Factor (in an experiment)

In experimental design a factor is an explanatory variable whose values the researcher sets rather than records. It has nothing to do with factors in arithmetic. The values chosen are the factor's levels, and the conditions actually applied to units are the treatments. With a single factor the levels are the treatments. With more than one, each treatment is one combination of levels, so the treatment count multiplies rather than adds.

A coffee study varies grind size at 2 levels, coarse and fine, and brew time at 4 levels: 2, 3, 4, and 5 minutes. That is 2 factors and $2 \times 4 = 8$ treatments, running from coarse at 2 minutes through to fine at 5 minutes. Add a third factor with 3 levels, water temperature, and the design carries $2 \times 4 \times 3 = 24$ treatments, each of which still needs several experimental units. That multiplication is why factors are expensive to add.

"The treatments are grind size and brew time" is the standard wrong answer. Those are the factors. A treatment is a complete specification of what one unit receives, so "fine grind, 4 minutes" is a treatment and "brew time" is not. The related slip is reporting the number of levels as the number of treatments: this design lists 6 levels across its two factors and applies 8 treatments to units.

Two things standing nearby are not factors. In an observational study the same variable is called an explanatory variable, because its value is recorded rather than imposed, and there are no treatments at all. A blocking variable is not a factor either: units already belong to their block by age or sex or location, nothing about it is assigned, and its categories are not treatments.

Nor is the response variable. Factors are inputs you set before the study runs; the response is the outcome measured on each unit afterward. Naming the factor, its levels, the treatments, and the response in that order is most of what a design description on the exam is asking for.