• Use the Empirical Rule to calculate percentages and probabilities.

The Empirical Rule

You may remember that the Empirical Rule helps us identify which values in a large data set are usual or unusual. The range of usual values can be calculated with just the mean and standard deviation, but the distribution of values is often represented using a graph. The Empirical Rule is appropriate when the data set is normally distributed. The normal distribution is a bell-shaped curve that is symmetrical and gradually approaches the [latex]x[/latex]-axis in both directions. The area under the curve represents the percentage of values that fall in that range.

The Empirical Rule states that, in a bell-shaped, unimodal distribution, almost all the observed data values, [latex]x[/latex], lie within three standard deviations, [latex]\sigma[/latex], to either side of the mean, [latex]\mu[/latex]. Specifically,

[latex]68[/latex]% of the observations lie within one standard deviation of the mean [latex]\left(\mu\pm\sigma\right)[/latex]

[latex]95[/latex]% of the observations lie within two standard deviations of the mean [latex]\left(\mu\pm2\sigma\right)[/latex]

[latex]99.7[/latex]% of the observations lie within three standard deviations of the mean [latex]\left(\mu\pm3\sigma\right)[/latex]

For this reason, the Empirical Rule is sometimes called the [latex]68-95-99.7[/latex] rule.

Recall that [latex]z[/latex]-scores represent distance from the mean. [latex]z[/latex]-scores can also be represented on the normal curve.