Week 11 · Chapter

Chapter 11 — Numbers in the News

Critical Thinking Free sample edition

The module's primary reading · ~40–50 minutes · Objective 6 (with Objective 1 carried forward) · Read it first this week; the lecture, the slides, the tutorial, the quiz, and the Argument Workshop all build on it.

In plain language

This week is about the numbers inside everyday claims. A rise from 50 percent to 60 percent can be described as 10 percentage points (the plain difference between two percents) or as a 20 percent increase, and both descriptions are true. A risk can double and still touch only a few people, so you learn to ask for the base (the "out of how many" behind a risk). You also learn why most positive results on a test for a rare condition can be false positives (positive results in people who do not have the condition). Finally, you learn to check a graph's axis before you trust its shape.

Why you already care about this

Three numbers cross your screen in one evening. A campus post says support for overnight library hours "jumped 20 percent". A store sign says sales are "up 100 percent". A health alert says a risk "rose by half". Each one sounds big, and each one is built to make you feel something.

You have probably repeated a number like that yourself. It came from somewhere that looked official, it had a percent sign, and it seemed to settle the matter. The trouble is that a number can be true and still leave you with the wrong picture. "Up 100 percent" can mean 2 sales became 4.

For ten weeks you tested arguments: their form, their words, their samples, and their causes. Last week you asked whether a pattern shows a cause. This week you ask what a number in a claim actually says, and how much it can carry.

By the end of this chapter you will be able to:

find the conclusion a number in a news paragraph is offered to support, or see that the paragraph only reports it;
turn a change between two percents into percentage points and into a percent change, and report both;
state a change in risk both as a relative change and as an absolute change, with the base named;
explain how a real relative reduction can be a small absolute one;
set out a screening problem as counts of people and read how many positives are real;
spot a truncated axis, a stretched scale, and a cherry-picked time window on a graph.

Carry this line with you: a number is a premise — ask what it measures before you weigh it.

Where we are

Already covered: Week 1 taught you to find an argument's premises and conclusion. Weeks 2 to 8 taught you to reconstruct arguments, test their form, and catch fallacies. Week 9 taught you to check a sample, and Week 10 taught you to check a causal claim for a confounding factor.
This week: you check the numbers that claims about samples and causes are stated in: percents, risks, screening results, and graphs.
How they connect: Week 9's surveys return with their change over a year, and Week 10's habit of asking "compared with what?" becomes the habit of asking "out of how many?"
Where we're going next: Week 12 turns from the number to its source, asking who stands behind a claim and how to trace it back to where it started.

Week 1 drew the line you lean on first. An argument offers premises as reasons to accept a conclusion. A report tells what happened and offers nothing as a reason for anything. A number can sit in either one. Week 9 gave you the two surveys you will meet again. The random sample of 400 students is the better guide to the campus, and the doorway sample of 400 is biased, because it asked only people already in the library late at night.

1. Find the claim first: numbers as premises

Picture a friend holding up a phone. "Look, 9 out of 10 people love this blender. You should buy it." Before you ask whether 9 out of 10 is right, you can ask a simpler question. What is the number being used for?

A number in a passage is a claim like any other. Often it is a premise, offered as a reason for a conclusion such as "so you should buy this" or "so the city should act". Sometimes the passage only reports the number and argues for nothing at all. Week 1's test still applies: what is the writer trying to get you to accept, and what is offered as the reason?

Worked example — the campus post (illustrative). A post on a campus page reads: "Support for overnight hours jumped 20 percent in one year, so the library should open around the clock."

Find the indicator. The word "so" signals a conclusion.
Find the conclusion. "The library should open around the clock."
Find the premise. "Support for overnight hours jumped 20 percent in one year."
Name the job. The number is the premise, and the conclusion rests on it.

In standard form, the way Week 2 taught:

P1: Support for overnight hours jumped 20 percent in one year.

So,

C: The library should open around the clock.

In words: the number is the premise and "the library should open around the clock" is the conclusion. Before you weigh the support, you check what "20 percent" means, and the next section does that.

Now compare a report. "This year's random survey found 60 percent of students in favor of overnight hours." Nothing is offered as a reason to accept anything else. It is a report, and you can still check its number, but there is no argument to evaluate yet.

