Empirical Rule 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].
Empirical Rule Calculator

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

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