About Chi-Square Distribution Calculator
This chi-square distribution calculator gives upper tail probabilities and critical values for any degrees of freedom k. It opens on k = 5 and finds the p-value of a statistic of 11.07, which is the 5% critical value. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
A chi-square variable is a sum of k squared standard normals, so it is never negative and is skewed to the right. The mean is k and the variance is 2k.
Goodness of fit and independence tests are right-tailed: the p-value is P(X ≥ statistic). Use the inverse at 0.95 to get the critical value for a 5% test.
How to use Chi-Square Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
Chi-square density
The density is a gamma distribution with shape k ÷ 2 and scale 2.
f(x) = x^(k/2 − 1) × e^(−x/2) / (2^(k/2) × Γ(k/2)), x > 0
mean = k
variance = 2kWorked examples
k = 5: P(X ≥ 11.07) gives 0.05001
About 0.05, so 11.07 is the 5% critical value for 5 degrees of freedom.
k = 1: 95th percentile gives 3.8415
3.841, which is 1.96 squared.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I find the p-value of a chi-square test?
Choose P(X ≥ x), enter the degrees of freedom and the chi-square statistic. The upper tail is the p-value.
How many degrees of freedom do I use?
For goodness of fit it is categories minus 1. For a contingency table it is (rows - 1) × (columns - 1).
Why is the chi-square curve not symmetric?
It is a sum of squares, so it cannot go below zero and has a long right tail. It becomes more symmetric as k grows.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Normal Distribution CalculatorEnter a mean and standard deviation to get tail probabilities, intervals and percentiles on a bell curve.
- Binomial Distribution CalculatorEnter the number of trials and the chance of success to get P(X = k), P(X ≤ k), P(X ≥ k) and ranges.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- Geometric Distribution CalculatorEnter the success probability to get the chance that the first success happens on or by trial k.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Student t Distribution CalculatorEnter the degrees of freedom to get t critical values, tail areas and p-values on a shaded t curve.
- F Distribution CalculatorEnter both degrees of freedom to get F p-values, critical values and tail areas.