Side-By-Side Boxplot vs Comparative Dotplot

A side-by-side boxplot puts one boxplot per group on a single common axis. Each box runs from $Q_1$ (the first quartile) to $Q_3$ (the third quartile) with the median marked inside, and each of the four sections, two whiskers and two halves of the box, covers about a quarter of that group's values. This site takes quartiles by the median-excluded (TI-84) rule.

Eleven commute times for group A are 20, 22, 24, 25, 26, 28, 30, 33, 38, 45 and 52 minutes, giving a five-number summary of 20, 24, 28, 38 and 52 and an $\text{IQR}$ of $38 - 24 = 14$. Group B's eleven times are 26, 28, 29, 30, 31, 32, 33, 34, 35, 36 and 40, summarizing to 26, 29, 32, 35 and 40 with an $\text{IQR}$ of 6. Read side by side: B's median sits 4 minutes higher and B is less than half as variable.

The misreading is about length: "A's upper whisker is far longer than its lower one, so most of A's data is at the high end." Length is spread, not count. A's lower whisker runs 20 to 24, four minutes, and the upper runs 38 to 52, fourteen minutes, but each of them covers about a quarter of the eleven values. The long upper whisker says the top quarter is stretched thin, which is the signature of right skew, and no more than that.

Five numbers per group is all a boxplot carries, so it cannot show a cluster, a gap, or a second peak. When shape matters, put a comparative dotplot next to it. Overlapping boxes are also not a test: separation on a graph is not a significance decision.

Topic 1.9 of the Fall 2026 course puts it plainly. Boxplots compare center, variability, outliers, and skew or symmetry, but they do not show clusters or gaps.

A comparative dotplot stacks two or more dotplots above a single shared scale. Every dot is still one observation sitting at its own value, so nothing is summarized away: repeats, gaps and clusters all survive into the picture. The shared scale is the requirement. Two dotplots drawn on different axes cannot be compared by eye no matter how they are arranged.

Two groups of nine make the point. Group A is 11, 12, 12, 13, 14, 15, 16, 16, 17 and group B is 11, 12, 12, 12, 14, 16, 16, 16, 17. Both have mean 14, median 14, and the same five-number summary of 11, 12, 14, 16 and 17 under the median-excluded (TI-84) rule this site uses, so their boxplots are identical. The dotplots are not. A puts a dot at every value from 11 to 17, while B piles three dots on 12, three on 16, and leaves 13 and 15 empty.

The error to name is reading height as center: "group B's tallest stack is at 12, so B is centered near 12." A stack height counts repeats, which makes it the mode. The center is a position in the ordered data, and B's median and mean are both 14, a value carrying a single dot. In a set with two clumps, the tallest stack can sit a long way from every measure of center.

The display fails as $n$ grows, because stacks of forty dots stop being countable and start being bars, at which point a histogram is the honest choice. If the groups differ in size, read each stack as a share of its own group rather than as a raw count.

Topic 1.9 of the Fall 2026 course names dotplots among the displays for comparing distributions, specifically because they show clusters and gaps that boxplots cannot.