Calculator
Example Data Table
| Category | Observed | Expected | Meaning |
|---|---|---|---|
| Red | 18 | 20 | Observed count is lower than expected. |
| Blue | 22 | 20 | Observed count is higher than expected. |
| Green | 20 | 20 | Observed count matches expectation. |
| Yellow | 25 | 20 | This row may add more evidence. |
| Purple | 15 | 20 | Observed count is lower than expected. |
Formula Used
The calculator uses the chi square test statistic formula:
x² = Σ ((O - E)² / E)
Here, O is each observed count. E is each expected count. The calculator sums every category contribution.
When continuity correction is enabled, the contribution uses this adjusted difference:
((|O - E| - 0.5)² / E)
The p value is estimated from the upper tail of the chi square distribution. The critical value is found from the selected alpha level and degrees of freedom.
How to Use This Calculator
- Enter observed counts separated by commas, spaces, or new lines.
- Enter expected counts, weights, or proportions in the same order.
- Add category labels for clearer output.
- Select alpha, rounding, and degrees of freedom options.
- Use scaling when expected values are weights or proportions.
- Press the calculate button.
- Review the statistic, p value, decision, and contribution table.
- Download CSV or PDF results when needed.
Understanding the Chi Square Test Statistic
The chi square test statistic measures how far observed counts move away from expected counts. It is useful when data comes as categories, not measurements. Common uses include goodness of fit checks, survey response testing, genetics ratios, quality control, and table based association studies.
What the Result Means
A small value means the observed pattern is close to expectation. A large value means the pattern has stronger disagreement. The calculator also estimates a p value from the chi square distribution. The p value shows how unusual the statistic is when the null assumption is true. Lower p values suggest stronger evidence against that assumption.
Degrees of Freedom
Degrees of freedom control the shape of the reference distribution. For a goodness of fit test, it is usually number of categories minus one. If expected values were estimated from sample data, subtract each estimated parameter. You may also enter a manual value when your study design needs a different rule.
Expected Counts Matter
Expected counts must be positive. Very small expected counts can make the approximation unstable. A common practical check is that most expected counts should be at least five. If counts are too small, combine categories, collect more data, or use an exact method where possible.
Using Advanced Options
This calculator accepts observed counts, expected counts, labels, alpha, decimal rounding, estimated parameters, and optional correction. It reports each contribution, residual, total statistic, critical value, p value, and decision. The CSV and PDF tools help you save results for notes, reports, and audit trails.
Interpreting Evidence Carefully
Statistical significance is not the same as practical importance. A very large sample can make small differences look important. Review the effect size, the contribution table, and the study context. The largest contribution rows often show which categories create most of the disagreement.
Best Practice
Enter clean count data only. Do not enter percentages unless they have been converted to counts. Keep categories independent. Match every observed value with its expected value. Then compare the statistic, p value, and context before drawing a final conclusion. When results are borderline, document assumptions and rerun checks with clear category notes. This makes later review easier for teachers, analysts, and clients.
FAQs
What does x² mean in this calculator?
x² means the chi square test statistic. It summarizes the total difference between observed and expected counts across all categories.
Can I use percentages as observed values?
No. Use actual counts for observed values. Percentages should be converted into counts before using the calculator.
What are expected counts?
Expected counts are the category counts predicted by your null hypothesis. They must be positive and match the observed category order.
What is a good p value?
A smaller p value means stronger evidence against the null assumption. Many studies compare it with 0.05, but your field may require another alpha.
When should I scale expected values?
Use scaling when expected entries are proportions, ratios, or weights. The calculator converts them to expected counts using the observed total.
What are degrees of freedom?
Degrees of freedom define the reference curve. For goodness of fit, use categories minus one, then subtract estimated parameters if needed.
What does the contribution table show?
It shows each category’s part of the total statistic. Larger contribution values identify categories with stronger disagreement from expectation.
Does this prove causation?
No. A chi square result shows statistical disagreement or association. It does not prove cause and effect by itself.