The 68–95–99.7 rule
In a normal distribution, about 68% of values lie within 1 SD of the mean, 95% within 2 SDs and 99.7% within 3 SDs. The exact figures are 68.27%, 95.45% and 99.73%. See the empirical rule explained.
Worked example
IQ-style scores with mean 100 and SD 15: about 68% lie between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145. So a score above 130 is rare, at roughly 2.3% of people.
Checking your own data
Paste values to compare your data with the rule. The 20 reaction times used elsewhere on this site have 90% within 1 SD, far from 68%, because two slow responses inflate the SD. That mismatch is a quick sign of non-normality.
Chebyshev’s inequality
For any distribution, at least 1 − 1/k² of values lie within k SDs: at least 75% within 2 SDs and 88.9% within 3. These minimums are much weaker but always hold. For exact areas at any cut-off, use the normal distribution calculator.
Frequently asked questions
What is the 68-95-99.7 rule?
In a normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
Does the empirical rule apply to all data?
No, only to roughly bell-shaped data. For any distribution, Chebyshev's inequality gives weaker guaranteed minimums.
How do I use the rule to spot unusual values?
A value more than 2 SDs from the mean is uncommon (about 5%), and more than 3 SDs is rare (about 0.3%) in normal data.