Empirical Rule Formula

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Definition:-The empirical rule is an important rule of thumb that is used to state the approximate percentage of values that lie within a given number of standard deviations from the mean of a set of data if the data are normally distributed.
The empirical rule is used only for three numbers of standard deviations 1σ,2σ, and 3σ.

Distance from the mean Values within Distance
µ±1σ 68%
µ±2σ 95%
µ±3σ 99.7%

If a set of data is normally distributed, or bell shaped
·         approximately 68% of the data values are within one standard deviation of the mean
·         95% are within two standard deviations, and
·         Almost 100% are within three standard deviation.

Example:-   Support a recent report states that for California, the average statewide price of a gallon of regular gasoline was $1.42. Suppose regular gasoline price varied across the state with a standard deviation of $0.08 and were normally distributed.

According to empirical rule,
Approximately 68% of the prices should fall within
µ±1σ= 1.42 ±1*0.08
         = (1.42- 0.08, 1.42+0.08)
          = (1.34, 1.5)

Similarly 95% of the prices should fall within
µ±2σ= 1.42 ±2*0.08
         = (1.42- 0.16, 1.42+0.16)
          = (1.26, 1.58)

And 99.7% of the prices should fall within
µ±3σ= 1.42 ±3*0.08
         = (1.42- 0.24, 1.42+0.24)
          = (1.18, 1.66)


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