What Is a Ratio? Comparing Two Quantities Without Units Getting in the Way
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
A ratio compares two quantities by division rather than by subtraction, which makes it independent of scale. That property is why ratios appear everywhere from recipes to finance to medicine, and why misreading one is among the commonest quantitative errors in public life.
Division rather than difference
Two quantities can be compared by asking how much larger one is, which is subtraction, or how many times larger, which is division. The second is a ratio and it has the useful property of being unchanged when both quantities are scaled, so a recipe using two parts flour to one part water works whether the parts are grams or kilograms. That scale independence is what makes ratios portable between contexts. A rate is a ratio between quantities in different units, such as distance over time, and it carries units accordingly. A proportion states that two ratios are equal, which is what allows a known relationship to be scaled to a new case and is the basis of most practical arithmetic outside a classroom. A percentage is a ratio expressed with a denominator of a hundred, and a fraction, a decimal, a percentage and a ratio are four notations for the same relationship, which is a fact that many people learn as four separate topics.
Where they mislead
The characteristic errors are consistent and appear constantly in reporting:
- •Relative risk stated without the absolute risk, so a treatment doubling a risk sounds alarming when the risk moves from one in a million to two
- •Percentage changes of percentages, where a rise from two percent to three percent is a fifty percent increase and a one point increase, and the choice of which to report is a rhetorical decision
- •Ratios with unstated denominators, where a count is compared without accounting for how many were exposed
- •Averaging ratios directly, which is generally wrong, since combining rates requires weighting by the underlying quantities
- •Simpson's paradox, where a ratio that favours one group in every subgroup reverses when the subgroups are combined, which is a genuine mathematical result and not an error
- •Base rate neglect, where a ratio is applied without regard to how common the underlying condition is, which is what makes screening test results so widely misunderstood
The screening example
The most consequential ratio confusion concerns medical tests and is worth working through. A test that correctly identifies ninety-nine percent of people with a condition and correctly clears ninety-nine percent of those without it sounds excellent. Applied to a population where one person in ten thousand has the condition, testing a million people finds about ninety-nine of the hundred who have it and also produces about ten thousand false positives among the nine hundred and ninety-nine thousand nine hundred who do not. A positive result therefore means roughly a one percent chance of actually having the condition. That result is counterintuitive and is misunderstood by patients and, in documented studies, by a substantial proportion of doctors asked the same question. The error is neglecting the base rate, and the standard remedy is to express the problem in natural frequencies rather than in percentages, which improves performance measurably in experiments and is now recommended in medical communication.
The ratios people rely on
Specific ratios do heavy lifting in several fields and each embeds assumptions worth noticing. Financial ratios including debt to equity and price to earnings compress a company into a comparable number and are only meaningful against a relevant benchmark, since the appropriate value differs entirely by industry. Body mass index is a ratio of weight to height squared, was devised for population statistics rather than individual assessment, and misclassifies muscular and elderly people systematically. Aspect ratios govern screens, images and paper. Gear ratios determine mechanical advantage. Staffing ratios are used in regulation of childcare and healthcare and are blunt instruments that ignore case complexity. In every case the ratio is a summary that discards information deliberately, which is what makes it usable and also what makes it misleading when the discarded information was the relevant part.
The takeaway
Comparing by division rather than subtraction makes the comparison independent of scale, which is why a recipe in parts works at any size. Relative figures reported without absolute ones mislead systematically, and a fifty percent increase may be one percentage point. A highly accurate test for a rare condition still produces mostly false positives, which is base rate neglect and which natural frequencies fix.