Calculator Inputs
Enter values separated by commas, spaces, semicolons, pipes, or new lines.
Formula Used
Chi-square statistic:
χ² = Σ ((Oᵢ - Eᵢ)² / Eᵢ)
Degrees of freedom:
df = k - 1 - m
Where: Oᵢ is observed count, Eᵢ is expected count, k is number of categories, and m is estimated parameters.
Effect size: Cohen's w = √(χ² / N)
How to Use This Calculator
- Enter observed category counts in the first box.
- Enter expected counts, probabilities, or percentages in the second box.
- Add category labels if you want clearer tables and charts.
- Select the expected value type.
- Choose whether expected values should be scaled or normalized.
- Enter the alpha level, such as 0.05.
- Add estimated parameter adjustment when needed.
- Click calculate, then review the result above the form.
- Use CSV or PDF buttons to save the report.
Example Data Table
| Category | Observed Count | Expected Count | Use Case |
|---|---|---|---|
| Red | 18 | 20 | Product color share |
| Blue | 22 | 20 | Product color share |
| Green | 20 | 20 | Product color share |
| Yellow | 25 | 20 | Product color share |
| Purple | 15 | 20 | Product color share |
Understanding the Goodness of Fit Test
A chi square goodness of fit test checks one categorical variable. It compares observed counts with expected counts. The expected counts may come from theory, history, market share, genetics, or a target distribution. The test asks whether the visible differences are small enough to be random variation.
Why the Test Matters
This calculator helps when data is grouped into classes. You can test dice rolls, survey choices, product defects, color ratios, website clicks, or customer segments. It shows the total statistic, degrees of freedom, p-value, and decision. It also shows each category contribution. That helps you see which classes drive the mismatch.
Reading the Results
A small chi square statistic means the observed pattern is close to the expected pattern. A large statistic means the gap is stronger. The p-value gives the chance of seeing a mismatch this large, assuming the expected model is true. If the p-value is below alpha, the result is statistically significant. The null model is rejected for that alpha level.
Using Residuals
Residuals make the output easier to inspect. A positive residual means the observed count is above the expected count. A negative residual means it is below the expected count. Larger absolute residuals deserve attention. They often identify categories that need review, cleaning, or deeper research.
Practical Checks
Use counts, not percentages, in the observed field. Expected counts should usually be at least five. Very small expected counts can weaken the test. Combine rare categories when it makes practical sense. Do not use overlapping categories. Every observation should belong to one class only.
Reporting Guidance
A clear report should include the statistic, degrees of freedom, p-value, alpha, sample size, and decision. Mention how expected counts were formed. If parameters were estimated from the same data, reduce degrees of freedom using the adjustment field. Export the table for records. Use the chart to explain results to nontechnical readers.
Limitations
The test does not prove why differences exist. It only signals whether the pattern departs from the expected model. Review sampling method, independence, and data quality before taking action. Strong statistics are useful, but context still matters in every report.
FAQs
1. What does this calculator test?
It tests whether observed category counts match an expected distribution. The output includes the chi-square statistic, degrees of freedom, p-value, critical value, residuals, and a decision based on the selected alpha level.
2. Should I enter counts or percentages?
Observed values should be counts. Expected values can be counts, probabilities, or percentages. Select the correct expected value type before calculating, so the page can convert expected values properly.
3. What is the null hypothesis?
The null hypothesis says the observed counts follow the expected distribution. A low p-value suggests the observed pattern is unlikely under that expected model.
4. What does a small p-value mean?
A small p-value means the observed and expected counts differ more than random variation would usually allow. If p is below alpha, reject the null hypothesis.
5. Why are residuals shown?
Residuals show which categories are above or below their expected values. Large positive or negative residuals identify categories that contribute heavily to the total statistic.
6. What if expected counts are below five?
Small expected counts can make the approximation less reliable. Consider combining rare categories when it is logical and does not damage the meaning of the analysis.
7. What is the estimated parameters adjustment?
Use it when expected values depend on parameters estimated from the same sample. Each estimated parameter usually reduces degrees of freedom by one.
8. Can I export the report?
Yes. After calculating, use the CSV button for spreadsheet work. Use the PDF button for a formatted report with summary statistics and category results.