Boxplot

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.