Boxplot vs Dotplot

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.

A dotplot puts one dot on a number line for every observation, at that observation's value, and stacks dots when values repeat. The horizontal axis is the variable's scale. The vertical direction is a count, not a second variable. Nothing is grouped and nothing is averaged, so the display loses no information: the whole data set can be read back off it.

Twelve students report how many siblings they have. Four say 0, five say 1, two say 2, and one says 3, so the plot shows stacks of height 4, 5, 2 and 1 above the marks 0, 1, 2 and 3. The mean is $\frac{0(4) + 1(5) + 2(2) + 3(1)}{12} = \frac{12}{12} = 1$ sibling. With $n = 12$ the median falls between the 6th and 7th values, both of which are 1, so the median is 1 as well.

The error that follows is reading the axis instead of the dots: "the values are 0, 1, 2 and 3, so the mean is $(0 + 1 + 2 + 3)/4 = 1.5$." That averages four axis labels. Each dot is one student, so every value has to be counted as many times as it has dots above it. The marks on the axis record which values occurred, not how often.

A dotplot conceals almost nothing, which is also where it breaks. With 400 measurements recorded to two decimals, nearly every value is unique, so you get 400 stacks one dot tall and no readable shape. That is the point at which you bin into a histogram and accept the loss. A dotplot also needs a number line, so it is the wrong display for a categorical variable.

For contrast, the five-number summary of those twelve counts is 0, 0, 1, 1.5, 3, and a boxplot drawn from it cannot show that the tallest stack sits at 1. Dotplots are one of the three displays in topic 1.5, Graphical Representations for One Quantitative Variable.