Misconception check — a number in a paragraph proves its conclusion. Students often treat a paragraph with a statistic in it as already argued: "it has data, so it must be right." That is tempting, because numbers look precise and official. But a number supports a conclusion only if it is read correctly and bears on that conclusion. Find the conclusion first, then ask whether the number, read correctly, carries it.

Memory hook: find the "so" first; then weigh the number.

2. Points versus percent: report both

Suppose your phone battery read 50 percent at lunch and 60 percent after you charged it for a few minutes. How much did it go up? You could say 10, because 60 minus 50 is 10. You could also say it went up by a fifth of where it started. Both are true, and they answer different questions.

Those two answers have names.

A change in percentage points is the plain difference between two percents, found by subtracting one from the other.

A percent change is the difference between a new value and a starting value, divided by the starting value and multiplied by 100.

The course rule is simple to state and easy to forget: whenever you give one, give the other.

Worked example — the random sample's change over a year (illustrative). Week 9's random sample drew 400 students from the enrollment list in two years.

Last year. The random sample found 200 of 400 in favor last year, and 200 ÷ 400 × 100 = 50 percent.
This year. This year it found 240 of 400, and 240 ÷ 400 × 100 = 60 percent.
Percentage points. 60 − 50 = 10 percentage points.
Percent change. (60 − 50) ÷ 50 × 100 = 20 percent.

In words: support rose 10 percentage points, which is a 20 percent increase, and a fair report gives both. So the campus post's "jumped 20 percent" is true, but it tells only half the story. A reader who hears "20 percent" may picture support going from 50 percent to 70 percent, which is 20 percentage points and did not happen.

The same arithmetic measures the gap between Week 9's two surveys.

Worked example — the doorway gap (illustrative). The doorway sample found 75 percent in favor, and the random sample found 60 percent.

Percentage points. 75 − 60 = 15 percentage points.
Percent. 15 ÷ 60 × 100 = 25 percent higher.

In words: the doorway sample reads 15 percentage points, or 25 percent, above the random sample. The gap comes from a biased sample, so neither number shows a rise in support. A correct calculation does not rescue a bad sample.

Percent changes can also grow huge when the starting value is tiny.

Worked example — the store sign (illustrative). A sign says sales of a new item are "up 100 percent".

The counts. Sales went from 2 units to 4 units.
The plain change. 4 − 2 = 2 more units.
The percent change. (4 − 2) ÷ 2 × 100 = 100 percent.

In words: "up 100 percent" means 2 more units were sold. A large percent can rest on a tiny base, so ask for the starting count.

Misconception check — a percentage-point change written as a percent. Students often write "support rose 10 percent" when support went from 50 percent to 60 percent. That is tempting, because both numbers are already percents, so subtracting them feels like it gives a percent. But 10 percent of 50 is 5, and support did not go from 50 to 55. The rise is 10 percentage points. The 20 percent figure is the percent change. Say "percentage points" for the plain difference and "percent" for the change divided by where it started.

Memory hook: points for the gap, percent for the growth — always say both.

3. Relative versus absolute risk: name the base

A friend reads a headline aloud: "Doing this doubles your risk!" Your stomach drops. Then you ask the question this section teaches: doubles it from what? Doubling a risk of 1 in 10,000 and doubling a risk of 250 in 1,000 are very different news.

Two kinds of risk statement answer two different questions.

Absolute risk is the chance of an outcome stated as a count out of a named base, such as 2 in 1,000 people; a change in absolute risk is the number of extra or fewer cases per that base.

Relative risk is a comparison of one risk with another stated as a ratio or a percent change, such as a 50 percent increase; without the base it leaves out how many people are affected.

The course writes every risk as "N in 1,000" (or "N in 10,000"), with the base named, and gives every change both ways.

Worked example — a risk that rises (illustrative). Suppose a risk rises from 2 in 1,000 to 3 in 1,000.

The absolute change. The absolute change is 3 − 2 = 1 more case per 1,000 people.
The relative change. The relative change is (3 − 2) ÷ 2 × 100 = 50 percent.

In words: a 50 percent relative increase here means 1 more case in every 1,000 people. A headline that says "risk up 50 percent" is true. But without the base, a reader cannot tell that 997 of every 1,000 people are still unaffected.

Worked example — a risk that falls (illustrative). Now suppose a risk falls from 4 in 1,000 to 3 in 1,000.

The absolute change. The absolute change is 4 − 3 = 1 fewer case per 1,000 people.
The relative change. The relative change is (4 − 3) ÷ 4 × 100 = 25 percent.

