# Empirical Rule Calculator

## Empirical Rule

### Need Mean & Standard Deviation? Use This

68% of Values Lie Between 8 and 12
95% of Values Lie Between 6 and 14
97.7% of Values Lie Between 4 and 16

# Empirical Rule Calculator: A Tool for Normal Distribution Analysis

The empirical rule calculator is a statistical tool used to identify the expected range of a normally distributed variable expressed as multiples of the observed standard deviation of the population. This calculator is useful for statisticians and data analysts who need to quickly analyze the normality of a given data set.

## Understanding the Empirical Rule

The empirical rule is a statistical rule that states that nearly all of the data in a normal distribution will fall within three standard deviations of the mean. This is also known as the three-sigma rule or the 68-95-99.7 rule. The empirical rule can be used as a quick and intuitive test of the normality of a sample distribution.

A normally distributed variable should have 68% of data points within one sigma of the center, 95% of data points within two sigmas of the center, and almost everything within three standard deviations. If the sample you are looking at is notably different from this pattern, you may be dealing with something other than a normal distribution.

Alternatively, you may be looking at a sample that would eventually converge on the normal distribution but is constrained by low sample sizes from doing this in practice. This latter pattern is quite common in certain supply chain problems where the average order size at the customer level is normally distributed, but you don't have enough homogeneous customers buying the same item for the joint distribution to converge on a normal pattern.

## Using the Empirical Rule Calculator

Using the empirical rule calculator is easy. Simply enter the mean value and the sample standard deviation for the given data set. The calculator will then generate the expected range of values for the normal distribution based on the empirical rule.

The empirical rule calculator is a useful tool for statisticians and data analysts who need to quickly analyze the normality of a given data set. It can be used in conjunction with other statistical tools such as the standard deviation calculator and the statistics calculator to gain a deeper understanding of the statistical data.

## Example Calculation

Let's say we have a data set with the following values

5, 7, 8, 10, 12, 15

To calculate the mean value, we first need to add up all of the values and divide by the sample size:

(5 + 7 + 8 + 10 + 12 + 15) / 6 = 9.5

Next, we calculate the sample standard deviation:

First, we calculate the variance:

(5 - 9.5)^2 = 21.25

(7 - 9.5)^2 = 6.25

(8 - 9.5)^2 = 2.25

(10 - 9.5)^2 = 0.25

(12 - 9.5)^2 = 5.25

(15 - 9.5)^2 = 30.25

Then, we sum up all of the squared differences:

21.25 + 6.25 + 2.25 + 0.25 + 5.25 + 30.25 = 65.5

Finally, we divide by the sample size minus one and take the square root of the result:

sqrt(65.5 / 5) = 3.055

Therefore, the sample standard deviation for this data set is 3.055.

Using the empirical rule calculator, we can calculate the expected range of values for this data set based on the empirical rule:

• About 68% of the data points will fall within one standard deviation of the mean, or between 6.445 and 12.555.
• About 95% of the data points will fall within two standard deviations of the mean, or between 3.39 and 15.61.
• About 99.7% of the data points will fall within three standard deviations of the mean, or between 0.335 and 18.665.

## Interpreting the Empirical Rule

The empirical rule is a statistical rule that provides valuable information about the normality of a given data set. If the data set follows a normal distribution, then the empirical rule can be used to estimate the expected range of values for the given data set. This can be useful for identifying outliers or other unusual observations within the data set.

However, it is important to note that the empirical rule is only applicable to normally distributed data. If the data set does not follow a normal distribution, then the empirical rule may not be accurate, and other statistical tools may need to be used instead.

Additionally, the empirical rule is based on probabilities and is not a definitive measure of the data set. While it can provide valuable insights into the normality of a given data set, it is important to exercise caution when interpreting the results and to consider other statistical measures as well.

## Conclusion

The empirical rule calculator is a powerful tool for statisticians and data analysts who need to quickly analyze the normality of a given data set. It provides valuable insights into the expected range of values for normally distributed data, and can be used in conjunction with other statistical tools to gain a deeper understanding of the data set.

Whether you are working with a sample or the entire population, the empirical rule calculator is a useful tool for any statistician or data analyst who needs to analyze statistical data and make informed decisions based on the results.

Our statistics calculator is also available on our website, along with other useful calculators such as the standard deviation calculator and the fraction calculator. These calculators are designed to make statistical calculations easier and more accessible for professionals in a variety of fields.