# 12 Empirical Rule Statistics Statistics, is the study of the collection of data, presentation of data, analysis and interpretation of the data. This is in the form of mathematics or we can say it’s a form of mathematical studies. This is also called as the methodological studies. Today, we will see about the Empirical Rule Statistics in statistical studies.

The Empirical Rule Statistics is used to calculate the normal distributions of the means. This can be calculated in three types, viz., 68-95-99.7, this form is also known as the “Three Sigma Rule” or “Three Sigma Statistics”.

The Empirical Rule can be given or stated as a formula in the form of standard deviations. These are the three standard deviations as follows :

• For 68% of data, the first standard deviation is used and the formula is µ ± σ
• For 95% of the data, the two standard deviation is used and the formula is µ ± 2σ
• Similarly for 99.7% of the data, the three standard deviation is used and the formula is µ ± 3σ

In the study of the statistics the Empirical Rule Statistics is used for the prediction calculations of the collected data or the surveyed data. This is the data that cannot be calculated by the means of simple normal mathematical calculations. This rule can give us how the data might look after the predictions.

Suppose there is variable X with the randomly collected data sets, which is the data set of the normal distribution, here we can use the Empirical Rule Statistics and can calculate the standard deviations and differences range. This rule is applicable only for the normal distribution and not for other distributions.

For Example :

Let us find the Empiricla Rule Statistics for the given data set {10,15,20,25,30,35,40,45}

Solution :

Step 1 : We need to calculate the mean first

∴ Mean (µ) = (10+15+20+25+30+35+40+45)/8

= 220/8

= 27.5

Step 2 : Now we can find the standard deviation, by using the formula

SD (σ) = 1÷(n-1)*((x1-µ)2+x2-µ2+…+xn-µ2)

∴σ=√(1÷8-1*(10-27.52+15-27.52+20-27.52+25-27.52+                        30-27.52+35-27.52+40-27.52+45-27.52)

∴σ = ((17)*(-17.52+-12.52+-7.52+-2.52+2.52+7.52+12.52+               17.52)

∴σ = √(17*(306.25+156.25+56.25+6.25+6.25+56.25+156.25+              306.25)

∴σ = √(17*1050

∴σ = 150

∴σ = 12.24

Step 3 :

In this step we can apply the Empirical Rule on the given set of data

1. For 68% of data

µ ± σ

µ – σ = 27.5 – 12.24 = 15.26

&

µ + σ = 27.5 + 12.24 = 39.74

Hence, for the 1st standard deviation the data can be in the range of 15.26 to 39.74

1. For 95% of data

µ ± 2σ

µ – 2σ = 27.5 – (2*12.24) = 27.5 – 24.48 = 3.02

&

µ + 2σ = 27.5 + (2*12.24) = 27.5 + 24.48 = 51.98

Hence, for the 2nd standard deviation the data can be in the range of 3.02 to 51.98

1. For 99.7% of data

µ ± 3σ

µ – 3σ = 274.5 – (3*12.24) = 27.5 – 36.72 = 9.22

&

µ + 3σ = 27.5 + (3*12.24) = 27.5 + 36.72 = 64.22 Hence, for the 3rd standard deviation the data can be in the range of 9.22 to 64.22 