Chi-Square Calculator
Calculate chi-square test statistics, p-values, and critical values. Perform goodness-of-fit and independence tests with step-by-step analysis.
Goodness of Fit Test
χ² = 13.9889
P-Value
0.0011
Critical Value
5.9369
Decision
Reject H₀
Conclusion
Reject null hypothesis: Observed differs significantly from expected
Contribution to χ² by Category
| Category | Observed | Expected | Contribution |
|---|---|---|---|
| 1 | 20 | 33.33 | 5.3312 |
| 2 | 30 | 33.33 | 0.3327 |
| 3 | 50 | 33.34 | 8.325 |
Chi-Square Test Assumptions
- • Independence: Observations must be independent
- • Sample Size: Expected frequency ≥ 5 in each cell (rule of thumb)
- • Random Sampling: Data should be randomly sampled
- • Categorical Data: Variables must be categorical (nominal or ordinal)
- • Mutually Exclusive: Each observation falls into exactly one category
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About This Calculator
The Chi-Square Calculator is an essential statistical tool for analyzing categorical data and testing hypotheses about expected versus observed frequencies. Whether you are conducting a goodness-of-fit test to determine if your sample data matches a theoretical distribution, performing a test of independence to examine relationships between categorical variables in a contingency table, or calculating critical values for hypothesis testing, this calculator provides accurate results with detailed step-by-step analysis. Chi-square tests are fundamental in research across fields including biology, psychology, sociology, market research, quality control, and medical studies. By comparing observed frequencies to expected frequencies, researchers can determine whether differences are statistically significant or simply due to chance. Our calculator handles both one-variable goodness-of-fit tests and two-variable tests of independence, automatically computing the chi-square statistic, degrees of freedom, p-value, and critical values at common significance levels. Perfect for students learning statistical analysis, researchers conducting surveys and experiments, data analysts examining categorical data, and professionals who need to validate hypotheses about categorical distributions.
How to Use the Chi-Square Calculator
- 1Select the test type: Goodness of Fit or Test of Independence.
- 2For Goodness of Fit: Enter observed frequencies and expected frequencies.
- 3For Test of Independence: Enter your contingency table data.
- 4Choose your significance level (alpha): 0.05, 0.01, or 0.10.
- 5Click Calculate to see the chi-square statistic and p-value.
- 6Review the step-by-step calculation breakdown.
- 7Compare the test statistic to the critical value to make your decision.
- 8Interpret the results to accept or reject the null hypothesis.
Understanding the Chi-Square Test
The chi-square test is a statistical hypothesis test that compares observed frequencies with expected frequencies.
Chi-Square Formula:
chi-square = Sum of [(Observed - Expected)^2 / Expected]
Key Components:
- Observed (O): The actual counts from your data
- Expected (E): The counts you would expect under the null hypothesis
- Degrees of Freedom: (rows - 1) x (columns - 1) for independence tests
When to Use Chi-Square:
- Testing if sample data fits a theoretical distribution
- Examining independence between two categorical variables
- Analyzing survey responses across categories
- Quality control and defect analysis
Critical Values Table
Common chi-square critical values for hypothesis testing:
| df | alpha = 0.10 | alpha = 0.05 | alpha = 0.01 |
|---|---|---|---|
| 1 | 2.706 | 3.841 | 6.635 |
| 2 | 4.605 | 5.991 | 9.210 |
| 3 | 6.251 | 7.815 | 11.345 |
| 4 | 7.779 | 9.488 | 13.277 |
| 5 | 9.236 | 11.070 | 15.086 |
| 10 | 15.987 | 18.307 | 23.209 |
| 20 | 28.412 | 31.410 | 37.566 |
Decision Rule: If chi-square statistic > critical value, reject the null hypothesis.
Pro Tips
- 💡Ensure all expected frequencies are at least 5 for reliable results.
- 💡Use degrees of freedom = (rows - 1) x (columns - 1) for contingency tables.
- 💡A larger chi-square value indicates greater difference from expected.
- 💡Compare your p-value to your chosen significance level (usually 0.05).
- 💡For 2x2 tables with small samples, consider Fisher's exact test instead.
Frequently Asked Questions
The chi-square test is used to determine if there is a significant association between categorical variables (test of independence) or if observed frequencies match expected frequencies (goodness of fit test). It is widely used in research, quality control, and data analysis.
