$X \sim \chi^2_{(\nu)}$

This applet computes probabilities and percentiles for the chi-square distribution: $$X \sim \chi^2_{(\nu)}$$

- Enter the degrees of freedom in the $\nu$ box.
- To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue (left) box and press "Tab" or "Enter" on your keyboard. The probability $P(X \lt x)$ will appear in the pink (right) box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability.
- To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink (right) box, select $P(X \lt x)$ from the drop-down box, and press "Tab" or "Enter" on your keyboard. The percentile $x$ will appear in the blue (left) box.

On the graph, the $x$ value appears in blue while the probability is shaded in pink.

- Probability density function $$f(x)=\frac{1}{2^{\nu/2}\Gamma\left(\frac{\nu}{2}\right)} x^{\nu/2-1} e^{-x/2}$$ where $x > 0$ and $\nu > 0$
- $\mu=E(X)=\nu$
- $\sigma^2=Var(X)=2\nu$
- $\sigma=SD(X)=\sqrt{2\nu}$