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← A Number's True Size

A Number's True Size

Four ways numbers in the news get bigger or smaller than they are: per cent against percentage points, relative against absolute risk, big totals per person, and a test result counted out in beans.

Spark & Whetstone · sparkwhetstone.org/learn/a-numbers-true-size · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Per cent and percentage points

When a rate goes from 4% to 6%, two different things are true. It rose by 2 percentage points: 6 minus 4. And it rose by 50%: the 2 points are half of the 4 it started at.

Both are honest, but they sound very different. A headline that wants a change to sound big tends to use the per cent; one that wants it to sound small uses the points.

When you see a change in a rate, ask for both, and ask what it started at.

Question. An interest rate goes from 4% to 6%. Which statement is correct?

  1. It rose 2 percentage points, which is a 50% increase.
  2. It rose 2%, and that is all you can say.
  3. It rose 50 percentage points.

Say it back. In your own words: what is the difference between "two per cent" and "two percentage points"?

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Lesson 2. Relative and absolute risk

Suppose something happens to 2 people in every 1,000. If a change makes it 4 in 1,000, the risk has doubled: a 100% rise. That is the relative risk.

The absolute change is 2 more people in every 1,000. Out of a thousand people, 996 are untouched either way.

"Doubles the risk" can be true and still describe a small change. The question that helps is: out of how many, and how many more?

Question. A headline says a habit "doubles the risk" of an illness that affects 2 people in 1,000. What does that mean in counts?

  1. About 4 in 1,000 instead of 2 in 1,000.
  2. Half of everyone who has the habit.
  3. About 200 in 1,000.

Say it back. Say it back: what two questions turn "doubles the risk" into something you can picture?

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Lesson 3. Big numbers per person

A billion is hard to picture. Dividing a big total by the number of people it is spread across often makes it ordinary.

Three billion dollars across 330 million people is about $9 each. The same total can be a fortune for one town and loose change for a country.

Before you react to a big number, ask: per what? Per person, per year, per household?

Question. A programme costs $3 billion, spread across 330 million people. About how much is that per person?

  1. About $9
  2. About $90
  3. About $900

Say it back. In your own words: why does it help to ask "per what?" about a big number?

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Lesson 4. A test in beans

These numbers are made up to show the idea; they are not any real test. Picture 1,000 people. 10 of them have a condition. A test finds 9 of those 10, and also wrongly flags about 5 in every 100 of the people who do not have it.

Counting it out: 9 true alarms, and about 50 false ones from the 990 who are well (5% of 990 is about 50). So about 59 people test positive, and only 9 of them have the condition.

In percentages this feels like a puzzle. In counts it is a bowl of beans. People get this kind of question right far more often when it is shown as counts of people, not as percentages.

Question. In the made-up example, of the people who test positive, about how many have the condition?

  1. About 1 in 6
  2. About 9 in 10
  3. About 1 in 2

Say it back. Say it back: why does a rare condition mean most positive results can be false ones?

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At the table: A thousand beans

Players: 2 to 6, a teenager welcome. You'll need: a bag of dried beans (or rice), a few coloured ones or a marker, two bowls.

  1. Count out 100 beans as a town of 100 (or weigh out a thousand if you are patient). Mark the few who "have it".
  2. Pick a headline from today's news with a percentage in it and act it out in beans.
  3. Each person says the headline back as a count: "this many out of this many". Vote on which way of saying it is fairest.

Answers

  1. It rose 2 percentage points, which is a 50% increase. — 6 − 4 is 2 percentage points. Two is half of four, so the rate is 50% higher than it was.
  2. About 4 in 1,000 instead of 2 in 1,000. — Doubling 2 in 1,000 gives 4 in 1,000. The relative change is large; the absolute change is 2 people in a thousand.
  3. About $9 — 3,000 million ÷ 330 million is a little over 9. The total is big; the share is small.
  4. About 1 in 6 — About 9 true positives out of about 59 positives in all: roughly one in six.