Normal Distribution

Standardize to Z, then read probabilities from .

The idea

The normal distribution is the bell-shaped distribution followed by quantities that many small independent influences push around — heights in a population, errors in a measurement, averages of large samples. Such quantities pile up around a central value in the same symmetric way, largely without regard to what the individual influences are.

A continuous random variable $X$ is normal with mean $\mu$ and variance $\sigma^{2}$, written $X \sim N(\mu, \sigma^{2})$, when its density is $f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^{2}/(2\sigma^{2})}.$ The density is symmetric about $\mu$, which is therefore both the mean and the median, and $\sigma$ sets the width.

This density has no elementary antiderivative, so we do not find probabilities by integrating it by hand. Instead we reduce every normal to a single one: subtracting the mean and dividing by the standard deviation gives $Z = \frac{X - \mu}{\sigma},$ which is standard normal, $Z \sim N(0, 1)$. A value of $Z$ counts standard deviations from the mean, so $z = 1.5$ means the same thing on every scale. Its cumulative distribution function $\Phi(z) = \mathbb{P}(Z \leq z)$ is tabulated once, and every question about every normal becomes a lookup in that one table.

Ways to work on it

• Walkthrough. Standardize a normal value and read its probability off the standard normal curve.
• Practice. Standardize a normal value to its z-score.
• Hardest. Find the probability that a normal variable lands in a given interval.

Not sure where to start? Take the ten-question placement test.