Empirical Rule Calculator (68-95-99.7 Rule)
Enter your mean and standard deviation to find the three empirical-rule intervals. The calculator shows every range at 1, 2 and 3 standard deviations from the mean, the percentage of data expected inside each band, and a bell-curve visual so you can see exactly where your distribution lies. Optionally enter any observed value to find which sigma band it falls in and what percentage of a normal distribution lies below it.
Formula
Worked example
IQ scores are normally distributed with mean 100 and standard deviation 15. Applying the empirical rule: 68% of people score between 85 and 115; 95% score between 70 and 130; 99.7% score between 55 and 145. A score of 130 has z-score (130 - 100) / 15 = 2.0, placing it in the top 2.28% of the distribution.
What is the empirical rule?
The empirical rule, also called the 68-95-99.7 rule or the three-sigma rule, describes how data in a normal (bell-shaped) distribution cluster around the mean. Specifically, about 68% of observations fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations. The rule was formalized in the 19th century alongside the normal distribution itself and remains one of the most widely cited facts in statistics because it gives an instant sense of how spread out or concentrated a dataset is without any complex calculation.
How to use this calculator
Enter the mean and standard deviation of your dataset. The three empirical-rule intervals update instantly: you will see the lower and upper bounds for the 68%, 95% and 99.7% bands. To look up a specific value, enter it in the optional "Observed value" field. The calculator will compute its z-score, tell you which sigma band it falls in, and give you an approximate percentile based on the normal distribution CDF. Results apply to any dataset that is approximately normally distributed; if your data is skewed or has heavy tails, the intervals will still be calculated but may not accurately represent the true data spread.
The formula and what each term means
The three intervals are calculated by multiplying the standard deviation by 1, 2 or 3 and adding or subtracting that product from the mean. For the 68% band: lower = μ - σ, upper = μ + σ. For the 95% band: lower = μ - 2σ, upper = μ + 2σ. For the 99.7% band: lower = μ - 3σ, upper = μ + 3σ. A z-score converts any raw value into "standard deviation units" by computing (x - μ) / σ, which makes it easy to compare values across distributions with different means and spreads. A z-score of +2 means the value is two standard deviations above the mean, falling in the upper half of the 95% band.
Real-world applications
The empirical rule appears across many fields. In quality control, manufacturers use the rule to set acceptance limits: parts within three standard deviations of the target dimension pass, while those outside (fewer than 3 in 1,000) are flagged. In finance, analysts model asset returns as approximately normal, using the rule to estimate how often a fund might post an unusually bad month. In education, standardized test scores such as IQ (mean 100, SD 15) or SAT scores (mean 1010, SD 210) use the rule to communicate where a student sits relative to the population. In medicine, reference ranges for blood tests are often set at mean plus or minus two standard deviations, covering the central 95% of healthy subjects.
When does the empirical rule not apply?
The rule is valid only for normal or approximately normal distributions. It breaks down for skewed data such as incomes or house prices, heavy-tailed distributions such as daily stock market moves, bimodal data with two distinct clusters, or any dataset with large outliers. Before applying the rule, verify normality with a histogram, Q-Q plot, or a formal test such as Shapiro-Wilk. For non-normal distributions, Chebyshev's inequality provides a more conservative but universally valid bound: at least 1 - 1/k^2 of data lies within k standard deviations of the mean for any distribution, regardless of shape.
Empirical rule summary (68-95-99.7 rule)
| Interval | Range | % within band | % outside band | Approx. 1 in N |
|---|---|---|---|---|
| μ ± 1σ | [lo1, hi1] | 68.27% | 31.73% | 1 in 3 |
| μ ± 2σ | [lo2, hi2] | 95.45% | 4.55% | 1 in 22 |
| μ ± 3σ | [lo3, hi3] | 99.73% | 0.27% | 1 in 370 |
| μ ± 4σ | [lo4, hi4] | 99.9937% | 0.0063% | 1 in 15,787 |
| μ ± 5σ | [lo5, hi5] | 99.99994% | 0.00006% | 1 in 1,744,278 |
| μ ± 6σ | [lo6, hi6] | 99.9999998% | 0.0000002% | 1 in 506,797,346 |
Expected percentages of data in each sigma band for a perfectly normal distribution.
Frequently asked questions
What is the 68-95-99.7 rule in statistics?
The 68-95-99.7 rule states that for any normally distributed dataset, approximately 68% of values lie within one standard deviation of the mean, 95% lie within two standard deviations, and 99.7% lie within three standard deviations. The rule is a shorthand summary of the normal distribution's bell-curve shape and lets you quickly estimate how unusual or typical any value is without looking up a z-table.
How do I find the empirical rule intervals?
Multiply your standard deviation by 1, 2 and 3, then add and subtract each result from the mean. For a mean of 50 and standard deviation of 10: the 68% interval is [40, 60], the 95% interval is [30, 70], and the 99.7% interval is [20, 80]. This calculator automates those three multiplications for any mean and standard deviation you enter.
What is a z-score and how is it related to the empirical rule?
A z-score measures how many standard deviations a value is from the mean: z = (x - μ) / σ. The empirical rule bands correspond directly to z-score cutoffs: values with z between -1 and +1 are in the 68% band, between -2 and +2 are in the 95% band, and between -3 and +3 are in the 99.7% band. A z-score of +2 means your value is two standard deviations above average.
Can I use the empirical rule for non-normal distributions?
The empirical rule gives accurate percentages only for normal or near-normal distributions. For any other shape, use Chebyshev's inequality instead: it guarantees that at least 75% of data lies within 2 standard deviations and at least 89% within 3 standard deviations, no matter how the data are distributed. The percentages are less precise but universally valid.
What percentage of data falls beyond 3 standard deviations?
Only about 0.27% of data in a normal distribution lies more than 3 standard deviations from the mean, or roughly 1 in every 370 observations. This is why a "three-sigma event" is considered highly unusual in manufacturing and finance. Beyond 4 sigma the probability drops to about 0.006%, and beyond 6 sigma (the goal of Six Sigma quality programs) it is about 2 in a billion.
How is the empirical rule used in quality control?
In manufacturing, process engineers use the empirical rule to set control limits. Control charts typically flag a data point as out-of-control when it falls beyond 3 standard deviations from the process mean (the 99.7% boundary). Because only about 0.27% of normal variation crosses that line, any point beyond 3 sigma is likely a genuine signal rather than random noise, prompting an investigation.