Three Sigma Rule in Statistics: Formula, Example, and Limits

What Is the Three Sigma Rule?

The three sigma rule says that, for an approximately normal distribution, about 99.73% of values fall within three standard deviations of the mean — mean − 3s to mean + 3s for sample data, or μ − 3σ to μ + 3σ for a known population. It is the third step of the 68–95–99.7 (empirical) rule: about 68.27% within ±1 standard deviation, 95.45% within ±2, and 99.73% within ±3.

Only about 0.27% of normal values — roughly 1 in 370 — land outside that band, so a value beyond it is unusual enough to investigate. It is a screening rule, not proof of an error.

To compute the mean and standard deviation of your own data, use the Standard Deviation Calculator, the Sample Standard Deviation Calculator or the Descriptive Statistics Calculator. Use the Z-Score Calculator when you already know the mean and standard deviation and want the z-score or tail probability for one value.

Three Sigma Limits Formula

Here x is the observation being checked, x̄ is the baseline mean and s is the baseline sample standard deviation. A value is outside the three sigma limits exactly when |z| > 3. For a known population, use μ and σ instead.

How to use the three sigma rule in 4 steps

STDEV.S divides by n − 1 and matches the calculator's Sample (n-1) mode; STDEV.P divides by N and matches Population (N) mode. See STDEV.S vs STDEV.P in Excel.

Worked Example: 3-Sigma Limits for Fill Weights

A packaging line fills bottles to a 500 g target. An analyst takes 12 recent fills from a stable run as the baseline (illustrative data):

Baseline fill weights (g): 499.4, 499.6, 499.8, 499.9, 500.0, 500.1, 500.2, 500.3, 500.4, 500.5, 500.7, 500.9

Sample mean x̄ = 500.15 g, sample standard deviation s = 0.442102 g. The calculator's sigma-limits panel shows these bands for the 12 baseline values:

Computed with this site's calculator on 2026-11-04. 8 of 12 baseline values (66.67%) sit inside ±1s and all 12 sit inside ±2s, close to the normal-model shares for a sample this small. The review band is 498.824 g to 501.476 g.

Checking new values against the limits

Three new bottles are weighed later and compared with the baseline limits:

The 501.7 g bottle is above the upper limit by 0.224 g. That is a statistical signal, not proof of a bad bottle: the next step is operational — check the filler setting, the scale calibration, product temperature, and whether nearby bottles show the same shift.

68–95–99.7 Percentages Behind the Rule

The percentages come from the standard normal distribution (see the NIST reference below). The rounded 68–95–99.7 figures are the ones usually quoted; the exact values are:

Using the Rule to Find Other Percentages

Because the normal curve is symmetric, the same bands give the share of values in any region between whole standard deviations. Values below come from this site's normal distribution function (the one behind the Z-Score Calculator):

The left side mirrors the right. Outside ±3σ is about 1 value in 370, so in a run of 1,000 in-control measurements you should expect roughly 2.7 beyond the limits by chance alone.

Outliers vs. Non-Normal Data

A value outside ±3s can mean three different things: a real special cause (a filler fault), a data error (a typo, a wrong unit), or simply that the data were never normal. Two calculation traps make the rule miss real outliers.

Masking: when the outlier hides itself

Add 501.7 g to the baseline and recompute: s grows from 0.442 g to 0.603 g, the upper 3-sigma limit moves out to 502.079 g, and the same bottle's z-score drops from 3.51 to 2.37. Inside the band — the suspect value has hidden itself. This is called masking.

Small samples: when |z| > 3 is impossible

With 10 or fewer values, no single observation can be more than 3 sample standard deviations from the sample mean, however extreme it is — the outlier inflates s. From n = 11 the limit is barely above 3. This is why the rule needs a baseline that excludes the value being tested, or a larger sample.

Skewed or heavy-tailed data

For data that are not roughly normal, the 99.73% figure does not hold. Chebyshev's inequality still guarantees that at least 1 − 1/3² ≈ 88.9% of values lie within three standard deviations for any distribution, but that bound is much weaker. Compare the two in Empirical Rule vs Chebyshev's Theorem, and consider IQR fences or a modified z-score for skewed data.

When to Use the Three Sigma Rule (and When Not To)

Use it: The baseline process is stable, the distribution is roughly bell-shaped, and the cost of reviewing a small number of signals is acceptable.

Use control charts: The data arrive over time and you need to distinguish routine variation from special-cause variation. Start with Control Charts and Process Control.

Use a z-score: You need to report how far a single value is from the mean in standard deviation units. See Z-Score Explained.

Use a robust method: The data are skewed, heavy-tailed, or already include extreme values. Compare Robust Statistics and Modified Z-Score Outlier Detection.

Use a tolerance limit: A customer, regulator, or engineering specification already defines the acceptable range. A three sigma signal can support investigation, but it does not replace the requirement.

Three Sigma Rule Checklist

• Is the baseline data from the same process, instrument, population, and time period as the value being checked?
• Was the suspicious value kept out of the baseline mean and standard deviation calculation?
• Does a histogram or normal probability plot look roughly normal?
• Are there enough observations for the standard deviation to be stable?
• Is the decision to investigate, quarantine, delete, or report the value defined before seeing the result?
• Would a domain limit, control chart, or robust statistic answer the practical question better?