What is the empirical rule in statistics?
The empirical rule says that in a bell-shaped (normal) distribution about 68% of the data fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. It is also called the 68-95-99.7 rule, and it only works for data that are roughly symmetric and bell-shaped.
Empirical Rule in a picture
Formal definition and notation
For a normal distribution with mean μ and standard deviation σ, P(μ − σ < X < μ + σ) ≈ 0.68, P(μ − 2σ < X < μ + 2σ) ≈ 0.95 and P(μ − 3σ < X < μ + 3σ) ≈ 0.997. Because the curve is symmetric, each interval splits evenly around the mean, which gives the band percentages 34%, 13.5%, 2.35% and 0.15% on each side. For data that only approximately follow a bell curve, the percentages are approximate.
| Symbol | Read it as | Meaning |
|---|---|---|
| μ | mew | the mean, the center of the bell |
| σ | sigma | the standard deviation, the width of one band |
| μ ± 2σ | mew plus or minus two sigma | the interval from 2 standard deviations below the mean to 2 above |
Key facts
- Band by band from the mean outward: 34%, 13.5%, 2.35%, 0.15% on each side.
- Half the data (50%) lie above the mean and half below, because the curve is symmetric.
- About 5% of values are more than 2 standard deviations from the mean: 2.5% in each tail.
- A value more than 3 standard deviations away is very rare (about 0.3% in all), so it may be an outlier.
- The rule is a quick estimate. For other cutoffs, use a z-score table or calculator.
- Skewed data, like incomes, do not follow the rule.
The 68-95-99.7 rule
- μ
- mean of the distribution
- σ
- standard deviation
Half of each band lies on each side of the mean: 34% + 34% = 68%, and 13.5% + 34% + 34% + 13.5% = 95%. The two tails beyond 2σ hold 2.5% each.
Empirical Rule examples, step by step
Example 1: Within one standard deviation
Adult heights in a group are bell-shaped with mean 64 inches and standard deviation 3 inches. Between which two heights are the middle 68% of the group?
- 68% means within 1 standard deviation of the mean.
- 64 − 3 = 61 and 64 + 3 = 67.
Answer: About 68% of the heights are between 61 and 67 inches.
Example 2: Percent in one tail
Test scores are bell-shaped with mean 70 and standard deviation 8. About what percent of scores are above 86?
- 86 = 70 + 2 × 8, so 86 is 2 standard deviations above the mean.
- About 95% of scores are within 2 standard deviations, so 100% − 95% = 5% are outside.
- The curve is symmetric, so half of that 5% is above 86: 5%2 = 2.5%.
Answer: About 2.5% of scores are above 86.
Example 3: Count values in a lopsided interval
The lifetimes of 1,000 batteries are bell-shaped with mean 500 hours and standard deviation 40 hours. About how many batteries last between 460 and 580 hours?
- 460 is 500 − 40, 1 standard deviation below. 580 is 500 + 2 × 40, 2 standard deviations above.
- From −1σ to the mean: 34%. From the mean to +1σ: 34%. From +1σ to +2σ: 13.5%.
- Add: 34% + 34% + 13.5% = 81.5%.
- 0.815 × 1000 = 815.
Answer: About 815 batteries last between 460 and 580 hours.
Common mistakes
The rule gives symmetric intervals. For one side, split the leftover tail percent in half.
68% covers both sides of the mean, from −1σ to +1σ.
The 68-95-99.7 percentages come from the normal curve, so they can be far off for other shapes.
Not to be confused with
A z-score measures how many standard deviations one value is from the mean, like z = 2. The empirical rule tells what percent of the data lie within 1, 2 or 3 of those standard deviations.
Chebyshev’s rule works for any shape but gives weaker guarantees: at least 75% within 2 standard deviations, compared with about 95% for bell-shaped data.
Quick check: 3 questions
For bell-shaped data, about what percent of values lie within 2 standard deviations of the mean?
Answer: B. Within 2 standard deviations is about 95%. 68% is within 1, 99.7% is within 3, and 47.5% is only one side, from the mean to +2σ.Show answer and explanation
Weights are bell-shaped with mean 150 g and standard deviation 10 g. About what percent weigh less than 140 g?
Answer: D. 140 is 1 standard deviation below the mean. Outside ± 1σ is 100% − 68% = 32%, and half of it is below: 16%. 32% counts both tails, 34% is the band from 140 to 150, and 68% is the middle.Show answer and explanation
A data set is bell-shaped with mean 50 and standard deviation 5. Which interval holds about 99.7% of the data?
Answer: A. 99.7% lies within 3 standard deviations: 50 ± 3 × 5, from 35 to 65. 45 to 55 is ± 1σ (68%) and 40 to 60 is ± 2σ (95%).Show answer and explanation
Empirical Rule: frequently asked questions
Why is it called the 68-95-99.7 rule?
The three numbers are the rounded percents of a normal distribution within 1, 2 and 3 standard deviations of the mean. More exact values are about 68.27%, 95.45% and 99.73%. The rounded versions are easy to remember and close enough for quick estimates.
When can you use the empirical rule?
Use it when the data are roughly symmetric and bell-shaped, with one peak in the middle, such as many measurements of heights, test scores or manufacturing errors. Look at a histogram first. If the data are skewed, have two peaks or have big outliers, the percentages can be badly off.
What percent of data is beyond 3 standard deviations?
About 0.3%, since 99.7% lies within 3 standard deviations. Split between the two tails, that is about 0.15% above μ + 3σ and 0.15% below μ − 3σ. In a group of 1,000 values you would expect only about 3 of them that far out.
Practice and tools
Sources
- OpenStax, Introductory Statistics 2e, 6.1 The Standard Normal Distribution
- OpenStax, Introductory Statistics 2e, 2.7 Measures of the Spread of the Data
- Common Core State Standards, High School Statistics and Probability, Interpreting Categorical and Quantitative Data (HSS-ID.A.4)
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-ID.A.4
Cite this entry
Effortless Math Team. “Empirical Rule.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/empirical-rule/
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