Z-Score and the Empirical Rule: Learn It 5

  • Calculate z-scores to explain the location of data points.
  • Compare observations using z-scores and the Empirical Rule.

The Empirical Rule

If a distribution of a variable [latex]X[/latex] is bell-shaped, unimodal, and symmetric, then we can estimate how many observations are within a certain number of standard deviations.

The Empirical Rule (also known as the [latex]68-95-99.7[/latex] rule) is a guideline that predicts the percentage of observations within a certain number of standard deviations.

empirical rule

The Empirical Rule states that:

  • About [latex]68\%[/latex] of observations in a data set will be within one standard deviation of the mean.
  • About [latex]95\%[/latex] of the observations in a data set will be within two standard deviations of the mean.
  • About [latex]99.7\%[/latex] of the observations in a data set will be within three standard deviations of the mean.

Graphically, the Empirical Rule can be expressed like this:

An image displaying the distribution set by the Empirical Rule.
Figure 1. A normal distribution curve showing the empirical rule: about 68% of data fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 standard deviations.

Now, try applying the Empirical Rule to the Movie Runtime data set.