Distributions
Chi-Square Distribution Calculator
Tail probabilities and critical values for the chi-square distribution — the p-value behind every chi-square test.
How to read this result
Chi-square lives on the positive numbers and is right-skewed. It is a sum of squared standard normals, so it cannot be negative, and its shape depends entirely on the degrees of freedom: sharply skewed at 1 or 2, increasingly symmetric as df grows.
The p-value is always the upper tail. Every chi-square test — independence, goodness of fit, variance — produces a statistic that grows under any departure from the null. All the evidence is in one tail, so there is no two-tailed version to choose.
Mean df, variance 2df. A quick sanity check: a chi-square statistic close to its degrees of freedom is unremarkable. χ² = 7 on 7 df is exactly average; χ² = 25 on 7 df is not.
With 1 degree of freedom it is the squared normal, which is why a two-tailed z-test and a 1-df chi-square test always give the same p-value. 1.96² = 3.84, the familiar 5% critical value.
The formula
- the degrees of freedom
- the gamma function
The upper tail is the regularised incomplete gamma function Q(k/2, x/2), computed directly rather than as one minus the CDF so that tiny p-values keep their digits.
Worked example
A chi-square of 12.3 on 7 degrees of freedom
A goodness-of-fit test across eight categories produces χ² = 12.3 with 7 degrees of freedom. Is that unusual?
- The expected value of the statistic is its degrees of freedom.mean = 7, SD = √14 = 3.74
- 12.3 is about 1.4 standard deviations above the mean — worth checking, not obviously extreme.(12.3 − 7)/3.74 = 1.42
- Compute the exact upper tail.P(X > 12.3) = 0.0911
- Compare with the 5% critical value on 7 df.χ²* = 14.07
A statistic 75% larger than its expected value still gives p = 0.09. Chi-square distributions are wide, and small tables have little power to detect anything but large departures.
Checked against R's pchisq(12.3, 7, lower.tail = FALSE).
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
The distribution itself assumes nothing — it is a mathematical object. What assumes things is the test that produces the statistic: independent observations, counts rather than percentages, and expected counts large enough for the approximation to hold. See the chi-square test for those.
Degrees of freedom are the input most often entered wrongly: (rows − 1)(columns − 1) for a table of independence, categories − 1 for goodness of fit, and one fewer for each parameter estimated from the data.
- You have a table rather than a statisticChi-square testComputes χ², the degrees of freedom and the p-value from the counts.
- You want the p-value from a different statisticP-value calculatorz, t, chi-square and F together.
- You are testing variancesLevene's testThe modern, more robust way to compare spreads across groups.
Questions people ask
Why is the chi-square test always one-tailed?
Because χ² is a sum of squares: any departure from the null, in any direction, makes it larger. A small χ² means the data fit the null better than chance would predict, which is not evidence against it — though a suspiciously small value can indicate fabricated data, which is how Fisher questioned Mendel's results.
What is the critical chi-square value for 5%?
It depends on the degrees of freedom: 3.841 at 1 df, 5.991 at 2, 11.070 at 5, 18.307 at 10. The critical-value mode computes it for any df with a table of common levels.
What is the relationship between chi-square and the normal distribution?
A chi-square with k degrees of freedom is the sum of k squared independent standard normals. With k = 1 that means χ² = z², which is why a 1-df chi-square test and a two-tailed z-test agree exactly.