Modified boxplot

A modified boxplot is the boxplot that shows outliers instead of burying them. The box is untouched: it runs from the first quartile $Q_1$ to the third quartile $Q_3$ with a line at the median. The whiskers are what change. Compute the fences at $Q_1 - 1.5 \times \text{IQR}$ and $Q_3 + 1.5 \times \text{IQR}$, mark every value beyond a fence as its own point, and stop each whisker at the most extreme value that is not an outlier.

Ten values: 12, 19, 22, 25, 27, 29, 31, 34, 36, 58. The quartiles are $Q_1 = 22$ and $Q_3 = 34$, so the IQR is 12 and the fences sit at $22 - 18 = 4$ and $34 + 18 = 52$. Only 58 lies beyond a fence, so it is drawn on its own. The upper whisker stops at 36 and the lower whisker reaches 12, the minimum, since nothing falls below 4.

"The whisker runs out to the fence at 52." It does not, and the gap is 16 units wide. A fence is a number you compute, not a place on the graph, and nothing is ever drawn at one. The whisker ends at 36 because 36 is the largest observation that survived the test. On a plain boxplot the whisker would instead run all the way to 58 and the outlier would vanish into it.

A modified boxplot is precise about position and blind to shape. The box says half the data lie between 22 and 34 and nothing about how they sit in there, so two clusters draw the same picture as a single peak. A histogram or dotplot is the better first look; the boxplot is the better tool for comparing several groups at once.

Topic 1.8 of the Fall 2026 course covers boxplots and states this rule directly: where there are outliers, the whiskers extend to the most extreme values that are not outliers, and the outliers are usually marked with an asterisk or another symbol. Boxplots return in topic 1.9 for comparing center, variability, outliers, and skewness.