Rates and Counts

A count answers how many. A rate answers how many for each — for each person, or each hundred, or each hundred thousand. Counts grow when a place grows, even if nothing gets worse. So comparing two places by count alone is unfair to the bigger one. Divide by the population first; then the comparison is between equals.

A rate is a count divided by the population it came from, scaled to a round base: per 1,000 or per 100,000. City B: 200 burglaries ÷ 40,000 residents = 0.005, times 100,000 gives 500 per 100,000. City A: 800 ÷ 400,000 × 100,000 = 200. The rate is the number a resident actually experiences — the chance the count reaches them. A story comparing cities, eras, or groups without one is comparing population sizes and calling the result news.

Two habits complete the toolkit. First, watch the denominator over time: a city that doubles its population should roughly double its counts even at constant risk, so “more crashes than a decade ago” may be arithmetic, not deterioration. Second, distrust percent change on a small base: from one case to two is a 100 percent increase and also exactly one case. Print both figures — the percent and the underlying counts — and neither can fool the reader.

Choose the denominator honestly, too. Injuries per resident, per rider, or per mile ridden tell three different stories about the same bike lane; the fair one divides by exposure — the amount of the activity — not merely by who lives nearby.

Big schools lose more mittens than small schools. More kids means more mittens. It does not mean big-school kids are more careless.

To compare fairly, ask: how many mittens lost for every hundred kids? Ten from a hundred kids is worse than twenty from a thousand.

Formally, a rate estimates risk: events divided by exposure. The choice of denominator is a modeling decision, and much statistical misdirection in news copy is a quietly wrong denominator — airport crime divided by residents instead of passengers. Small counts add a second trap: noise. A department that sees two homicides a year will regularly post 100 percent swings by chance alone; the honest report shows the multi-year series, not the latest jump.

Comparisons across groups invite composition effects: an overall rate can move because the mix of subgroups changed, not because anyone's risk did — the aggregation error known as Simpson's paradox. When a rate shifts, ask what happened to the denominator's composition before crediting the numerator.

Treat reported counts as draws from an approximately Poisson process: the standard deviation of a count near n scales as √n, so relative noise runs as 1/√n — enormous when n is small. A jurisdiction with four annual cases carries roughly 50 percent relative sampling noise before any real change; year-over-year percent changes there are close to meaningless without pooling across years.

Serious comparisons also standardize: age-adjustment against a reference population removes the confound that older age structures inflate crude mortality rates. And rates inherit the record-generating process — a rise can be surveillance (more testing, a new hotline) rather than incidence. The reporting question from folio 2 applies to every denominator: who counted, and how would they know?