Modified Boxplot vs 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.

A boxplot draws exactly five numbers: the minimum, $Q_1$ (Q one, the first quartile), the median, $Q_3$, and the maximum. The box runs from $Q_1$ to $Q_3$, so its length is the interquartile range. In the modified version, the one normally drawn, any value beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$ is plotted as its own point and the whiskers stop at the most extreme values inside those fences. Quartiles here follow the median-excluded (TI-84) rule.

Two sets of eight values show what those five numbers cost. Set A is 10, 20, 30, 40, 60, 70, 80, 90. Set B is 10, 24, 26, 27, 73, 74, 76, 90. Both give minimum 10, $Q_1 = 25$, median 50, $Q_3 = 75$, maximum 90, an IQR of 50, no outliers, and a mean of 50, so the two boxplots are identical. Set A is spread evenly. Set B is two tight clumps with a hole through the middle, which a dotplot shows at a glance. A boxplot says nothing about where values sit inside a quarter, so it cannot show a second peak.

The other misreading appears whenever two boxplots share an axis: "group A's box sits entirely above group B's, so every student in A scored higher than every student in B." Take group A as 41, 62, 68, 72, 78, 82, 86, 95 and group B as 30, 44, 48, 55, 59, 61, 63, 85. A's box runs 65 to 84 and B's runs 46 to 62, so the two boxes never touch. B's top score of 85 still beats six of A's eight scores. A box covers the middle half of its group, not the group's range.

A boxplot also hides sample size completely: 8 observations and 800 with the same five numbers draw the same picture, so report $n$ beside it. When $n$ is small the five numbers nearly are the data, and a dotplot then shows everything the boxplot would and more. Boxplots are topic 1.8.