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Empirical Rule Formula

The empirical rule formula (or a 68 95 99 rule formula) uses normal distribution data to find the first standard deviation, second standard deviation and the third standard deviation deviate from the mean value by 68%, 95%, and 99% respectively. It also indicates that all of the data (99%) fall under the range of third standard deviation (either above or below the mean value). The empirical rule formula is explained below along with the solved examples.

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What is the Empirical Rule Formula?

The empirical rule formula is used to calculate the first, second, and third standard deviation and it also predicts the percentage chances of the data falls under that deviation. The empirical rule formula of a given sequence can be expressed as,

first standard deviation = µ - σ to µ + σ  (68% data)
second standard deviation = µ - 2σ to µ + 2σ  (95% data)
third standard deviation = µ - 3σ to µ + 3σ  (99% data)

Where,

  • µ = mean
  • σ = standard deviation

 

Solved Examples Using Empirical Rule Formula.

 

  1. Example 1: People arriving at a store are normally distributed with a mean and the standard deviation as 25 and 3 respectively. Find the percentage chances of people arriving are 19 to 31 at the store.

    Solution: 

    To find: percentage chance of 19 to 31 people at the store

    Given:

    µ = 25
    σ = 3
    Now, using the Empirical rule formula,

    first standard deviation = µ - σ to µ + σ 

    = (25-3) to (25+3)

    = 22 to 28

    second standard deviation = µ - 2σ to µ + 2σ

    = (25 - 2 × 3) to (25 + 2 × 3)

    = (25 - 6) to (25 + 6)
    = 19 to 31
    chances of 20 people arriving at the store are 95%

    Answer: The percentage chance of 19 to 31 people at the store is 95%.

  2. Example 2: For a town, the IQ level of the people is normally distributed with mean and standard deviation as 120 and 5 respectively. Find the range in which 99% of people’s IQ level falls.

    Solution 

    To find: Range in which 99% of people’s IQ level falls

    Given:

    µ = 120
    σ = 5

    Using the Empirical rule formula,

    third standard deviation = µ - 2σ to µ + 2σ

    = (120 - 3*5) to (120 + 3*5)

    = (120 -15) to (120 +15)

    = 105 to 135

    Answer: The range in which 99% of people’s IQ level falls is 105 to 135

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