Chi-Square Distribution Calculator

Chi-square density, cumulative probability, and quantile for any degrees of freedom.

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About this tool

Compute the chi-square density, cumulative probability or quantile for given degrees of freedom.

How to use

1
Step 1
Choose PDF, CDF or inverse CDF.
2
Step 2
Enter df.
3
Step 3
Enter x or a probability p and press Calculate.

How it works

Chi-square is the sum of squared standard normal variables, so it is positive and skewed right, with mean equal to df. The 95th percentile is the right-tailed critical value at α = 0.05.

p (right-tail) = 1 − CDF(χ²)      mean = df, variance = 2·df

Worked example

df = 8

Input
CDF, x = 10
Output
0.734974
Walkthrough
The right-tail p-value for χ² = 10 is 1 − 0.734974 = 0.265. The inverse CDF at 0.95 gives 15.507313.

Limitations

  • Central chi-square only.
  • Probability input must be strictly between 0 and 1 for the quantile.

Questions

What is the 95th percentile of chi-square with 8 df?
15.5073.
What is P(χ² ≤ 10) for 8 df?
0.7350.
What is the mean of a chi-square distribution?
Its degrees of freedom.

Features

  • PDF, CDF, quantile
  • Plotted curve
  • Runs in your browser

Related resources

Guides and videos from around the web—opens on YouTube or the publisher’s site.

  • Student's t-distribution

    Khan Academy

    Why t-tables differ from z for small samples.

    Watch on YouTube

  • Hypothesis testing and p-values

    StatQuest with Josh Starmer

    Intuition for t/z/χ² decision rules.

    Watch on YouTube

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