Read categorical displays with their denominators

Try

A synthetic travel survey asks each of 36 respondents to choose exactly one usual mode: walking, bus or bicycle. The counts are 18, 12 and 6. Before drawing, identify the observational unit (one respondent), the variable (usual travel mode), and the denominator (36 respondents). The labels are categories, not positions along a numerical measurement scale.

Read the supplied count bars. Which mode is twice as common as bicycle? Which mode accounts for half the sample? A bar’s height answers a count question only because its vertical scale is labeled as respondents and begins at zero.

Predict

A pie chart displays shares of the same whole. Predict which sector should span 180 degrees. Then calculate the bus share and its angle before examining the next display: 12/36=1/3, and one third of a full 360-degree turn is 120 degrees. Bicycle occupies the remaining 60 degrees.

Experiment

Now investigate a different synthetic dataset with two variables: group and a yes/no response. Group A contains 80 people, of whom 60 answer yes; group B contains 20 people, of whom 5 answer yes. Complete the row percentages before comparing the displays.

Group	Yes	No	Group total	Yes within group
A	60	20	80	60/80=75%
B	5	15	20	5/20=25%
Whole sample	65	35	100	65/100=65%

An equal-width segmented bar gives each group the same visual width and divides its height using within-group proportions. A mosaic instead assigns widths proportional to group sizes, while retaining those conditional heights. Predict which cell dimensions change when switching from one representation to the other.

Understand

The denominator determines the question. Among A respondents, 75% answer yes; among B respondents, 25% do. Among everyone, 60% are both A and yes and 5% are both B and yes. These statements are all compatible. Calling B’s yes rate “5%” without saying “of the whole sample” changes the intended denominator.

In the mosaic, A’s yes cell occupies width 0.8 times height 0.75, or area 0.60 of the full rectangle. B’s yes cell occupies 0.2 times 0.25, or 0.05. Thus areas recover joint frequencies while heights compare conditional distributions. The yes-height ratio is three, but the yes-count ratio is twelve because A is four times as large. Equal-width segmented bars deliberately suppress that width difference to emphasize response distributions; their areas should not be read as total-sample shares.

Different conditional response distributions are evidence of association in this sample. Neither graphic establishes a causal effect of group membership or a representative population estimate. Those claims require information about assignment, sampling and possible confounders, not a different color palette.

A one-variable pie requires categories that partition the stated whole. If people can select several modes, respondent-based category counts may add beyond the number of people. A pie of all selections can still be meaningful if its denominator is clearly labeled as selections, but it no longer answers what fraction of people chose each category. Bar charts can display overlapping counts if that fact is made explicit.

Apply

Construct the fresh preference charts in the practice bank. Label the count-axis baseline, sample total and pie-sector angles. Then construct a segmented-bar and mosaic specification for the mixed case, where the smaller group has the higher conditional satisfaction rate. Preserve the distinction between “more satisfied people” and “a greater proportion satisfied.”

Use a sketch or complete dimensional description: group widths, segment heights, category labels and denominators must be recoverable. A statement such as “the blue region is bigger” supplies neither a reproducible construction nor a statistical comparison.

Continue

Categorical displays aggregate counts by labels. The next lesson uses a keyed stem-and-leaf display to preserve individual numerical observations, then asks how ties affect percentile statements. Keep the variable type and the denominator visible in both kinds of analysis.