Normal Distribution & the Empirical Rule
Big Idea
The normal distribution is the most important probability distribution in statistics. It describes how data naturally clusters around a central value, forming the iconic "bell curve." The Empirical Rule (also called the 68-95-99.7 Rule) tells us exactly how data spreads out in any normal distribution.
Formal Definition
A continuous random variable $X$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$ if its probability density function (PDF) is:
$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \, e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$
We write this as $X \sim N(\mu, \sigma^2)$.
- $\mu$ (mu) — the mean, median, and mode (center of the curve)
- $\sigma$ (sigma) — the standard deviation (controls the spread)
- $e \approx 2.71828$ — Euler's number
- $\pi \approx 3.14159$ — pi
Key Properties
- Symmetric about the mean $\mu$
- Bell-shaped — unimodal curve
- Mean = Median = Mode = $\mu$
- Total area under the curve = 1 (100%)
- The curve never touches the x-axis (asymptotic)
- Inflection points occur at $x = \mu \pm \sigma$
- Completely determined by $\mu$ and $\sigma$