Empirical Rule Calculator
Apply the empirical rule to any normal distribution. Enter the mean and standard deviation (any units — test scores, heights, measurements) and get the ranges that hold about 68%, 95%, and 99.7% of values: μ ± 1σ, μ ± 2σ, and μ ± 3σ.
Example: with Mean (μ) 100 · Standard deviation (σ) 15 → About 68% of values (μ ± 1σ): 85 to 115.
- About 95% of values (μ ± 2σ)70 to 130
- About 99.7% of values (μ ± 3σ)55 to 145
- Beyond three sigmaOnly about 0.3% of values fall outside 55 to 145 — roughly 1 in 370
Computed by the calculator below using its default values. Change any input to see your own numbers.
The 68-95-99.7 shares are properties of the normal curve — exact values 68.27%, 95.45%, 99.73%. The rule is a rounded convention for quick estimates.
What the 68-95-99.7 rule buys you
For anything roughly bell-shaped, the standard deviation converts directly into coverage: about 68% of values sit within one σ of the mean, 95% within two, and 99.7% within three. With IQ-style scores (mean 100, σ 15), that means 85 to 115 holds about two-thirds of people, 70 to 130 holds 19 in 20, and scores beyond 55 to 145 are rare — about 1 in 370.
The rule also works in reverse as a sanity check. If a "normal" dataset shows 10% of values beyond three sigma, the distribution is not normal — it is skewed or heavy-tailed, and the empirical rule will mislead. Exact normal shares are 68.27%, 95.45%, and 99.73%; the rule rounds them for mental math.
How it’s calculated
Ranges are μ ± kσ for k = 1, 2, 3: with mean 100 and σ 15 they are 85 to 115, 70 to 130, and 55 to 145. Coverage percentages come from the normal distribution: exactly 68.27%, 95.45%, and 99.73%, which the empirical rule rounds to 68, 95, and 99.7. The outside-3σ share is 100 − 99.73 = 0.27%, about 1 in 370.
The rule assumes an approximately normal (symmetric, bell-shaped) distribution — for skewed data the percentages can be badly off, and only the weaker Chebyshev bounds are guaranteed.
Empirical rule with mean 100, σ = 15
| Band | Range | Share inside | Share outside |
|---|---|---|---|
| μ ± 1σ | 85 to 115 | 68.27% | 31.73% |
| μ ± 2σ | 70 to 130 | 95.45% | 4.55% |
| μ ± 3σ | 55 to 145 | 99.73% | 0.27% |
Exact shares from the standard normal distribution; the rule rounds them to 68, 95, and 99.7.
Common mistakes
- Applying the rule to skewed data — income, house prices, and wait times are not bell-shaped, and the percentages fail.
- Reading 95% as μ ± 1.96σ exactly: the empirical rule uses 2σ (95.45%); the 1.96 figure belongs to exact confidence intervals.
- Using the range ÷ 2 as σ — estimate the standard deviation properly before applying the rule.
- Assuming half of the outside-2σ 5% sits on each tail only works for symmetric distributions (it is 2.5% per tail for a true normal).
Frequently asked questions
What is the empirical rule?
For a normal distribution, about 68% of values fall within 1 standard deviation of the mean (μ ± 1σ), about 95% within 2 (μ ± 2σ), and about 99.7% within 3 (μ ± 3σ). Exact normal shares are 68.27%, 95.45%, and 99.73%.
How do I use it with real numbers?
Add and subtract multiples of σ from the mean. Scores with mean 100 and σ 15: 68% land in 85 to 115, 95% in 70 to 130, 99.7% in 55 to 145.
What share of values falls above 2 standard deviations?
About 2.5%. Outside μ ± 2σ lies roughly 5% total, split evenly between the two tails because the normal curve is symmetric.
Does the empirical rule work for any dataset?
No — it is a property of the normal curve. For arbitrary distributions, Chebyshev’s inequality only guarantees at least 75% within 2σ and 89% within 3σ, much weaker than the rule’s 95% and 99.7%.
Is the empirical rule the same as the 68-95-99.7 rule?
Yes — the two names describe the same three coverage bands. It is also called the three-sigma rule, especially in quality control.