Enter a mean and standard deviation to see where 68%, 95% and 99.7% of a bell-shaped data set falls, drawn on a shaded bell curve. You can also paste raw data to check how closely it follows the rule, or find the share of values in any range.
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Leave blank for −∞ (everything below b)
Leave blank for +∞ (everything above a)
Enter your values to see the result.
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How to use the Empirical Rule Calculator
Choose your starting point: a known mean and standard deviation, or a list of raw data values.
Enter the mean (μ) and standard deviation (σ), or paste your data and pick sample or population SD.
Read the three ranges: 68% within one SD, 95% within two and 99.7% within three.
Optionally enter a lower and/or upper value to get the share of values in that range.
Check the shaded bell curve and the table, then copy the results or download the table as CSV.
The 68-95-99.7 rule explained
In a normal distribution the mean sits in the middle and the standard deviation (σ) measures the spread. The empirical rule turns those two numbers into quick ranges: about 68% of values fall within μ ± σ, about 95% within μ ± 2σ and about 99.7% within μ ± 3σ. Values more than 3 standard deviations from the mean are rare (about 3 in 1,000), which is why they are often treated as unusual or as possible outliers.
Range
Empirical rule
Exact normal value
Outside the range
μ ± 1σ
68%
68.27%
31.73% (about 1 in 3)
μ ± 2σ
95%
95.45%
4.55% (about 1 in 22)
μ ± 3σ
99.7%
99.73%
0.27% (about 1 in 370)
Percent in each band of the bell curve
Split the curve at every standard deviation and the rule gives the share in each slice. Add the slices you need: the share below μ + σ is 50% + 34% = 84%, and the share between μ − σ and μ + 2σ is 34% + 34% + 13.5% = 81.5%.
Band (each side of the mean)
Share
Example: IQ, μ = 100, σ = 15
Mean to 1σ
34%
100 to 115 (and 85 to 100)
1σ to 2σ
13.5%
115 to 130 (and 70 to 85)
2σ to 3σ
2.35%
130 to 145 (and 55 to 70)
Beyond 3σ
0.15%
above 145 (and below 55)
Tips for homework and real data
Check the shape first. A histogram or stem-and-leaf plot should look roughly symmetric with one peak before you rely on the rule.
Use the raw data mode as a normality check. If far fewer or far more than 68% of your values land within one SD, the data is probably skewed or has outliers.
Empirical rule vs Chebyshev. Chebyshev's theorem (at least 1 − 1/k² within k SDs) works for any distribution but is much looser: at least 75% within 2σ instead of 95%.
Need an exact percentile? For values that are not whole SDs from the mean, use the z-score calculator.
Frequently asked questions
What is the empirical rule?
The empirical rule, also called the 68-95-99.7 rule or three-sigma rule, says that for 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.
How do I use the empirical rule with a mean and standard deviation?
Add and subtract multiples of the standard deviation from the mean. With mean 100 and SD 15: 68% of values are between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145.
What percentage is above or below one standard deviation?
Because the curve is symmetric, 34% lies between the mean and one SD on each side. So about 16% of values are below μ − σ and 16% above μ + σ; about 2.5% are below μ − 2σ and 2.5% above μ + 2σ.
How accurate are the 68, 95 and 99.7 figures?
They are rounded. The exact normal-distribution values are 68.27%, 95.45% and 99.73%. The calculator shows both so you can use the rounded figures for homework and the exact ones when precision matters.
When can I not use the empirical rule?
Only use it for roughly symmetric, bell-shaped data. For skewed data or data with heavy tails the percentages can be far off. For any distribution, Chebyshev's theorem guarantees at least 75% within 2 SDs and at least 88.9% within 3 SDs.
How do I find the percent between two values that are not whole standard deviations away?
Convert each value to a z-score, z = (x − μ) ÷ σ, and use the normal distribution, since the empirical rule only covers whole numbers of SDs. The calculator does this automatically and labels the answer as an exact normal probability.
Should I use the sample or population standard deviation for raw data?
Use the sample SD (dividing by n − 1) when your data is a sample from a larger group, which is the usual case. Use the population SD (dividing by N) only when your data includes every member of the group.
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