Normal distribution
High school
A random variable has a normal distribution if its values cluster symmetrically around a mean $\mu$, following a bell-shaped curve whose width is set by the standard deviation $\sigma$.
Undergraduate
A random variable $X$ has a normal distribution with mean $\mu$ and variance $\sigma^2 > 0$, written $X \sim \mathcal{N}(\mu, \sigma^2)$, if $X$ has the probability density
\[f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right).\]Examples
Worked examples
-
Customers arriving at a shop independently of each other at an average rate of $3$ per hour
: The number of customers in $100$ hours: approximately $\mathcal{N}(300, 300)$
Undergraduate
For customers arriving at an average rate of $3$ per hour, the number of customers in $100$ hours is Poisson with mean $300$, which is approximately normal with mean $300$ and variance $300$ by the central limit theorem.
Properties
- By the central limit theorem, the sum of many independent random variables with finite variance is approximately normal, which is why measurement errors are often modeled as normal.
- About $68\%$ of the probability of a normal distribution lies within one standard deviation of the mean, and about $95\%$ lies within two.