Measures of spread and box plots

Same center, different spread

Two classes both average \(75\) on a test. In one, every score is between \(70\) and \(80\); in the other they range from \(40\) to \(100\). Measures of spread capture that difference.

Range and quartiles

The range is the maximum minus the minimum. It's simple but depends only on the two most extreme values.

Quartiles split ordered data into four equal parts:

• \(Q_1\), the first quartile, is the median of the lower half;
• \(Q_2\) is the median;
• \(Q_3\), the third quartile, is the median of the upper half.

Box plots and outliers

A box plot draws a box from \(Q_1\) to \(Q_3\) with a line at the median, and "whiskers" out to the smallest and largest values that aren't outliers.

In the example, the fences are \(7.5 - 18 = -10.5\) and \(19.5 + 18 = 37.5\): there are no outliers.

Variance and standard deviation

The standard deviation measures the typical distance of the data from the mean.

Practice