Picturing a categorical variable

A categorical variable sorts observational units into named groups: blood type, transportation to school, favorite genre. A bar chart displays its distribution by drawing one bar per category, with the bar's height (or length) equal to that category's frequency (its count) or its relative frequency (its proportion of the whole).

Everything readable from the chart follows from that one rule. If 40 students were surveyed and the bar for "podcast" reaches 12, then 12 of the 40 units sit in that category, and the heights of all the bars together sum to 40. Switch to relative frequency and the same bar reaches $\frac{12}{40} = 30\%$, with all bars summing to 100%.

The axis carrying the categories is a set of labels, not a scale. Nothing sits "between" walk and bike, and no category is numerically larger than another. That has consequences the last section spells out.

Here is the trap this topic exists to close. Suppose 80 of 200 students in class A chose tacos and 30 of 50 students in class B did. The count bar for A towers over the count bar for B, and the wrong conclusion writes itself: A likes tacos more.

But A is four times the size of B. The comparison that respects that is proportion within each group:

1. Find each group's own total: 200 for A, 50 for B.
2. Divide each count by its own group's total: $\frac{80}{200} = 40\%$ and $\frac{30}{50} = 60\%$.
3. Compare the relative frequencies: 60% beats 40%, so the tendency is stronger in B.

The count winner and the share winner are different groups, and the share is the one that answers "which group leans harder toward tacos". A relative frequency bar chart bakes this correction in: each group's bars are drawn as percentages of that group, so groups of any sizes share one fair scale.

A pie chart displays the same distribution a different way: each slice's area, as a fraction of the whole circle, is that category's relative frequency, and the slices together account for 100% of the units. That "together they make one whole" condition is not decoration. It is the requirement.

It fails in two common situations:

1. Overlapping categories. If respondents could pick more than one streaming service, the category percentages sum past 100%, and no circle can hold slices that overlap-count the same person twice. A bar chart, where bars are independent of each other, handles this fine.
2. Comparing two groups. Two pies drawn side by side hide the group sizes completely: a pie for 40 people and a pie for 4000 are the same size circle. Side-by-side relative frequency bars, labeled with each group's $n$, keep both the shares and the scale visible.

When a pie works, it works because the question is "how does one whole split up". When the question is anything else, reach for bars.

Bar charts and histograms are both rectangles standing on an axis, and that is where the resemblance ends. On a bar chart the axis holds category labels: the bars are separated by gaps, their order is a choice (alphabetical, by size, whatever reads best), and rearranging them changes nothing about the data. On a histogram the axis is a number line: each bar covers an interval of values, adjacent bars share edges, and the left-to-right order is fixed by arithmetic.

So the vocabulary that describes a quantitative distribution has no meaning here. A bar chart cannot be "skewed right": slide the tallest bar to the other end by reordering the labels and the "skew" reverses while the data sit unchanged. What a bar chart supports is category talk: which categories are most and least common, and what share of the whole each one holds.

The choice of display starts with the variable's type. Named groups take a bar chart or a pie chart. Measured numbers take the displays of Topic 1.5. A number-looking label, like an area code, is still a label: arithmetic on it means nothing, so it takes the categorical displays.