68-95-99.7 Empirical Rule Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
In a normal distribution, about 68% of values lie between μ−σ and μ+σ, about 95% between μ−2σ and μ+2σ, and about 99.7% between μ−3σ and μ+3σ. For a mean of 100 and a standard deviation of 15 (like an IQ score), 68% of values lie between 85 and 115.
Explanation
The empirical rule, also called the 68-95-99.7 rule, describes how the values of a normal distribution (the famous "bell curve") are distributed around their mean, as a function of the standard deviation. It is a precise mathematical result (the area under the normal curve between −1 and +1, −2 and +2, −3 and +3 standard deviations), not an empirical approximation despite its name: about 68% of observations lie within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% — almost all — within three standard deviations. This rule is widely used to quickly interpret a standard deviation without complex calculation: an IQ score of 130 (2 standard deviations above a mean of 100), for example, is in the top 2.5% of a population, since the central 95% stops at that value. It assumes, however, that the distribution studied is actually normal (symmetric, bell-shaped): it becomes misleading on an asymmetric or multi-peaked distribution, where the real proportions can deviate significantly from these benchmarks. To convert a precise value into a number of standard deviations (rather than the reverse), see our z-score calculator, and for estimation from a sample, our confidence interval calculator.
Example: IQ score, mean 100, standard deviation 15
Inputs
Mean: 100. Standard deviation: 15.
Calculation
68% range: 100 − 15 = 85, to 100 + 15 = 115. 95% range: 100 − 30 = 70, to 100 + 30 = 130. 99.7% range: 100 − 45 = 55, to 100 + 45 = 145.
Result
About 68% of IQ scores lie between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145.
Frequently asked questions
Does this rule work for all data distributions?
No, it specifically assumes a normal distribution (symmetric, bell-shaped). On an asymmetric, multi-peaked distribution, or one with frequent extreme values, the real proportions of values in each range can deviate significantly from 68%, 95%, and 99.7% — you must then refer to the actual distribution of the data rather than this general rule.
What is the difference from the z-score calculator?
This calculator starts from a mean and a standard deviation to give the value ranges corresponding to 1, 2, and 3 standard deviations. The z-score calculator does the reverse: from a precise value, it tells how many standard deviations it is from the mean — both calculations rest on the same mathematical relationship, in opposite directions.
Why 99.7% and not 100% at 3 standard deviations?
A theoretical normal distribution extends in principle to infinity in both directions, with a probability that never becomes exactly zero, even far from the mean. The proportion of values beyond 3 standard deviations (about 0.3%) therefore always stays non-zero, however small — hence 99.7% rather than 100%.