Counts into proportions

A frequency table shows the number of observational units in each category of a categorical variable. One column lists the categories; the other lists the frequency, or count, in each. Built from a raw list of responses, the procedure is short:

1. List every category the variable takes.
2. Tally each unit into exactly one category.
3. Check that the frequencies sum to the number of units. If 250 students each chose one entree, the counts must reassemble into 250.

That last check is not optional. A frequency table is a partition of the data set: every unit lands somewhere, and nowhere twice. A missing or double-counted unit shows up as a total that misses, which is how one smudged cell in a table can be recovered by subtraction.

A relative frequency table shows the proportion of observational units in each category: each count divided by the total number of units. If 70 of 250 students chose a sandwich, the relative frequency is $\frac{70}{250} = 0.28$. Because the categories partition the whole, relative frequencies always sum to 1, just as the counts sum to the total. That sum is the fastest error check in the course.

Percentages, relative frequencies, and ratios all provide the same information as proportions: $0.28$, $28\%$, and 28 out of every 100 are three spellings of one fact. Moving between them is formatting, not analysis - which also means a report can be checked in whichever form is easiest, and an impossible set of percentages convicts the counts behind it.

A count answers how many. A relative frequency answers how common. The two agree only when the groups being compared are the same size, and real groups almost never are.

Suppose 84 of a school's 600 seniors and 45 of its 180 juniors ride the bus. The counts say seniors, 84 to 45. The proportions say juniors: $\frac{84}{600} = 0.14$ against $\frac{45}{180} = 0.25$. Riding the bus is nearly twice as common among juniors; the seniors' bigger count is a fact about there being more seniors.

Neither number is wrong. They answer different questions, and the error is answering how common with the one built for how many. Whenever group sizes differ, divide each count by its own group's total before comparing anything.

Counts and relative frequencies of categorical variables reveal information that can be used to justify claims about the variable in context. The discipline has three parts:

1. Name the variable and the group measured. Whether a bin was sorted correctly, among the 500 bins audited - not recycling in general.
2. Give the share with its count. 325 of 500, or 65%: the count shows the evidence, the share carries the comparison.
3. Stop at the data's edge. The table describes the units counted; a claim about a larger population is an estimate and should be worded as one.

A number with no group attached, or a count paraded where a share is needed, is not yet a claim - it is a fragment waiting to mislead. The table is small; the sentence built from it is where the statistics happens.