Using the Normal Distribution

A percentile is simply a value with a given share of the data below it: the 90th percentile of SAT scores is the score that 90% of students fall under. This lesson is about translating both ways, since "the top 10% of scores" and "a score of 1306" are two descriptions of the same spot on the bell curve. The formula $x = \mu + z\sigma$ is short and friendly: start at $\mu$ (the average), then step $z$ (how many standard deviations) times $\sigma$ (the size of one step). So the 90th percentile, $z = 1.282$, is about one and a quarter steps above average. That's the whole idea.

Value to probability: this is what you did in the last lesson. Standardize, $z = \frac{x - \mu}{\sigma}$, then find the area.

Probability to value (percentiles): now run it backward. Find the $z$ with the right area to its left, then un-standardize. Notice that $x = \mu + z\sigma$ is just $z = \frac{x - \mu}{\sigma}$ solved for $x$; nothing new is going on. Useful $z$ values:

Draw a sketch every time: shade the region you want and decide whether it's a left tail, right tail or middle.

A common mistake. For "the top 5%," it's tempting to look up the $z$ with 5% to its left. That's $-1.645$, the cutoff for the bottom 5%. The top 5% has 95% to its left, so you want $z = 1.645$. The sketch settles it in a second.

Clothing makers size garments from height percentiles, schools set cutoffs from score distributions, and factories set tolerances so only a tiny fraction of parts fail.