Example Card: Identifying and Displaying Outliers

Sometimes one data point is far from the others; let's see how to find it. This example uses the second key idea, identifying outliers.

Example Problem The daily commute times (in minutes) for 10 employees are: $15, 18, 20, 21, 21, 25, 28, 30, 31, 55$. Display the data and identify any outliers.

Step by Step 1. Find the median, $Q 1$, $Q 3$, and the interquartile range (IQR). The data is already in order. The two middle numbers are $21$ and $25$. The median is their mean: $\frac{21+25}{2} = 23$. The first quartile, $Q 1$, is the median of the lower half $(15, 18, 20, 21, 21)$, which is $20$. The third quartile, $Q 3$, is the median of the upper half $(25, 28, 30, 31, 55)$, which is $30$. The interquartile range is $IQR = Q 3 Q 1 = 30 20 = 10$. 2. Use the outlier formulas to find the boundaries. A value $x$ is an outlier if $x < Q 1 1.5(IQR)$ or $x Q 3 + 1.5(IQR)$. Lower boundary: $20 1.5(10) = 5$. Upper boundary: $30 + 1.5(10) = 45$. 3. Check for outliers. The value $55$ is an outlier because it is greater than the upper boundary of $45$. There are no outliers on the low end. 4. When drawing the plot, the upper whisker extends only to the highest value that is not an outlier, which is $31$. The outlier at $55$ is marked separately with an asterisk ( ).