Whenever you are shown a count ("4,000 complaints", "12 deaths", "9 out of 10 dentists"), ask "out of how many?", because a numerator on its own cannot tell you whether something is common, rare, rising, or falling.

How to do it

1. 1Notice the tell: a raw count, a list of cases, or a vivid example offered as proof of a rate. Counts are numerators; rates need a denominator.
2. 2Ask "out of how many?" and "over what period?" Four thousand complaints about a product sold forty million times is one in ten thousand.
3. 3When two groups are compared, ask for the denominator of each. More incidents in a bigger city, a bigger age group, or a more widely used product is what you would expect from size alone; compare per capita or per use.
4. 4Ask who was counted and who was left out: "9 out of 10" of those surveyed, of those who replied, or of those the company chose to ask?
5. 5For success stories, ask how many tried. Ten testimonials from a program with ten thousand participants say nothing about your odds.
6. 6If no denominator can be supplied, treat the number as an anecdote with a digit attached.

When to use it

• •News stories built on a total ("crimes committed by", "adverse events reported after") with no population or exposure figure.
• •Marketing claims using survey fractions, testimonials, or "thousands of satisfied customers".
• •Income or success claims from a course, franchise, or multi-level marketing pitch.
• •Comparisons between places or groups of very different size.

Counters

Evidence and how strong it is

Denominator neglect is well established in the laboratory. Denes-Raj & Epstein (1994) found that many participants preferred to draw from a bowl with 7 winning beans in 100 rather than 1 in 10, while knowing the odds were worse, because the count of winners felt larger. Yamagishi (1997) found people rated a cause of death that kills 1,286 out of 10,000 as riskier than one that kills 24.14 out of 100. Reyna & Brainerd (2008) review this literature and show that the error appears even in numerate adults and in medical judgments. Huff (1954) catalogued the same omission in advertising and journalism. Evidence strength: strong (replicated experiments) that people neglect denominators; the defensive habit of asking for one is a logical corrective and a teaching practice, not a separately trialled intervention.

1. Reyna, V. F., & Brainerd, C. J. (2008). Numeracy, Ratio Bias, and Denominator Neglect in Judgments of Risk and Probability. Learning and Individual Differences, 18(1), 89-107
   Review showing that attention to numerators at the expense of denominators distorts risk judgments even in numerate adults.
2. Denes-Raj, V., & Epstein, S. (1994). Conflict Between Intuitive and Rational Processing: When People Behave Against Their Better Judgment. Journal of Personality and Social Psychology, 66(5), 819-829
   The jelly-bean experiments demonstrating preference for larger numerators despite worse odds.
3. Yamagishi, K. (1997). When a 12.86% Mortality Is More Dangerous than 24.14%: Implications for Risk Communication. Applied Cognitive Psychology, 11(6), 495-506
   Evidence that risks stated with larger counts are judged more dangerous than larger risks stated with smaller counts.
4. Huff, D. (1954). How to Lie with Statistics. W. W. Norton
   The classic popular catalogue of counts without bases, selected samples, and unanchored comparisons in advertising and news.