Empirical Rule Calculator (68–95–99.7 rule ranges)

Understanding the empirical rule calculator — concept and principle

This calculator applies the 68–95–99.7 rule to a bell-shaped data set: from the mean and standard deviation it gives the ranges that hold about 68 %, 95 % and 99.7 % of the values. It also places a value of your choice, giving its z-score and the share of the normal curve below it. Use it for test scores, measurements and quality checks.

68–95–99.7: the share of a normal distribution within 1, 2 and 3 SD.

How it is calculated

About 68 % within μ ± σ; about 95 % within μ ± 2σ; about 99.7 % within μ ± 3σ

Exact normal shares: 68.27 %, 95.45 %, 99.73 %

z = (x − μ) ÷ σ; share below x = Φ(z), the standard normal cumulative probability

The principle

For a normal distribution, the share of values within a given number of standard deviations of the mean is the same, whatever the mean and spread. That is why three easy numbers describe any bell curve. The z-score translates a value into "how many standard deviations from the mean", which lets values from different scales be compared. The rule is only as good as the bell shape: skewed or heavy-tailed data can be far off, and Chebyshev's inequality gives a guaranteed but looser alternative.

Worked example

Default inputs: μ = 70, σ = 8, value 86.

About 68 % between 70 ± 8: 62 – 78.
About 95 % between 70 ± 16: 54 – 86.
About 99.7 % between 70 ± 24: 46 – 94; exact shares 68.27 % / 95.45 % / 99.73 %.
z for 86 = (86 − 70) ÷ 8 = 2, which is on the edge of the 2σ range (the page reports between 1σ and 2σ from the mean).
Share below 86 = Φ(2) = 97.7 %.

Each point is the calculator re-run with only “Mean μ” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Mean μ	Share below your value (normal model)
17.5	100%
35	100%
52.5	100%
70	97.7%
87.5	42.6%
105	0.9%
140	0%

How to read the result

A score of 86 here is better than about 97.7 % of a normal population. Values beyond 2σ occur about 1 time in 20 overall (about 1 in 40 on each side), and beyond 3σ about 3 times in 1,000, so they are worth a second look as possible errors or genuine outliers.

Common mistakes

Using the rule for clearly skewed data, such as incomes.
Mixing up the population σ with a rough guess.
Thinking 95 % lies above μ − 2σ; it lies between μ − 2σ and μ + 2σ.
Forgetting that the percentages are approximate.

Keep in mind: The rule assumes a normal distribution. Check a histogram first; for small samples the spread itself is uncertain.

Frequently asked questions

What is the 68–95–99.7 rule?

In a normal distribution about 68 %, 95 % and 99.7 % of values lie within 1, 2 and 3 standard deviations of the mean.

What percentile is a z-score of 2?

About the 97.7th percentile.

What if my data are not normal?

Use Chebyshev's theorem, which guarantees at least 75 % within 2σ for any distribution.

Sources: OpenStax, Introductory Statistics 2e (2023), §6.1 The Standard Normal Distribution

Checked 9 Nov 2026. OpenStax, Introductory Business Statistics 2e, §6.1 The Standard Normal Distribution (empirical rule: about 68%, 95% and 99.7% of values within 1, 2 and 3 standard deviations; z-scores); exact shares from the standard normal CDF.

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