### Normal Distribution$X \sim N(\mu, \sigma)$

 $\mu=$ $\sigma=$ $x=$ P(X > x) = P(X < x) = 2P(X > |x|) = P(-|x| < X < |x|) =

This applet computes probabilities and percentiles for normal random variables: $$X \sim N(\mu, \sigma)$$

#### Directions

• Enter the mean $\mu$ and standard deviation $\sigma$.
• To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue box and press "Tab" or "Enter" on your keyboard. The probability $P(X \lt x)$ will appear in the pink 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 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 box.

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

#### Details

• Probability density function $$f(x)=\frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{1}{2\sigma^2}(x-\mu)^2}$$ where $-\infty \lt x \lt \infty$, $-\infty \lt \mu \lt \infty$, and $\sigma \gt 0$
• $\mu=E(X)$
• $\sigma^2=Var(X)$
• $\sigma=SD(X)$