χ² Chi-Square Test Calculator
Calculate a chi-square goodness-of-fit statistic from observed and expected category counts.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Calculate a chi-square goodness-of-fit statistic from observed and expected category counts.
| Concept | Formula or rule |
|---|---|
| Chi-Square Test Calculator | χ² = Σ(O−E)²/E |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: Compare observed counts 24,36,40 with equal expected counts.
✓ df = categories − 1 − estimated constraints
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Using percentages instead of counts
The standard test uses frequency counts.
The standard test uses frequency counts.
⚠️
Allowing very small expected counts
Sparse categories can make the approximation unreliable.
Sparse categories can make the approximation unreliable.
⚠️
Ignoring how expected values were estimated
Estimated parameters reduce degrees of freedom.
Estimated parameters reduce degrees of freedom.
Practice Questions
1. With 4 categories and no estimated parameters, what is df?
Answer: 3
2. If observed equals expected exactly, what is χ²?
Answer: 0
3. Must expected counts sum to observed counts?
Answer: Usually yes for goodness-of-fit.
Frequently Asked Questions
What is the null hypothesis?▾
Observed frequencies follow the expected distribution.
Can expected counts be zero?▾
No.
What makes χ² large?▾
Large squared differences relative to expected counts.