?
Use Yates correction for some 2×2 tables, especially when counts are small. It slightly reduces the chi-square value and makes the test more conservative.
6. What is Cramer's V?
Cramer's V is an effect size measure for categorical association. It helps describe strength after significance is tested, with larger values showing stronger relationships.
7. What if expected counts are under five?
That can weaken the usual chi-square approximation. Review the assumption warning carefully. In some cases, combining categories or using an exact method may be better.
8. Can this calculator handle larger tables?
Yes. This version supports tables from 2×2 up to 6×6. That covers many practical independence testing tasks.