AP® Statistics · Unit 2 · Topic 2.11
The Normal Curve and the Empirical Rule
The normal curve is a symmetric bell-shaped model where data cluster around the mean, and the empirical rule says about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of it.
Lots of natural measurements pile up near an average and taper off evenly on both sides: adult heights, measurement errors, standardized test scores. That happens because each value is pushed around by many small, independent random factors, some nudging it up, some down, and those nudges tend to cancel out near the center and get rarer the further you go. The result is the same bell shape over and over, described by one mathematical curve: the normal distribution, set by just two numbers, the mean (center) and the standard deviation (spread).
The empirical rule gives quick landmarks on that curve. Mark the mean, then mark off one, two, and three standard deviations on each side. About 68% of the data sits within 1 SD of the mean, about 95% within 2 SD, and about 99.7% within 3 SD. The curve is symmetric, so each rule splits evenly: half the middle 68% lies above the mean, half below, and the thin tails beyond 3 SD hold almost nothing, about 0.3% combined.
This rule is really a shortcut version of finding area under the normal curve, which later gets more precise with z-scores and tables once you need percentages that don't land neatly on whole standard deviations. It also sets up sampling distributions later in the unit: when a statistic is approximately normal, the same 68-95-99.7 landmarks describe how much it varies from sample to sample. Change the standard deviation and the landmarks move: double the SD and the curve flattens and stretches, keeping the same 68-95-99.7 percentages but spreading them over twice the distance from the mean.
Key terms
- Normal distribution
- A specific symmetric, bell-shaped probability model defined entirely by its mean and standard deviation.
- Empirical rule
- The guideline that about 68%, 95%, and 99.7% of normal data fall within 1, 2, and 3 standard deviations of the mean.
- Standard deviation
- A measure of spread showing the typical distance of values from the mean.
- Mound-shaped
- Describes a distribution with a single peak that tapers off on both sides, without specifying exact tail behavior.
Watch it explained
- How it worksWalk me through the 68-95-99.7 rule drawn onto a normal curve.
- Worked exampleFind the percent of adult heights between 64 and 74 inches when the mean is 69 and SD is 2.5.
- Common mistakeWhy isn't every symmetric mound-shaped distribution a normal distribution?
- Going furtherWhat happens to the curve when the standard deviation doubles but the mean stays put?
Check yourself
End of topic.