Statistics Reference
The Empirical Rule (68-95-99.7) Explained
Two numbers — a mean and a standard deviation — describe any normal distribution completely. The empirical rule is the cheat code that turns them into ranges: 68% of values within one standard deviation, 95% within two, 99.7% within three. This guide shows where those percentages come from, works a complete example, and marks the boundary where the rule stops applying.
The Rule and Its Exact Values
| Range | Rule of thumb | Exact normal area | Left outside (both tails) |
|---|---|---|---|
| μ ± 1σ | 68% | 0.6827 | 31.7% — about 1 in 3 |
| μ ± 2σ | 95% | 0.9545 | 4.6% — about 1 in 22 |
| μ ± 3σ | 99.7% | 0.9973 | 0.27% — about 1 in 370 |
The percentages are areas under the bell curve: P(−1 < Z < 1) = 0.6827 and so on — the same numbers a z-table lookup produces for z = 1, 2, 3. The empirical rule is a three-row z-table committed to memory. (A close cousin: 95% exactly corresponds to z = 1.96, which is why that constant rules confidence intervals.)
Worked Example: IQ Scores
IQ scores are designed to be normal with μ = 100 and σ = 15. The rule immediately maps the population:
- μ ± 1σ = 85 to 115: about 68% of people.
- μ ± 2σ = 70 to 130: about 95%.
- μ ± 3σ = 55 to 145: about 99.7%.
Symmetry then answers tail questions with arithmetic alone:
- Above 115? Outside the 1σ band lies 32%, split evenly: 16%.
- Above 130? (100% − 95%)/2 = 2.5%.
- Between 85 and 130? Half of each band: 68%/2 below the mean + 95%/2 above it = 34% + 47.5% = 81.5% (the exact normal area is 81.9%).
- Below 55? (100% − 99.7%)/2 = 0.15% — about 1 person in 667.
Any boundary that is not a whole number of standard deviations — “above 120,” say — needs a z-score (120 is z = 1.33) and a real lookup: the z-score calculator or the normal distribution calculator picks up exactly where the rule leaves off.
Using the Rule as a Normality Check
The rule also runs in reverse: compare your data's actual coverage against 68-95-99.7 to judge whether a normal model is reasonable. Compute the mean and standard deviation, count the fraction of observations inside each band, and compare. A right-skewed data set might put 80% inside one sigma on the short side and leak far beyond three sigma on the long side — a loud warning that normal-based shortcuts (like the rule itself, or z-based control limits) will mislead.
This is exactly the logic of “three-sigma” process control: under a stable normal process, points beyond μ ± 3σ occur once per 370 samples, so their appearance signals a real change rather than noise.
When Data Is Not Bell-Shaped: Chebyshev's Floor
The empirical rule is a property of the normal shape, not of data in general. For any distribution whatsoever, Chebyshev's inequality guarantees at least 1 − 1/k² of values within k standard deviations:
| Within | Normal (empirical rule) | Any distribution (Chebyshev) |
|---|---|---|
| ±2σ | 95.45% | at least 75% |
| ±3σ | 99.73% | at least 88.9% |
The gap between the columns is the price of dropping the normality assumption. Skewed incomes, waiting times (see the exponential distribution), and bounded percentages all live between the two columns — which is why quoting 68-95-99.7 for arbitrary data overstates certainty.
Try the Empirical Rule Calculator
Enter a mean and standard deviation to get the 1σ, 2σ, and 3σ ranges with their percentages — the worked example automated.
Frequently Asked Questions
What is the empirical rule in simple terms?
For data shaped like a bell curve, about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. It turns the two numbers that describe a normal distribution - the mean and the standard deviation - into concrete ranges you can reason about without any further calculation.
Why 68, 95, and 99.7 specifically?
They are areas under the standard normal curve: P(-1 < Z < 1) = 0.6827, P(-2 < Z < 2) = 0.9545, and P(-3 < Z < 3) = 0.9973, conventionally rounded to 68%, 95%, and 99.7%. Nothing is special about the whole numbers 1, 2, 3 except convenience - the exact areas come straight from the normal distribution's shape.
Does the empirical rule apply to every data set?
No - it is a property of approximately normal (bell-shaped, symmetric) distributions. Skewed data like incomes, bounded data piled near a limit, or bimodal data can deviate wildly from 68-95-99.7. For completely arbitrary distributions, Chebyshev's inequality gives the guaranteed floor: at least 75% within 2 standard deviations and at least 89% within 3, no shape assumptions needed.
How do I use the empirical rule to find percentages?
Express your boundaries as standard deviations from the mean, then combine the three ranges with symmetry. With IQ scores (mean 100, SD 15): above 115 is (100% - 68%)/2 = 16%; between 70 and 130 is 95%; below 55 is (100% - 99.7%)/2 = 0.15%. For boundaries that are not whole numbers of SDs, use a z-table or normal distribution calculator instead.
What is the difference between the empirical rule and a z-score?
A z-score measures how many standard deviations one value sits from the mean; the empirical rule tells you what fraction of a normal distribution lies within the whole-number z-ranges. The rule is essentially a memorized three-row z-table. Any z that is not exactly 1, 2, or 3 needs the full table or a calculator.
Is a value outside 2 or 3 standard deviations an outlier?
Being outside 2 SDs happens to about 1 normal value in 22, so it is uncommon but expected in any decent-sized data set. Outside 3 SDs is about 1 in 370 - genuinely rare, and a common screening flag. Whether to call either an outlier depends on sample size and context: in 10,000 normal observations, about 27 beyond 3 SDs are entirely normal.