Many histograms of real measurements (heights, reaction times, errors in a lab scale) are single, symmetric mounds. The normal curve is a smooth model of that shape. It is described completely by two numbers, its mean $\mu$ (the center) and its standard deviation $\sigma$ (the spread), and it works hand in hand with z-scores.
Key idea
For a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. This is the 68-95-99.7 rule (the empirical rule).
Why it works
The area under a density curve is a proportion: the whole area is 1, and the area over an interval is the fraction of values in it. Every normal curve is the same bell, slid to its own mean and stretched by its own $\sigma$. Converting to z-scores undoes the slide and the stretch, so "within 2 standard deviations" covers the same area, about 95%, on every normal curve. (More exactly: 68.27%, 95.45%, 99.73%.)
Splitting the curve into pieces
By symmetry, each band splits in half. Between the mean and 1 SD: 34%. Between 1 and 2 SD: $(95 - 68)/2 = 13.5\%$. Between 2 and 3 SD: $(99.7 - 95)/2 = 2.35\%$. Beyond 3 SD on one side: 0.15%.
| $z$ from | to | percent |
|---|---|---|
| $-\infty$ | $-3$ | 0.15 |
| $-3$ | $-2$ | 2.35 |
| $-2$ | $-1$ | 13.5 |
| $-1$ | $0$ | 34 |
| $0$ | $1$ | 34 |
| $1$ | $2$ | 13.5 |
| $2$ | $3$ | 2.35 |
| $3$ | $\infty$ | 0.15 |
Worked example: Within two standard deviations
Men's heights are normal with mean 70 inches and SD 3 inches. About what percent are between 64 and 76 inches?
- $64 = 70 - 2(3)$ and $76 = 70 + 2(3)$: two standard deviations on each side.
- The rule says about 95%.
Answer: About 95%.
Worked example: A tail
About what percent are taller than 76 inches?
- Within 2 SD: 95%, so 5% is outside, split equally between the two tails.
- Above 76: $5\% \div 2 = 2.5\%$.
Answer: About 2.5%.
Worked example: An uneven interval
About what percent are between 67 and 76 inches?
- $67$ is at $z = -1$ and $76$ is at $z = 2$.
- Add the pieces: $34 + 34 + 13.5 = 81.5$.
Answer: About 81.5%.
Common mistake
Applying the rule to data that are not bell-shaped. Incomes, waiting times and house prices are strongly skewed, and for them "within 2 SD" can hold a very different share than 95%. Check the histogram first: the rule is for single, roughly symmetric mounds.