Discrete variable

A discrete variable is a quantitative variable whose possible values can be listed one at a time, with nothing possible in between two neighboring entries on that list. The list may be finite, like the number correct on a 20-question quiz, or endless, like 0, 1, 2, 3 and onward, which is why countable is the right word rather than few. Counting is the usual source: how many, how often, how many out of $n$.

Eight households report their size: 1, 2, 2, 3, 3, 4, 4 and 6 people. The values sit on separate integers with nothing possible between them, and the mean is $25/8 = 3.125$ people. Note that a summary of a discrete variable need not be one of its possible values, since no household holds 3.125 people.

The wrong sentence is "whole numbers are discrete and decimals are continuous," and it fails in both directions. A quiz marked in half points out of 20 has 41 possible scores, 0, 0.5, 1 and so on up to 20: decimals throughout, all listable, nothing possible at 3.2, so discrete. An age recorded in whole years is written as an integer every time, and the quantity behind it is elapsed time, which is continuous. Neither the shape of the number nor its precision decides this.

What decides it is the set of values the quantity could take, not the set your instrument reports. Every measurement is rounded somewhere, so every recorded column is a finite list of numbers, and reading that as evidence of discreteness would make everything discrete. Money is the case people argue about: a price is a whole number of cents, so strictly its possible values are countable, and it is still usually modeled as continuous because a cent is tiny next to the amounts involved.

Topic 1.2 asks you to say whether a quantitative variable is discrete or continuous. The binomial distribution, topic 2.10, is a discrete model, since it counts successes in $n$ trials.