What does the empirical rule say?
See how much of your data lies within 1, 2, and 3 standard deviations of the mean, and compare it with the 68-95-99.7 rule.
- Within 1 standard deviation
- 60%
60% of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 lie within 1 standard deviation of the mean (2.627719 to 8.372281), 100% within 2, and 100% within 3.
- Within 2 standard deviationsnormal: about 95%
- 100%
- Within 3 standard deviationsnormal: about 99.7%
- 100%
- Mean (μ)
- 5.5
- Standard deviation
- 2.872281
- Mean − 1 SD
- 2.627719
- Mean + 1 SD
- 8.372281
- Numbers within 1 SD
- 6
- Mean − 2 SD
- −0.244563
- Mean + 2 SD
- 11.244563
- Numbers within 2 SD
- 10
- Mean − 3 SD
- −3.116844
- Mean + 3 SD
- 14.116844
- Numbers within 3 SD
- 10
- Count
- 10
Within 1 standard deviation: 60%. 60% of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 lie within 1 standard deviation of the mean (2.627719 to 8.372281), 100% within 2, and 100% within 3.
How are your numbers spread?
How to calculate
Computes the ranges within 1, 2, and 3 standard deviations of the mean of a list of numbers, and the percent of the numbers in each, to compare with the 68-95-99.7 rule.
Example with the default inputs (Your numbers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Standard deviation Population (σ)): 60% of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 lie within 1 standard deviation of the mean (2.627719 to 8.372281), 100% within 2, and 100% within 3.
Method: Find the mean and standard deviation; for k = 1, 2, 3 the range is mean ± k × SD, and the share is the count of numbers with |x − mean| ≤ k × SD, divided by n.
- The population standard deviation (divide by n) is the default, as on the old page. Pick sample (divide by n − 1) for a sample.
- A number on the end of a range counts as inside it, within a relative 2^-49 of |mean| + k × SD for rounding.
- The 68%, 95%, and 99.7% figures hold for a normal distribution. Other shapes can differ a lot.
Worked examples
Each example is checked against the calculator on every build.
- Your numbers 2, 4, 4, 4, 5, 5, 7, 9, Standard deviation Population (σ) gives Mean (μ) 5, Standard deviation 2, Mean − 1 SD 3, Mean + 1 SD 7, Numbers within 1 SD 6, Within 1 standard deviation 75%, Mean − 2 SD 1, Mean + 2 SD 9, Numbers within 2 SD 8, Within 2 standard deviations 100%, Mean − 3 SD -1, Mean + 3 SD 11, Within 3 standard deviations 100%.Source: OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution
- Your numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Standard deviation Population (σ) gives Mean (μ) 5.5, Standard deviation 2.872281, Mean − 1 SD 2.627719, Mean + 1 SD 8.372281, Numbers within 1 SD 6, Within 1 standard deviation 60%, Within 2 standard deviations 100%, Within 3 standard deviations 100%.Source: OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution
- Your numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Standard deviation Sample (s) gives Standard deviation 3.02765, Mean − 1 SD 2.47235, Mean + 1 SD 8.52765, Numbers within 1 SD 6, Within 1 standard deviation 60%.Source: OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution
How it works
For a list of n numbers x₁, …, xₙ:
- Find the mean: μ = (x₁ + … + xₙ) ÷ n.
- Find the standard deviation. Population (the default): σ = √(Σ(xᵢ − μ)² ÷ n). Sample: s = √(Σ(xᵢ − μ)² ÷ (n − 1)).
- For k = 1, 2, 3, the range is mean − k × SD to mean + k × SD.
- Count the numbers inside each range: those with |xᵢ − μ| ≤ k × SD. The share is that count ÷ n × 100%.
For a normal distribution, the shares are about 68%, 95%, and 99.7% (exactly 68.27%, 95.45%, and 99.73%, from the standard normal table). The calculator shows your data's shares next to them.
Assumptions
- The population standard deviation is the default, as on the old page. Choose Sample when the numbers are a sample from a larger group.
- A number exactly on the end of a range counts as inside it. Decimal numbers and the square root carry rounding error, so the test is |xᵢ − μ| ≤ k × SD + 2⁻⁴⁹ × (|μ| + k × SD): a number within about 8 units in the last place of an end counts as on it. For example, 0.1 and 0.2 are each exactly 1 SD from their mean, so both are inside the 1 SD range (100%).
- The list needs at least 2 numbers.
Worked examples by hand
2, 4, 4, 4, 5, 5, 7, 9 (population). The mean is 40 ÷ 8 = 5. The squared differences add up to 32, so σ = √(32 ÷ 8) = 2.
- 1 SD: 3 to 7. The numbers 4, 4, 4, 5, 5, 7 are inside, 6 of 8 = 75%.
- 2 SD: 1 to 9. All 8 are inside, 100%.
- 3 SD: −1 to 11. All 8, 100%.
1, 2, …, 10 (the default list, population). The mean is 5.5 and σ = √8.25 = 2.8723. 1 SD runs from 2.6277 to 8.3723, which holds 3, 4, 5, 6, 7, 8: 6 of 10 = 60%. 2 SD runs from −0.2446 to 11.2446 and holds all 10, 100%, and so does 3 SD.
The same list, sample. s = √(82.5 ÷ 9) = 3.0277. 1 SD runs from 2.4723 to 8.5277, which still holds 3 to 8: 60%.
Other questions people ask
What is the Empirical Rule?
The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical principle that states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
When should I use the Empirical Rule?
Use the Empirical Rule when you have data that follows a normal (bell-shaped) distribution. It's useful for understanding data spread, identifying outliers, and making predictions about where most of your data points should fall.
What does it mean if my data doesn't follow the Empirical Rule?
If your data doesn't closely match the 68-95-99.7 percentages, it may indicate that your data is not normally distributed. This could mean your data is skewed, has multiple peaks, or follows a different distribution pattern.
How do I interpret the results?
Compare the actual percentages in your data to the expected 68%, 95%, and 99.7%. If they're close, your data likely follows a normal distribution. Large differences suggest non-normal data or the presence of outliers. With only a few numbers, the percentages move in big steps, so expect differences.
What is the difference between population and sample standard deviation?
Population standard deviation divides by N (total count), while sample standard deviation divides by N-1 (degrees of freedom). This calculator uses population standard deviation by default. If your numbers are a sample from a larger group, choose Sample.
Can I use this calculator for any type of data?
While you can input any numeric data, the Empirical Rule is most meaningful for data that follows a normal distribution. For skewed or non-normal data, the percentages may not match the expected values.
What are outliers in the context of the Empirical Rule?
Outliers are data points that fall outside the expected ranges. According to the Empirical Rule, only about 0.3% of data should fall beyond 3 standard deviations from the mean in a normal distribution.
How accurate is the Empirical Rule?
The Empirical Rule provides approximate percentages. For an exact normal distribution the shares are 68.27%, 95.45%, and 99.73%. In practice, real data may vary slightly from these percentages even for normal distributions. The rule is most useful as a general guideline for understanding data spread.
What's the relationship between the Empirical Rule and z-scores?
Z-scores measure how many standard deviations a data point is from the mean. The Empirical Rule corresponds to z-scores of ±1 (68%), ±2 (95%), and ±3 (99.7%). This calculator shows the actual ranges and percentages for your data.
Can I use this for quality control or process monitoring?
Yes! The Empirical Rule is commonly used in quality control to identify when processes are out of control. Data points falling outside the expected ranges may indicate process changes or defects that need attention.