Categorical and Quantitative Variables

In Grade 6 you learned that a statistical question expects answers that vary. Statistics starts by collecting those answers. Each person, animal or thing we collect data about is an individual, and each characteristic we record about it is a variable. A class survey might record each student's height, favorite sport and number of siblings: three variables.

Key idea

A categorical variable places each individual into a group or category (eye color, favorite sport, ZIP code). A quantitative variable takes number values that measure or count something, so arithmetic with them makes sense (height in cm, number of siblings).

Why it works

The type decides what you can do next. With quantitative data you can find a mean, a standard deviation and a histogram, because adding and subtracting the values means something. With categorical data you can only count how many fall in each category and compare those counts or percents. Using the wrong tools gives answers that look like numbers but mean nothing, like "the average eye color."

A quick test

Ask: would it make sense to find the average? The average height of a class is a useful fact, so height is quantitative. The average favorite sport is nonsense, so favorite sport is categorical.

Use the "average" test on each variable.
Variable	Type	Reason
Time to run a mile	Quantitative	Measured in minutes; differences make sense
Number of pets	Quantitative	A count
Blood type	Categorical	A label: A, B, AB or O
Shirt size (S, M, L)	Categorical	Ordered labels, but you cannot average them
Jersey number	Categorical	A name written with digits

Worked example: A club survey

A club records each member's age in years, favorite snack, and whether they own a bike. Classify each variable.

Age in years: the average age of the club is meaningful, so age is quantitative.
Favorite snack: each answer is a name, like "popcorn," so it is categorical.
Owns a bike (yes or no): a two-category label, so it is categorical.

Answer: Age is quantitative; favorite snack and bike ownership are categorical.

Worked example: Numbers that are really labels

A store records each customer's ZIP code and the number of items they bought. Classify each.

ZIP codes are written with digits, so they look quantitative.
But ask the average test: the average of 90210 and 10001 is 50105.5, which is not a place and tells you nothing.
A ZIP code only names a region, so it is categorical.
Number of items bought is a count, and the average number of items per customer is useful: quantitative.

Answer: ZIP code is categorical; number of items is quantitative.

Common mistake

Calling a variable quantitative just because it is written with digits. ZIP codes, phone numbers, jersey numbers, room numbers and ID numbers are labels: player 24 is not "twice" player 12. Check by imagining the average. If the average is meaningless, the variable is categorical.

Quantitative variables come in two kinds. Discrete ones are counts with gaps between possible values (0, 1, 2 siblings). Continuous ones are measurements that can take any value in a range (a height of 162.37 cm).

Sources and further reading

OpenStax, Introductory Statistics 2e, section 1.1 Definitions of Statistics, Probability, and Key Terms

OpenStax, Introductory Statistics 2e, section 1.2 Data, Sampling, and Variation in Data and Sampling

Common Core State Standards for Mathematics, S-ID.A.1 (Represent data with plots on the real number line)

Categorical and Quantitative Variables

A quick test

Practice

Builds on

Leads to

Sources and further reading