Proportion

A proportion is a part of a whole. Write it $\hat{p} = x / n$, where $\hat{p}$ (p-hat) is the sample proportion, $x$ is how many observations fall in the category and $n$ is how many observations there are altogether; the population version is written $p$. Two properties follow straight from that definition and both are checks you can run. The numerator counts a subset of what the denominator counts, so $0 \le \hat{p} \le 1$. And the units cancel, so a proportion is a bare number with nothing attached.

In a neighborhood of 9,000 residents, 58 reported a bicycle stolen last year. The proportion is $58 / 9000 = 0.006444$, which is 0.64 percent of residents.

"58 out of 9,000 is a theft rate of 6.44." The 6.44 is a real number and it is not the proportion. It is that proportion multiplied by 1,000, so the whole statement is 6.44 thefts per 1,000 residents per year, and dropping the base leaves a figure a thousand times its own meaning. The bound is the fastest check there is: a proportion of 6.44 would mean 644 percent of residents, so any part-to-whole answer above 1 is an arithmetic or labeling error rather than a finding. Convert on purpose. $0.006444 \times 1000 = 6.444$, and 0.006444, 0.6444 percent and 6.44 per 1,000 are three names for one quantity.

The part has to sit inside the whole. Change the wording from "58 residents reported a theft" to "58 thefts were reported" and $58 / 9000$ stops being a proportion, because one resident can be robbed twice and the numerator is now counting events instead of people. That version is a rate, and unlike a proportion it has no ceiling of 1.

Unit 3, Inference for Categorical Data: Proportions, is built on $\hat{p}$ as the estimator of $p$, which are the sample and population versions of the same summary. See parameter vs statistic for that distinction on its own.