Chi-Square p-Value Calculator
Enter your chi-square statistic and degrees of freedom to compute the right-tail p-value.
This calculator returns the right-tail p-value: p = P(Χ²df ≥ observed χ²).
What this chi-square p-value calculator does
This page helps you quickly find the p-value for a chi-square test when you already have: (1) a chi-square statistic and (2) the degrees of freedom. The output is the probability of seeing a chi-square value at least as large as your observed value if the null hypothesis is true.
In practical terms, a smaller p-value means your data would be less likely under the null model. Researchers commonly compare this p-value to a significance threshold such as α = 0.05.
How to use the calculator
- Enter your test statistic in the χ² field (must be 0 or higher).
- Enter your degrees of freedom (positive integer).
- Optionally set your significance level α (for interpretation).
- Click Calculate p-value.
The tool reports the p-value and a quick decision statement: reject or fail to reject the null hypothesis at your chosen α.
Core formulas behind the result
1) Chi-square test statistic
For many chi-square tests, the statistic is:
where Oᵢ is the observed count and Eᵢ is the expected count for category or cell i.
2) p-value from the chi-square distribution
The p-value is the right-tail area under a chi-square distribution:
Internally, this calculator evaluates the regularized incomplete gamma function for accurate numerical results.
Choosing the correct degrees of freedom
Goodness-of-fit test
If you compare observed counts to expected counts across k categories:
where m is the number of parameters estimated from the data (often 0 in simple cases).
Test of independence (r × c table)
with r rows and c columns.
Worked example
Suppose your chi-square test produced χ² = 10.828 with df = 4. Enter those values and click calculate. The p-value is approximately 0.0286.
- If α = 0.05, then p < α, so you reject the null hypothesis.
- If α = 0.01, then p > α, so you fail to reject the null hypothesis.
Same data, different α, different final decision. That is why reporting the exact p-value is useful.
Interpretation guide
- p < 0.05: evidence against the null hypothesis at the 5% level.
- p ≥ 0.05: not enough evidence to reject the null at the 5% level.
- Very small p-values: data are highly inconsistent with the null model.
Remember: a p-value is not the probability that the null hypothesis is true.
Assumptions and best practices
- Observations should be independent.
- Expected counts in cells should generally not be too small (common rule: most ≥ 5).
- Use correct df formula for your test type.
- Report χ², df, p-value, and context—not just “significant” or “not significant.”
Common mistakes
- Using observed counts instead of expected counts when computing χ² manually.
- Wrong degrees of freedom from miscounted categories or table dimensions.
- Interpreting p-value as effect size.
- Ignoring sample size and practical significance.
Quick FAQ
Can this calculator compute χ² from raw tables?
This version is focused on p-value calculation from an existing χ² statistic and df. If you need raw-table processing, compute χ² first, then use this tool.
Is the p-value one-tailed or two-tailed?
Chi-square tests use a right-tail probability because χ² values are nonnegative and larger values indicate greater deviation from the null model.
Do I need to round inputs?
No. You can enter decimal χ² values. Keep as much precision as you have from your statistical output.