The 68-95-99.7 Rule: How the Normal Distribution Works
October 21, 2026
For data that follows a normal (bell-shaped) distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is called the empirical rule, or the 68-95-99.7 rule. On an IQ scale with a mean of 100 and standard deviation of 15, about 68% of people score between 85 and 115, and about 95% between 70 and 130.
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Take any normal distribution with mean μ and standard deviation σ. The intervals μ ± σ, μ ± 2σ, and μ ± 3σ contain approximately 68.27%, 95.45%, and 99.73% of the data. The remaining values lie in the tails, split evenly between the high and low ends.
That means values more than three standard deviations from the mean occur only about 0.27% of the time, or roughly 3 in 1,000.
| Range | Share of data | IQ example (mean 100, SD 15) |
|---|---|---|
| μ ± 1σ | 68.27% | 85 to 115 |
| μ ± 2σ | 95.45% | 70 to 130 |
| μ ± 3σ | 99.73% | 55 to 145 |
Using It to Spot Unusual Values
The rule gives you a quick way to judge whether a value is typical. A score of 130 is two standard deviations above the mean, which only about 2.3% of people exceed (half of the 4.55% outside the two-sigma range). A score of 145 is three standard deviations out, exceeded by only about 0.13%.
This is the logic behind z-scores and six-sigma quality control: the further a value is from the mean in standard deviations, the rarer it is.
When the Rule Doesn't Apply
Many real datasets are not normal. Incomes and house prices are skewed to the right, so a few huge values pull the mean up. Financial returns have fatter tails than a normal curve, so extreme moves happen more often than the rule suggests.
Before applying the rule, check the shape of your data with a histogram. For non-normal data, Chebyshev's inequality guarantees that at least 75% of values lie within two standard deviations and 89% within three, whatever the shape.
Frequently Asked Questions
What does the 68-95-99.7 rule mean?
It says that in a normal distribution, about 68%, 95%, and 99.7% of data lie within 1, 2, and 3 standard deviations of the mean.
Does the empirical rule work for any dataset?
No. It applies to approximately normal, symmetric, bell-shaped data. Skewed or heavy-tailed data can differ a lot.
What percentage is beyond 2 standard deviations?
About 4.55% in total, or roughly 2.3% in each tail.
How do I find the standard deviation of my data?
Calculate the mean, find each value's squared difference from it, average those squares, and take the square root. A calculator does this instantly.