The Normal Distribution

The normal distribution is simply the bell curve: most values pile up near the middle, with fewer and fewer as you go out on either side. Don't read too much into the name: so many real measurements (heights, test scores, measurement errors) roughly follow it that it's the standard reference shape, but other shapes aren't "abnormal." The density formula is probably the scariest-looking thing in this course, but it's only a few parts. $(x - \mu)^2$ (how far you are from the center, squared), divided by $2\sigma^2$ (measured relative to the width), inside $e^{-(\dots)}$ (which makes the curve fall off quickly in both directions), all times $\frac{1}{\sigma\sqrt{2\pi}}$ (just a constant so the total area is 1). You'll almost never plug into it by hand, so don't let it scare you.

One curve to rule them all. Every normal curve has the same shape; only the center and the width change. That's why the z-score $z = \frac{x - \mu}{\sigma}$ is so useful. Subtracting $\mu$ slides the curve so it's centered at 0, and dividing by $\sigma$ squeezes or stretches it to width 1. After that, every normal problem is a question about the same standard normal $Z$.

Using z-scores. To find $P(X < x)$ for any normal $X$, convert to $z$ and look up the area to the left of $z$ in a standard normal table (or use a calculator). Areas are probabilities, just as in the last few lessons.

Tricks:

• Right tails: $P(Z > z) = 1 - P(Z < z)$.
• Between: $P(a < Z < b) = P(Z < b) - P(Z < a)$.
• Symmetry: $P(Z < -z) = P(Z > z)$.

The empirical rule from the descriptive statistics unit is a special case: 68%, 95% and 99.7% within 1, 2 and 3 standard deviations.

A common mistake. Tables give the area to the left of $z$. It's tempting to read the table value straight off for a question like $P(Z > 1.5)$, but that gives $0.9332$, the chance of being below 1.5. For a right tail you need $1 - 0.9332$. A quick sketch with the region shaded catches this every time.

Measurement errors, test scores, heights and many manufacturing tolerances are close to normal, and the central limit theorem explains why averages almost always are.