Count Data vs Proportion

A count is a whole number, zero or more, saying how many. Counts are quantitative and discrete: you can add them, average them, and 3 really is one more than 2. What makes them feel categorical is where they come from, since most counts are produced by sorting units into categories and tallying.

Two different objects get called counts, and the observational unit tells them apart. Eight households report 0, 0, 1, 1, 1, 2, 2 and 3 pets. The unit is a household, the count is a variable measured on each one, and $\bar{x} = 10/8 = 1.25$ pets is a real average of eight measurements. Now tally 100 students by favorite sport and get 40, 25 and 35. The unit is still a student, but those three numbers summarize one categorical variable rather than measuring three things: they add to $n = 100$, and their average of about 33.3 is the mean category size, which describes no student at all.

The wrong sentence is "the counts 40, 25 and 35 are categorical data." Favorite sport is the categorical variable. The tallies are frequencies, and frequencies are numbers. The type belongs to the thing being counted, not to the tally, which is exactly why a chi-square procedure runs on counts while the variable it is testing is categorical.

A count depends on how many units you counted, so raw counts mislead across groups of different sizes. 45 walkers in a 300-student school is a larger share than 60 walkers in a 600-student school, since $45/300 = 0.15$ against $60/600 = 0.10$, and converting to a proportion or a rate is the fix. The exception runs the other way: chi-square needs the counts themselves, because the expected counts it compares against are counts, so feeding it percentages produces a statistic that answers nothing.

Tables and summary statistics for one categorical variable are topic 1.3. The chi-square tests for homogeneity and independence, topics 3.14 and 3.15, are where counts carry the inference.

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.