Categorical Variable vs Nominal Variable

A categorical variable, also called a qualitative variable, records which group each observational unit belongs to. Its values are names, so the only arithmetic that means anything is counting how many units carry each name. What decides the type is what the values stand for, not what they look like: a variable stored as digits is still categorical when the digits are labels.

Take 250 shoppers, each paying one way: 150 by card, 60 by cash, 40 by phone. That is the whole summary. Three counts adding back to 250, and three proportions, $150/250 = 0.60$, $60/250 = 0.24$ and $40/250 = 0.16$, adding to 1. There is no mean payment method and no standard deviation of one.

The wrong sentence is "the survey stored yes as 1 and no as 2, so the mean answer is 1.4." With 100 people, 60 yes and 40 no, those codes do give $(60(1) + 40(2))/100 = 1.4$. Recode yes as 0 and no as 1 and the same 100 answers give 0.4. The number moved because the labels moved and nobody changed their answer, which is exactly what it means for arithmetic on the values to be meaningless.

Ordered categories are still categorical. Small, medium and large have a real order and no measured gap between neighbors, so you can name the most common size and not an average one. One coding is the honest exception: for a two-category variable coded 0 and 1, the mean of the codes is the proportion of 1s, which is a genuine summary. That is a count in disguise, not an average of the labels.

Naming the type is topic 1.2. Tables and summary statistics for one categorical variable follow at 1.3, and its graphs at 1.4, where the bars stand apart because the axis is a list of labels rather than a number line; see histograms vs bar graphs. Unit 3, Inference for Categorical Data: Proportions, is the inference built for exactly this kind of variable.

A variable is nominal when its categories are names and nothing else, so no arrangement of them is more correct than any other. The test is not what the values look like. It is whether swapping two categories around in a list loses information. Blood type, home state, and brand of phone all pass that test: shuffle the categories and the data still say what they said.

Two hundred blood donors type as O 92, A 76, B 22, AB 10. The summaries you may report are counts, proportions, and the mode: O is the most common type at $92 / 200 = 0.46$, and the four proportions add to exactly 1. There is no median blood type and no mean blood type, because both would need an ordering this variable does not have.

"The values are numbers, so the variable is quantitative." Five players wear jersey numbers 3, 12, 21, 34 and 55. Their mean is $125 / 5 = 25$, a number that identifies nobody and that would change if the team reissued numbers without a single roster change. Zip codes, area codes, student ID numbers and the codes a survey assigns to answer choices are nominal for the same reason: the digits are labels. The general test is in categorical vs quantitative variables.

The reverse trap is just as common, and words do not make a variable nominal. Poor, fair, good and excellent are words carrying a real order, which makes them ordinal and puts a median category back on the table. Nominal is the case where the order genuinely is absent. Because the category order is arbitrary, the bars of a bar graph of a nominal variable can be sorted by height with nothing lost.

The Fall 2026 AP Statistics framework never uses the words nominal or ordinal. It defines one type, the categorical variable, also called qualitative, as one that takes on category names or group labels. Treat the split as useful vocabulary rather than tested content.