# What does the empirical rule say?

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.

- Page: https://www.acalculator.org/statistics/empirical-rule-calculator
- JSON spec: https://www.acalculator.org/statistics/empirical-rule-calculator.json
- Version: 17e7d6582853

## Default answer

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.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| input | Your numbers | The numbers, separated by commas, spaces, semicolons, or new lines. At least 2. |
| sd | Standard deviation | Population (divide by n) for a whole group, or sample (divide by n − 1) for a sample of a larger group. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| share1 | Within 1 standard deviation | The percent of your numbers within 1 standard deviation of the mean. A normal distribution has about 68%. |
| share2 | Within 2 standard deviations | The percent of your numbers within 2 standard deviations of the mean. A normal distribution has about 95%. |
| share3 | Within 3 standard deviations | The percent of your numbers within 3 standard deviations of the mean. A normal distribution has about 99.7%. |
| mean | Mean (μ) | The sum divided by the count. |
| sd | Standard deviation | The population or sample standard deviation, as chosen. |
| low1 | Mean − 1 SD | The low end of the range within 1 standard deviation of the mean. |
| high1 | Mean + 1 SD | The high end of the range within 1 standard deviation of the mean. |
| count1 | Numbers within 1 SD | How many of your numbers lie within 1 standard deviation of the mean, ends included. |
| low2 | Mean − 2 SD | The low end of the range within 2 standard deviations of the mean. |
| high2 | Mean + 2 SD | The high end of the range within 2 standard deviations of the mean. |
| count2 | Numbers within 2 SD | How many of your numbers lie within 2 standard deviations of the mean, ends included. |
| low3 | Mean − 3 SD | The low end of the range within 3 standard deviations of the mean. |
| high3 | Mean + 3 SD | The high end of the range within 3 standard deviations of the mean. |
| count3 | Numbers within 3 SD | How many of your numbers lie within 3 standard deviations of the mean, ends included. |
| count | Count | How many numbers are in the list. |

## 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.

## Assumptions

- 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

1. input = 2 or 4, sd = population gives mean = 5, sd = 2, low1 = 3, high1 = 7, count1 = 6, share1 = 75%, low2 = 1, high2 = 9, count2 = 8, share2 = 100%, low3 = -1, high3 = 11, share3 = 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.
2. input = 1 or 2, sd = population gives mean = 5.5, sd = 2.872281, low1 = 2.627719, high1 = 8.372281, count1 = 6, share1 = 60%, share2 = 100%, share3 = 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.
3. input = 1 or 2, sd = sample gives sd = 3.02765, low1 = 2.47235, high1 = 8.52765, count1 = 6, share1 = 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.

## FAQ

### 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.

## Sources

- 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
- OpenStax, Introductory Statistics 2e, §2.7 Measures of the Spread of the Data (standard deviation and variance). https://openstax.org/books/introductory-statistics-2e/pages/2-7-measures-of-the-spread-of-the-data