Empirical Rule

Empirical ( or 68-95-99) Rule for Data with a Bell-Shaped Distribution

If the data has a distribution that is approximately bell-shaped then

- about 68% of all the data is within 1 standard deviation of the mean.

- about 95% of all the data is within 2 standard deviations of the mean.

- about 99.7% of all the data is within 3 standard deviations of the mean.

Note that a value v is within 1 standard deviation of the mean if

We say that v is within 2 standard deviations of the mean if

Example

Adult IQ scores have a bell-shaped distribution with a mean of 100 and standard deviation of 15. Use the empirical rule to estimate the percentage of adults with IQ scores between 70 and 130.

Solution .

Note that 130 = 100 + 2s and 70 = 100 - 2s where s is the standard deviation of 15. Since 95% of the data is within 2 standard deviations of the mean, 95% of adults have IQs between 70 and 130.