The empirical rule – also called the 68–95–99.7 rule or the three-sigma rule – is a quick way to describe a normal (bell-shaped) distribution without any tables. About 68% of the values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three.
That makes it handy for sanity checks: if a value sits more than three standard deviations from the mean, it happens less than 0.3% of the time and is worth a second look as a possible outlier or error. It also gives a rough idea of the spread when you only know μ and σ, and a test of whether data look normal – if far fewer or far more than 68% of your values fall inside μ ± σ, the data are probably not bell-shaped.
The calculator shows the three ranges, a table with the exact normal percentages (including four standard deviations), and a chart with each band shaded. Enter an optional value to see its z-score, its exact percentile and which band it falls in.
How to use the Empirical Rule Calculator
- Mean (μ)
- Standard deviation (σ)
- Value to locate (optional)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The ranges are μ ± kσ for k = 1, 2, 3. The exact share inside each range is 2Φ(k) − 1 from the standard normal distribution, and a value x is located with z = (x − μ)/σ and its percentile Φ(z).
Worked example
IQ scores have μ = 100 and σ = 15. One standard deviation either side gives 85 to 115 (about 68% of people), two give 70 to 130 (95%) and three give 55 to 145 (99.7%). A score of 130 has z = 2: it is at the 97.72nd percentile, so roughly 1 person in 44 scores higher. A score of 145 or more (z ≥ 3) is reached by only 0.13% of people.
Empirical rule versus Chebyshev
| Within | Normal data (empirical rule) | Any data (Chebyshev, at least) |
|---|---|---|
| μ ± 1σ | 68.27% | no guarantee |
| μ ± 2σ | 95.45% | 75% |
| μ ± 3σ | 99.73% | 88.9% |
| μ ± 4σ | 99.994% | 93.75% |
Formula
68% within μ ± σ · 95% within μ ± 2σ · 99.7% within μ ± 3σ · exact: 68.27%, 95.45%, 99.73%
Worked example
Inputs — Mean (μ): 100; Standard deviation (σ): 15; Value to locate (optional): 130.
Results — About 68% of values lie between = 85 to 115; About 95% of values lie between = 70 to 130; About 99.7% of values lie between = 55 to 145; Exact shares by number of standard deviations = Range From To Inside (exact) Outside Outside rate
─────────────────────────────────────────────────────────────
±1σ 85 115 68.2689% 31.7311% 1 in 3
±2σ 70 130 95.45% 4.55% 1 in 22
±3σ 55 145 99.73% 0.27% 1 in 370
±4σ 40 160 99.9937% 0.0063% 1 in 15,787; z-score of your value = 2; Percentile of your value = 97.72 %.
Key terms
- Three-sigma rule
- Another name for the empirical rule; 3σ covers 99.7% of a normal distribution.
- Standard deviation (σ)
- The typical distance of values from the mean.
- Chebyshev's inequality
- A rule valid for any distribution: at least 1 − 1/k² of values lie within k standard deviations.
- Outlier
- A value unusually far from the rest, often defined as more than 3σ from the mean.
Frequently asked questions
What does the 68–95–99.7 rule say?
For a normal distribution, about 68% of values are within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.
What are the exact percentages?
68.27%, 95.45% and 99.73%. The rule rounds them for easy memory.
What share of values lie above μ + 2σ?
Half of the 4.55% outside ±2σ, about 2.28%; above μ + 3σ it is only 0.13%.
For IQ (mean 100, SD 15), what are the ranges?
85–115 (68%), 70–130 (95%) and 55–145 (99.7%).
Does the rule work for any data?
Only for roughly bell-shaped, symmetric data. For any distribution Chebyshev's inequality guarantees at least 75% within 2 SD and 88.9% within 3 SD.
Can I use it to find outliers?
Yes – values beyond ±3σ occur about 1 time in 370 under normality, so they are commonly flagged.
Should I use the sample or population standard deviation?
Use whichever describes the data; for a sample, the sample standard deviation (dividing by n − 1) is the usual choice.