4.9 : Empirical Method to Interpret Standard Deviation

The empirical rule, also known as the three-sigma rule, allows a statistician to interpret the standard deviation in a normally distributed dataset. The rule states that 68% of the data lies within one standard deviation from the mean, 95% lies within two standard deviations from the mean, and 99.7% lies within three standard deviations from the mean. Additionally, this rule is also called the 68-95-99.7 rule.

This rule is used widely in statistics to calculate the proportion of data values given the standard deviation. Consider a normal distribution of students’ test scores in a class. The mean test score is 70, and the standard deviation is 10. Using the empirical rule, we can find out the percentage of students with test scores within the range of 50 and 90.

Using data given in the example, observe that test scores 50 and 90 are two standard deviations away from the mean:

50 = 70 - 2*10

90 = 70 + 2*10

Further, the empirical rule states that 95% of the values in a normally distributed dataset lie within two standard deviations from the mean. So, for the above example, we can say that 95% of the students in the class have test scores within the range of 50-90.

The empirical rule is essential for understanding the upper and lower control limits for statistical quality control. Furthermore, this rule is used by economists to predict stock prices and forex rates.

Though this rule is helpful, it has a significant drawback– it applies only to normally distributed data.

This text is adapted from Openstax, Introductory Statistics, Section 6.1 The Standard Normal Distribution.