Empirical Rule Calculator
The empirical rule, or 68-95-99.7 rule, says that in a normal distribution about 68% of values sit within one standard deviation of the mean, 95% within two, and 99.7% within three. Give it a mean and a standard deviation and this tool draws the three bands, tells you which band any value lands in, and works backwards from a percentage to the range that covers it. Everything runs in your browser.
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How this is computed
pnorm(); the reverse range uses its inverse, qnorm().
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The 68-95-99.7 rule at a glance
For a normal distribution the area inside k standard deviations of the mean is the same regardless of the mean or the spread. That is what makes the rule portable across any bell-shaped variable.
| Within | Band | Exact area | Rounded |
|---|---|---|---|
| 1 SD | μ ± 1σ | 68.27% | 68% |
| 2 SD | μ ± 2σ | 95.45% | 95% |
| 3 SD | μ ± 3σ | 99.73% | 99.7% |
The neat round "95%" that fills confidence intervals is not exactly 2 SD; it is 1.96 SD. Exactly 2 SD covers a touch more, 95.45%. Switch to the range mode above to get the SD multiple for any coverage you want.
When the rule breaks
The percentages are only right when the data is close to normal. Read them as a sanity check, not a guarantee.
Skewed data (income, waiting times), heavy tails (financial returns), bounded scores, and counts near zero all violate the rule, sometimes badly: for a strongly right-skewed variable far more than 0.3% of values can sit beyond 3 SD on the long side. If you need a bound that holds for any distribution, Chebyshev's inequality guarantees at least 75% within 2 SD and about 89% within 3 SD, weaker figures, but true whatever the shape. Check normality first with a histogram or a QQ plot before trusting 68-95-99.7.
Frequently asked questions
What is the empirical rule?
The empirical rule, also called the 68-95-99.7 rule, describes how values spread out in a normal distribution: about 68% land within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Enter a mean and standard deviation in the first mode to see the three bands and their exact ranges.
How do I calculate the 68-95-99.7 bands?
Each band is the mean plus and minus one, two and three standard deviations. For a mean of 100 and an SD of 15 the bands are 85 to 115, 70 to 130 and 55 to 145. The areas inside them are pnorm(1) - pnorm(-1) = 0.6827, then 0.9545 and 0.9973.
Which band does a specific value fall in?
Standardise it: z = (x - mean) / SD. If the absolute z is below 1 the value is in the central 68% band, between 1 and 2 it is in the 95% band, between 2 and 3 the 99.7% band, and past 3 it is a rare outlier. The middle mode returns the exact share of the distribution that lies within and beyond that value.
Why 1.96 SD for 95% instead of 2?
Exactly two standard deviations cover 95.45%, slightly more than 95%. To capture exactly 95% you need 1.96 SD. The range mode inverts the normal curve for any coverage you type, the same calculation as R's qnorm(), so 95% returns 1.96, 99% returns 2.576, and so on.
Does the empirical rule work for any data?
No. It assumes the data is roughly normal and symmetric. Skewed, heavy-tailed, bounded or multimodal data break it. For a bound that always holds, use Chebyshev's inequality: at least 75% of any distribution is within 2 SD and about 89% within 3 SD.
Does this match R?
Yes. Every area is the same value R's pnorm() returns and every reverse range uses qnorm(), agreeing to nine decimal places, including the deep tails past three standard deviations.