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.