This empirical rule calculator can be employed to calculate the share of values that fall within a specified number of standard deviations from the mean. It also plots a graph of the results. Simply enter the mean (M) and standard deviation (SD), and click on the "Calculate" button to generate the statistics.

## The Empirical Rule

The Empirical Rule, which is also known as the three-sigma rule or the 68-95-99.7 rule, represents a high-level guide that can be used to estimate the proportion of a normal distribution that can be found within 1, 2, or 3 standard deviations of the mean. According to this rule, if the population of a given data set follows a normal, bell-shaped distribution in terms of the population mean (M) and standard deviation (SD), then the following is true of the data:

- An estimated 68% of the data within the set is positioned within one standard deviation of the mean; i.e., 68% lies within the range [M - SD, M + SD].
- An estimated 95% of the data within the set is positioned within two standard deviations of the mean; i.e., 95% lies within the range [M - 2SD, M + 2SD].
- An estimated 97.7% of the data within the set is positioned within three standard deviations of the mean; i.e., 99.7% lies within the range [M - 3SD, M + 3SD].

## Example

Let's say the scores of an exam follow a bell-shaped distribution that has a mean of 100 and a standard deviation of 16. What percentage of the people who completed the exam achieved a score between 68 and 132?

**Solution:** 132 – 100 = 32, which is 2(16). As such, 132 is 2 standard
deviations to the right of the mean. 100 – 68 = 32, which is 2(16). This means that
a score of 68 is 2 standard deviations to the left of the mean. Since 68 to 132 is
within 2 standard deviations of the mean, **95%** of the exam participants
achieved a score of between 68 and 132.

You may also be interested in our Z-Score Calculator or/and P-Value Calculator

This empirical rule calculator can be employed to calculate the share of values that fall within a specified number of standard deviations from the mean. It also plots a graph of the results. Simply enter the mean (M) and standard deviation (SD), and click on the "Calculate" button to generate the statistics.

## The Empirical Rule

The Empirical Rule, which is also known as the three-sigma rule or the 68-95-99.7 rule, represents a high-level guide that can be used to estimate the proportion of a normal distribution that can be found within 1, 2, or 3 standard deviations of the mean. According to this rule, if the population of a given data set follows a normal, bell-shaped distribution in terms of the population mean (M) and standard deviation (SD), then the following is true of the data:

- An estimated 68% of the data within the set is positioned within one standard deviation of the mean; i.e., 68% lies within the range [M - SD, M + SD].
- An estimated 95% of the data within the set is positioned within two standard deviations of the mean; i.e., 95% lies within the range [M - 2SD, M + 2SD].
- An estimated 97.7% of the data within the set is positioned within three standard deviations of the mean; i.e., 99.7% lies within the range [M - 3SD, M + 3SD].

## Example

Let's say the scores of an exam follow a bell-shaped distribution that has a mean of 100 and a standard deviation of 16. What percentage of the people who completed the exam achieved a score between 68 and 132?

**Solution:** 132 – 100 = 32, which is 2(16). As such, 132 is 2 standard
deviations to the right of the mean. 100 – 68 = 32, which is 2(16). This means that
a score of 68 is 2 standard deviations to the left of the mean. Since 68 to 132 is
within 2 standard deviations of the mean, **95%** of the exam participants
achieved a score of between 68 and 132.

You may also be interested in our Z-Score Calculator or/and P-Value Calculator