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Empirical Rule Calculator

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This empirical rule calculator is best tool to check the normal distribution of data within 3 ranges of standard deviation. Sometimes, this tool is also referred to as a three-sigma rule calculator or the 68 95 and 99.7 rule calculator. Just enter the mean and standard deviation if you select summary data or the sample or population if you select raw data to get the mean values for 68%, 95% and 99.7% of data within 3 SD ranges.

What is the Empirical Rule?

The empirical rule implies that for a normal distribution almost all data lies within 3 standard deviations of the mean. According to this 68 95 99 rule, 68% of the data lies within the first standard deviation. Ninety-five percent of the data is to be kept within second standard deviations. While 99.7% of the data lies within third standard deviations. While you're dealing with a usual catering of data, you can use this normal distribution empirical rule because of its ability to estimate probabilities. The empirical rule graph exhibits the three categories of the rule which are shown below:

empirical rule graph

What is The Empirical Rule Formula?

From the given algorithm you will come to know about the formula, our empirical rule calculator also uses the same formula to calculate the normal distribution of data within 3 ranges of standard deviation.

Calculate the mean using: μ = (Σ xi) / n

  • ∑ - indicates the sum of all given values
  • xi - each value from the data
  • N – total number of terms

Find the standard deviation using: σ = √ (∑ (xi – µ) ² / (n – 1))

What is The Percentage Rules?

Empirical Rule is categorized into three percentages, 68, 95, and 99.7. therefore, it is also known as the 68 95 and 99.7 rule. Different categories of the rule are:

  • In a normal distribution, 68% of the data values will rest among 1 standard deviation (within 1 sigma) of the mean.
  • In a normal, bell-shaped, distribution 95% of the data will fall into 2 standard deviations (within 2 sigma) of the mean.
  • 99.7% of the data will be kept among 3 standard deviations (within 3 sigma) of the mean in a normal bell-shaped distribution. While the empirical rule percentages graph is given below representing the percentages accordingly:

empirical rule statistics

How to Use the Empirical Calculator?

To Calculate a Normal Distribution for Summary Data:

If you have summary data, then you must select a ‘summary data’ option. In this case, you have the value of mean and standard deviation for your data. Follow the simple steps to check the data distribution.

Input

  • From the “calculation form” just Select the summary data.
  • Now enter the mean value of your data.
  • Very next you have to enter the standard deviation.
  • Hit the calculate button.

Output:

  • The very first output will be your entered mean. It will remain the same and represented by (x̅)
  • As a second output, you will have your value of standard deviation represented by (s)
  • Values will be given for 68% data falls between the first standard deviation.
  • Values will be given for 95% data falls between the second standard deviation.
  • Values will be given for 99.7% data falls between the third standard deviation.
  • Also, this bell shaped distribution calculator shows you the complete bell shaped empirical rule graph

To Calculate a Normal Distribution for Raw Data:

If you have your data in sample or population then you need to select the raw data option from the drop-down menu. The empirical calculator will do the rest.

Input:

  • From the “calculation form” just Select the raw data.
  • The first input is dataset type and you just have to Select the population or sample.
  • Now enter numbers according to your data.
  • Hit the calculate button

Output:

  • The very first output will be your mean represented by (x̅)
  • you will have your value of standard deviation represented by (s)
  • Values will be given for 68% data falls between the first standard deviation.
  • Values will be given for 95% data falls between the second standard deviation.
  • Values will be given for 99.7% data falls between the third standard deviation.
  • This Empirical rule calculator will show the bell-shaped empirical rule corresponding to empirical rule statistics

How to Calculate Empirical Rule?

If the values of Standard Deviation and mean are known anyone can calculate the value of empirical rule using empirical rule formula. Down below is the empirical rule example to better understanding the method. Empirical Rule Formula derivation:

  • 68% of values lies => mean ± sd
  • 95% of value lies => mean ± 2 sd
  • 99.73% of value lies => mean ± 3 sd

Here;

  • sd represents standard deviation for the value that is given
  • Mean represents the Arithmetic Mean for the given values

Example:

Let’s dig a little deep and find empirical rule for the values {12,32,45,53,21,43}

Part 1: The 1st step is to Calculate Mean

  • mean = (12+32+45+53+21+43)/6
  • mean = 206/6
  • mean = 34.33

Part 2: Then find the value for Standard Deviation

  • SD(σ)=&radical;(1/(N-1) *((x1-xm)2+(x2-xm)2+. +(xn-xm)2))
  • SD(σ)=√ (1/ (6-1) ((12-34.33)2+(32-34.33)2+(45-34.33)2+(53-34.33)2+(21-34.33)2+(43-34.33)2))
  • SD(σ)=√ (1/5((-22.34)2+(-2.33)2+(10.67)2+(18.67)2+(-13.34)2+(8.67)2))
  • SD(σ)=√ (1/5((498.76) +(5.43) +(113.78) +(348.45) +(177.77) +(75.12)))
  • SD(σ)=√ (243.87)
  • SD(σ)=15.62

References:

From the source of investopedia - Understanding the Empirical Rule - By WILL KENTON – Example of The Empirical Rule A Wiley Brand – Form dummies - Employing the Empirical Rule in Statistical Problems – Statistics Problems

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