# 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.

Empirical Rule Calculator

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## What is Empirical Rule Calculator

The Empirical Rule, also known as the 68-95-99.7 rule or the three-sigma rule, is a statistical guideline that describes the approximate distribution of data in a normal distribution. It states that for a normal distribution:

• Approximately 68% of the data falls within one standard deviation of the mean.
• Approximately 95% of the data falls within two standard deviations of the mean.
• Approximately 99.7% of the data falls within three standard deviations of the mean.

To use an Empirical Rule calculator, you'll need the mean and standard deviation of your dataset. Here's how you can apply the Empirical Rule using a calculator:

1. Calculate the mean (μ) and standard deviation (σ) of your dataset using appropriate statistical methods.

2. Use the calculated mean and standard deviation to determine the ranges based on the Empirical Rule:

• One standard deviation from the mean:

• Range: (μ - σ, μ + σ)
• Approximately 68% of the data falls within this range.
• Two standard deviations from the mean:

• Range: (μ - 2σ, μ + 2σ)
• Approximately 95% of the data falls within this range.
• Three standard deviations from the mean:

• Range: (μ - 3σ, μ + 3σ)
• Approximately 99.7% of the data falls within this range.

By applying these ranges, you can analyze how much of your data falls within each interval according to the Empirical Rule.

Please note that the Empirical Rule assumes a normal distribution, and its accuracy may vary for datasets that do not follow a normal distribution.

If you provide me with a specific dataset, mean, and standard deviation, I can demonstrate the application of the Empirical Rule using that dataset.

## 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