5 Empirical Rule Formula

The Empirical Rule Formula is used in the study of statistics for the calculation of the normal distributions of the mean. The empirical rule can be written in the three parts, which is as follows:

  • 68% of the data is in the form of the first standard deviation of mean and this data can be stated as or calculated as µ ± σ
  • 95% of the data is in the form of two standard deviations, this data is stated as µ ± 2σ
  • 99.7% of the data is in the form of three standard deviations, this data is stated as µ ± 3σ

This rule is also called the “Three Sigma Rule” or “68-95-99.7”.

The empirical rule is used in statistics to calculate the predictions of the data. This rule is used to get the difficult data to get by normal calculations. This empirical rule gives you the normal estimation that how the surveyed collected data might look.

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This rule can be applied to any random variable, suppose the variable is X, which is the normal distribution or we can say the Bell-Curve. Then with the mean, “µ” and the standard deviation “σ”. This rule is not directly applicable to the distribution other than the normal distribution. To use this rule for other than normal distribution we can use the “Chebyshev’s Theorem”

Let’s see an example for the Empirical Rule Formula:

  • Find the empirical rule for {12,24,36,48,60,72,84}

Solution : 

1st Step : Calculate the mean

Mean (µ) = (12+24+36+48+60+72+80)/7

          = 336/7

          = 48

2nd Step : Find the standard deviation.

Standard deviation can be calculated as

SD (σ) = 1÷(n-1)*((x1µ)2+x2-µ2+…+xnµ2)

      =√(1÷7-1*(12-482+24-482+36-482+48-482+                        60-482+72-482+84-482)

      = ((16)*(-362+-242+-122+02+122+242+362))

      = √(16*1296+576+144+0+144+576+1296)

            = √(16*4032)

            = 672

SD (σ) = 5.091

3rd Step : Applying empirical rule

  1. For 68% of data

µ ± σ 

µ – σ = 48 – 5.091 = 42.909


µ + σ = 48 + 5.091 = 53.091 

Hence, for the 1st standard deviation the data can be in the range of 42.909 to 53.091

  1. For 95% of data

µ ± 2σ 

µ – 2σ = 48 – (2*5.091) = 48 – 10.182 = 37.818


µ + 2σ = 48 + (2*5.091) = 48 + 10.182 = 58.182 

Hence, for the 2nd standard deviation the data can be in the range of 37.818 to 58.182

  1. For 99.7% of data

µ ± 3σ 

µ – 3σ = 48 – (3*5.091) = 48 – 15.273 = 32.727


µ + 2σ = 48 + (3*5.091) = 48 + 15.273 = 63.273 

Hence, for the 3rd standard deviation, the data can be in the range of 32.727 to 63.273 

  By the solution of the above example, in this was the empirical rule formula can be applied in statistics to study the surveyed data. This can be also represented graphically on the graph of the X and Y axis to understand this rule easily. 

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