For any normal distribution, the data clusters around the mean in a strikingly regular way. The empirical rule — also called the 68–95–99.7 rule — says how much of the data falls within one, two, and three standard deviations of the mean:

• about 68% of the data lies within 1\sigma of the mean,
• about 95% lies within 2\sigma,
• about 99.7% lies within 3\sigma.

Almost everything — 99.7\% — sits within three standard deviations of the centre. Values further out than that are genuinely rare.

Three nested bands

The picture below is a standard normal curve with three shaded bands. The innermost band spans \mu \pm 1\sigma and holds 68\% of the area; widening to \mu \pm 2\sigma captures 95\%; and \mu \pm 3\sigma captures 99.7\%.

Because the curve is symmetric, each percentage splits evenly across the two tails. For example, 95\% inside 2\sigma leaves about 5\% outside — roughly 2.5\% in each tail.

Using the rule

The rule turns the standard deviation into a quick yardstick. Suppose adult heights are normal with mean \mu = 170\,\text{cm} and standard deviation \sigma = 10\,\text{cm}. Then about 95\% of people are within 2\sigma = 20\,\text{cm} of the mean — that is, between 150\,\text{cm} and 190\,\text{cm}.