Statistics
Chi-Square Distribution Calculator
A chi-square table you can query exactly.
Enter degrees of freedom and a chi-square value to get the lower and upper tail areas (the upper tail is the p-value of a chi-square test), or start from a probability or α to get the matching value. The page also reports the distribution's mean, variance, and mode.
k - 1 categories for goodness of fit; (rows - 1)(cols - 1) for a table.
Try an example
Result
Result
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- P(χ² ≤ x)
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- P(χ² ≥ x)
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- Density f(x)
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- Mean (= df)
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- Variance (= 2df)
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- Mode
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Study path
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What the chi-square distribution calculator solves
A chi-square table you can query exactly. Enter degrees of freedom and a chi-square value to get the lower and upper tail areas (the upper tail is the The chance of seeing results this extreme if nothing real were going on. of a chi-square test), or start from a probability or α to get the matching value. The page also reports the distribution's mean, variance, and mode.
Formula
How to use it
- Enter the degrees of freedom for your test.
- Choose a mode: a χ² value gives both tail areas, a probability gives the matching value, α gives the critical value.
- For a chi-square test, the p-value is the upper tail P(χ² ≥ statistic).
- Compare the statistic with the critical value: larger means reject H0.
How to read the answer
The chi-square distribution is right-skewed and starts at 0. The upper-tail area is what hypothesis tests report as the p-value. Critical values grow with df because the statistic adds more squared terms; for df = 1 the 5% cutoff is 3.841, for df = 10 it is 18.307.
Common mistakes and edge cases
- Reading the lower tail as the p-value; chi-square tests always use the upper tail.
- Using the number of categories instead of categories minus one for df.
- Entering a negative value; a chi-square statistic is never negative.
- Confusing χ² with the reduced chi-square (χ²/df) used in curve fitting.
Worked examples
Upper tail at the 5% cutoff, df = 3
Result
0.04999 (5.00%)
Large statistic with df = 1
Result
0.0009987 (0.10%)
Inverse: x with lower tail 0.95, df = 3
Result
7.8147
Inverse from the upper tail, df = 10
Result
23.2093
Critical value α = 0.05, df = 1
Result
3.8415
Zero statistic
Result
1 (100.00%)
Impossible: negative statistic
Result
Error
Frequently asked questions
How do I find the p-value from a chi-square statistic?+
Use the probability mode with your statistic and df. The upper tail P(χ² ≥ x) is the p-value. For χ² = 7.815 with df = 3 it is 0.05.
What is the critical value of chi-square at 0.05?+
It depends on df: 3.841 (df = 1), 5.991 (df = 2), 7.815 (df = 3), 9.488 (df = 4), 11.070 (df = 5). Use the critical mode for any other df.
Why is the chi-square distribution skewed?+
It is the distribution of a sum of df squared standard normal values, which cannot be negative. With few terms the sum piles up near 0; as df grows the skew fades and the shape becomes nearly normal with mean df and variance 2df.
What is the mean of a chi-square distribution?+
The mean equals the degrees of freedom, the variance is 2df, and the mode is df - 2 for df ≥ 2.
Can degrees of freedom be a decimal?+
The formula works for any positive df because it uses the incomplete gamma function, though tests almost always produce whole numbers.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026