The 68-95-99.7 rule
Chilean boys aged 15 to 18 in 2009 to 2010 averaged 170 cm with a standard deviation of 6.28 cm, so about 95% were between 157.44 and 182.56 cm.
Every normal curve, however wide and wherever it stands, has the same areas when you measure distance from the mean in standard deviations. That gives a rule small enough to carry in your head. About 68% of the values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. OpenStax calls it the empirical rule; many people just call it the 68-95-99.7 rule.
Within one standard deviation of the mean: about 68%. Within two: about 95%. Within three: about 99.7%.
Using it with real numbers
OpenStax gives the heights of Chilean males aged 15 to 18 in 2009 to 2010 as N(170, 6.28) in centimetres. One standard deviation either side runs from 170 − 6.28 = 163.72 to 170 + 6.28 = 176.28 cm, so about 68% of these young men were in that range. Two standard deviations run from 157.44 to 182.56 cm and cover about 95%. Three run from 151.16 to 188.84 cm and cover about 99.7%, which leaves only 3 in 1,000 outside. For the same group in 1984 to 1985, N(172.36, 6.34), the book gives the ranges 166.02 to 178.7, 159.68 to 185.04 and 153.34 to 191.38 cm.
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The exact numbers and the tails
The rule rounds. Computed exactly, the areas are 68.27%, 95.45% and 99.73%. Because the curve is symmetric, whatever lies outside a band is split equally between the two tails. Outside one standard deviation is 31.73%, so 15.87% lies above μ + σ and 15.87% below μ − σ. Above μ + 2σ lies about 2.28%. Between one and two standard deviations above the mean lies 95.45% ÷ 2 − 68.27% ÷ 2 ≈ 13.59%. With these few numbers you can answer many questions without a table.
Checking real data against the rule
Quetelet's 5,738 chests have mean 39.83 and standard deviation 2.05 inches. Counting directly, 66.8% of the men lie within one standard deviation of the mean, 95.3% within two and 99.5% within three. Those are close to 68, 95 and 99.7, which is another way of saying the data are approximately normal. The rule does not hold for skewed data. For a distribution like waiting times, with a long right tail, the share within one standard deviation can be far from 68%; you will see this in the last unit.
Spotting unusual values and working backwards
The rule also tells you how rare a value is. Outside two standard deviations lies 4.55% of a normal population, about 1 value in 22, and outside three lies 0.27%, about 1 in 370. A Chilean 15 to 18 year old of 190 cm in 2009 to 2010 has z = (190 − 170) ÷ 6.28 ≈ 3.18, so heights like that are rare in that group. The rule can also run backwards. OpenStax describes a college entrance exam with mean 52 and standard deviation 11: about 68% of scores lie between 41 and 63, about 95% between 30 and 74 and about 99.7% between 19 and 85. Given the 68% range alone, its midpoint is the mean and half its width is the standard deviation.
| Within 1σ | About 68% (68.27%) |
| Within 2σ | About 95% (95.45%) |
| Within 3σ | About 99.7% (99.73%) |
| Above μ + σ | About 16% (15.87%) |
| Above μ + 2σ | About 2.5% (2.28%) |
- N(170, 6.28): lower end of the 68% range?
- N(170, 6.28): upper end of the 95% range?
- N(172.36, 6.34): lower end of the 99.7% range?
- N(50, 6): about 95% of values lie between 38 and what?
- About what percent of a normal population lies above μ + σ? (two decimals)
- Percent between μ + σ and μ + 2σ? (two decimals)
- Out of 1,000 values from a normal distribution, about how many lie more than 3σ from the mean?
- Quetelet chests (mean 39.83, s 2.05): lower end of the 2σ range, two decimals?
- Counting the chests gives 66.8%, 95.3% and 99.5%. What does that suggest?
- The data are approximately normal
- The data are skewed
- The rule is wrong
- There are outliers only
- Does the 68-95-99.7 rule apply to strongly skewed data?
- Yes, to all data
- No, it is for approximately normal data
- Only for 99.7
- Only for small samples
- About 68%, 95% and 99.7% of a normal distribution lie within 1, 2 and 3 standard deviations of the mean.
- Exact values: 68.27%, 95.45%, 99.73%; the outside splits equally between the tails.
- Use the rule only when the data are approximately normal.
Sources for this course (7)
- OpenStax, Introductory Statistics 2e by Barbara Illowsky and Susan Dean (, per the page notice on 2026-10-17): 2.2 frequency tables, relative and cumulative frequency, frequency polygons (calculus final test scores 5, 10, 30, 40, 15; ages of US presidents at inauguration in six classes)
- the 100 soccer player heights, the 50 students' books, the 25 classmates' weekend video game hours and the 22 New York restaurant staff counts (openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs)
- 6.1 the standard normal distribution, z-scores, the empirical rule, X ~ N(5, 6)
- heights of Chilean males aged 15 to 18 N(170, 6.28) for 2009 to 2010 and N(172.36, 6.34) for 1984 to 1985, SAT verbal 2012 N(496, 114) with 1,664,479 test takers, Jerome N(16, 4) (openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution)
- De Moivre and the normal approximation to coin flips: D. Lane, Online Statistics Education, 7.2 History of the Normal Distribution (stats.libretexts.org, Introductory Statistics (Lane) 7.2)
- EBSCO Research Starters, "De Moivre Describes the Bell-Shaped Curve" (1733)
- Quetelet's chest measurements of 5,738 Scottish militiamen (Quetelet 1846, from an 1817 source; Stigler 1986 corrections): R package HistData, ChestSizes and ChestStigler (vincentarelbundock.github.io/Rdatasets/doc/HistData/ChestSizes.html).
Checked: 2026-10-17