Segmented Bar Chart vs Mosaic Plot

A segmented bar chart draws one bar per category of the first variable and cuts each bar into pieces showing how that group splits across the second variable. The segments of a single bar are that group's conditional distribution, so within any one bar they account for the whole group. When each bar is rescaled to a common height of 100 percent, every bar's segments sum to 1, and only then can segment sizes be compared from bar to bar.

Take 200 students classified by year and by how they get to school. Among the 120 freshmen, 48 walk, 30 take the bus and 42 drive, giving segment heights of $48 / 120 = 0.40$, $0.25$ and $0.35$. Among the 80 seniors, 16 walk, 32 take the bus and 32 drive, giving $0.20$, $0.40$ and $0.40$.

Now the misreading, which is about counts rather than about the chart: "42 freshmen drive and only 32 seniors do, so driving is more common among freshmen." The two groups are not the same size. As shares of their own year the numbers are $42 / 120 = 0.35$ and $32 / 80 = 0.40$, so driving is in fact the more common choice among seniors. Dividing each group by its own total before drawing is exactly what the 100 percent version of this chart does, and it is the whole reason to draw one.

Only the bottom segment of each bar rests on a common baseline. Middle segments float, so a small difference three segments up is genuinely hard to see, and that is a real limit of the display rather than a reading error. Bars that split in identical proportions show no association in the sample, which on its own says nothing about the population.

Topic 2.1 of the Fall 2026 course covers two-way tables together with side-by-side and segmented bar charts and mosaic plots.

A mosaic plot is a segmented bar chart drawn to 100 percent with one extra rule: each bar's width is proportional to that group's share of all the observations. So a height inside a bar is a conditional proportion, a width across bars is a marginal proportion, and their product, the area of a tile, is that cell's share of the entire table.

Take 250 patients treated at two clinics, 150 at clinic A and 100 at clinic B, of whom 90 and 70 improved. The bar widths are $150 / 250 = 0.60$ and $100 / 250 = 0.40$. The improved segments have heights $90 / 150 = 0.60$ and $70 / 100 = 0.70$. The tile areas are therefore $0.60 \times 0.60 = 0.36$ and $0.40 \times 0.70 = 0.28$, which are exactly $90 / 250$ and $70 / 250$.

That gap between height and area is where the reading goes wrong: "clinic B's improved tile is taller, so more patients improved at B." Taller means a higher rate, 70 percent against 60 percent, and that much is right. It does not mean more patients. B's improved tile has area 0.28 against A's 0.36, which is 70 improved patients against 90. Height answers what share of this group, area answers what share of everybody, and those two questions have opposite answers here.

The widths are both the reason to use a mosaic plot and its cost. A small group gets a narrow bar, so its segment boundaries are hardest to read precisely when its rate is the interesting one, and with several categories on each variable the tiles shrink until labels no longer fit.

Topic 2.1 of the Fall 2026 course names mosaic plots alongside two-way tables and segmented bar charts for displaying two categorical variables.