Quantitative variable

A quantitative variable, also called a numerical variable, records a number that measures an amount: how much or how many, usually with units attached. Arithmetic on the values means something because the numbers are the quantity rather than a name for it, so a total, a mean $\bar{x}$ (x-bar) and a standard deviation are all defined. Quantitative variables split again into discrete ones, whose possible values can be listed, and continuous ones, whose possible values fill an interval.

Five commutes take 12, 18, 9, 25 and 16 minutes. They total 80, so $\bar{x} = 80/5 = 16$ minutes. The units survive the arithmetic: the answer is 16 minutes, not 16, and switching to hours divides every value and the mean by 60. Being able to say that sentence at all is the practical signature of a quantitative variable.

The wrong sentence is "I grouped the ages into 0 to 9, 10 to 19 and 20 to 29, so age is categorical now." Grouping changes the display, not the variable. The bins still sit in order on a number line at a fixed width, so the correct picture is a histogram with touching bars, and a bar graph with gaps would claim the groups were unordered labels. You can always bin a quantitative variable; you can never take the order and the spacing away from it.

The edge case is numbers that rank rather than measure. Finishing 1st, 2nd and 3rd puts runners in order, but the gap between 1st and 2nd need not match the gap between 2nd and 3rd, so an average finishing position summarizes the ranks and not the times behind them. The reverse edge is worth knowing too: a categorical variable produces quantitative summaries, since the tally in each category is itself a discrete quantitative variable, which is what count data means.

Graphs, descriptions and summary statistics for one quantitative variable are topics 1.5, 1.6 and 1.7, and Unit 4, Inference for Quantitative Data: Means, is the inference built on them.