Hong Life CalculatorsUnderstanding the chi-square distribution calculator — concept and principle
A chi-square distribution calculator converts a χ² test statistic and its degrees of freedom into a right-tail p-value, and returns the critical χ² value for a chosen significance level α. It is used with the chi-square goodness-of-fit test, the test of independence for contingency tables, and tests and confidence intervals for a variance.
How it is calculated
- p-value (right tail) = 1 − F(χ²; df), the probability of a statistic at least as large as the one observed if the null hypothesis is true.
- Critical χ² is the value c with F(c; df) = 1 − α, found by numerical search.
F is the chi-square cumulative distribution function, computed with the regularized lower incomplete gamma function: F(x; k) = P(k/2, x/2). Degrees of freedom are k − 1 for a goodness-of-fit test with k categories (minus any estimated parameters) and (r − 1)(c − 1) for an r × c table. The statistic cannot be negative.
The principle
If Z₁, …, Z_k are independent standard normal variables, the sum Z₁² + … + Z_k² follows a chi-square distribution with k degrees of freedom. In a contingency-table or goodness-of-fit test, each term (observed − expected)² ÷ expected behaves approximately like a squared standard normal when expected counts are large enough, so their sum is approximately χ². The distribution is skewed to the right, has mean k and variance 2k, and becomes more symmetric as k grows. Only large values signal a poor fit, which is why the test uses the right tail.
Worked example
- Inputs: χ² = 7.8, df = 3, α = 0.05.
- F(7.8; 3) = P(1.5, 3.9) ≈ 0.94967.
- p-value = 1 − 0.94967 ≈ 0.0503.
- Critical value: F(c; 3) = 0.95 gives c ≈ 7.8147.
| χ² statistic | p-value (right tail) |
|---|---|
| 1.95 | 0.5828489711 |
| 3.9 | 0.2724667583 |
| 5.85 | 0.1191405311 |
| 7.8 | 0.0503310979 |
| 9.75 | 0.0208152531 |
| 11.7 | 0.0084847789 |
| 15.6 | 0.0013694899 |
How to read the result
Here p ≈ 0.0503 is slightly above 0.05 and χ² = 7.8 is slightly below the critical 7.815, so the null hypothesis is not rejected at the 5% level, by a very small margin. Reference critical values at α = 0.05: 3.841 (df 1), 5.991 (df 2), 7.815 (df 3), 9.488 (df 4). A statistic close to df is what you would expect under the null hypothesis.
Common mistakes
- Using the total number of cells as df instead of (r − 1)(c − 1).
- Running the test on percentages instead of raw counts.
- Ignoring small expected counts; a common rule of thumb asks for expected counts of at least 5.
Frequently asked questions
Why is the chi-square test one-tailed?
The statistic grows with the mismatch between observed and expected counts, so only large values count as evidence against the null hypothesis.
What does a very small χ² mean?
The fit is unusually close. Extremely small values can sometimes point to data that are too good to be true.
What is the mean of a chi-square distribution?
It equals the degrees of freedom, and the variance is twice that.
Sources: Pearson, K., Philosophical Magazine, Series 5, 50(302), 1900 · NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.6.6 Chi-Square Distribution