Chi-Square Distribution
Chi-square PDF, CDF, and critical values
About this calculator
The Chi-Square Distribution Calculator draws the χ²(k) density curve for any degrees of freedom and shades the area you choose — the left tail P(X ≤ x), the right tail P(X ≥ x), or the critical value for an upper-tail area α. It reports the density f(x), both tail probabilities, and the distribution's mean and variance. Useful for chi-square goodness-of-fit and independence tests, p-values, and confidence intervals for a variance.
How to use the Chi-Square Distribution calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- χ²(3): the 95th percentile is 7.815, so P(X ≥ 7.815) = 0.05
- χ²(1): P(X ≤ 3.841) = 0.95 (the one-degree critical value)
- χ²(10): mean = 10, variance = 20, P(X ≤ 10) ≈ 0.560
- Critical value for α = 0.05 with k = 4 is 9.488
- χ²(2) is an exponential curve: P(X ≥ 5.991) = 0.05
Frequently asked questions
How does the Chi-Square Distribution calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Chi-Square Distribution calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.