In words: a 25 percent relative decrease here means 1 fewer case in every 1,000 people. Notice that both changes are 1 case per 1,000, yet one sounds like 50 percent and the other like 25 percent. The relative number depends on where the risk started.

Here are the two changes side by side.

The illustrative risk pair, per 1,000 people: the risk before and after, the absolute change, and the relative change
Change	Before	After	Absolute change	Relative change
Rising risk	2 in 1,000	3 in 1,000	1 more per 1,000	50 percent increase
Falling risk	4 in 1,000	3 in 1,000	1 fewer per 1,000	25 percent decrease

Misconception check — a 50 percent increase in risk is a large absolute one. Students often hear "50 percent higher risk" and picture half of everyone harmed. That is tempting, because 50 percent is a big number and the word "risk" sounds alarming. But a relative change multiplies whatever the starting risk was. When the starting risk is 2 in 1,000, a 50 percent rise adds 1 case per 1,000 people. Ask for the base before you react.

An honesty note: a small absolute change is not the same as an unimportant one. One more case per 1,000 people, across a large population, can mean many people. The point is not that relative numbers lie. The point is that you need both numbers to know what a change means.

Memory hook: half of what? Name the base — then react.

4. A real case in words: the lung-screening trial

So far every number in this chapter has been illustrative. Here is a real case, taught in words, from a federal health agency. No figure from it appears on any quiz.

The National Cancer Institute explains that in a large lung-screening trial a 20 percent decrease in the relative risk of dying of lung cancer translated to an absolute reduction of about 0.4 percent.

Worked example — reading the case both ways (real, taught in words). The source is the National Cancer Institute's explainer on what screening statistics mean.

The relative number. Screening was linked to a 20 percent decrease in the relative risk of dying of lung cancer. That compares the screened group's risk with the other group's risk.
The absolute number. The same result is an absolute reduction of about 0.4 percent. That is the plain difference in risk itself.
Why they differ. A 20 percent relative cut that equals only about 0.4 percent in absolute terms tells you the starting risk itself was small. A fifth of a small risk touches few people.

In words: the relative number is real and so is the absolute one; they answer different questions. The relative number tells you how much screening changed the risk compared with no screening. The absolute number tells you how many people out of a group that change touched.

The agency's point is the same as this week's rule. A report that gives only the 20 percent sounds larger than the result is. A report that gives only the 0.4 percent can sound like nothing happened. A fair report gives both.

Misconception check — the smaller number is the "real" one. Students who learn about relative and absolute risk sometimes swing too far and decide that relative numbers are tricks. That is tempting, because the relative number is often the one used to impress. But both numbers describe the same result. The relative number is useful for comparing options, and the absolute number is useful for picturing how many people are affected. Neither one is the lie; leaving one out is the problem.

A caveat: this case is taught to show how the two kinds of number relate. It is not advice about any screening decision. Questions about a person's own screening belong with a health professional who knows that person's situation.

Memory hook: both numbers are true; the missing one misleads.

5. Count the people: base rates and false positives

Imagine a phone alert that warns of a security problem. It is right most of the time it fires. Yet most of the alerts you see turn out to be nothing. How can both be true? Real problems are rare. So even a small share of false alarms, spread over many safe phones, can outnumber the real ones.

A screening test works the same way, and three terms describe it.

The base rate is how common a condition or outcome is in the whole group before any test or new evidence, such as 1 in 100 people.

A false positive is a positive test result in a person who does not have the condition the test looks for.

A false negative is a negative test result in a person who does have the condition the test looks for, a case the test misses.

The clearest way to work such a problem uses natural frequencies. That is a way of setting out a probability problem as counts of people in a group of round size, such as 9 of the 108 people who test positive, instead of as percents or probabilities. Gigerenzer and Hoffrage (1995), "How to improve Bayesian reasoning without instruction: Frequency formats", Psychological Review, found that people reason about such problems more easily in counts than in percents.

Worked example — the screening table (illustrative). A test for a condition is given to 1,000 people.

People with the condition. 1,000 people are tested, and the condition affects 1 in 100, so 1,000 ÷ 100 = 10 have it. The test detects 9 of them and misses 10 − 9 = 1.
People without the condition. 1,000 − 10 = 990 do not have the condition. The test wrongly flags 10 percent of them, so 990 × 10 ÷ 100 = 99 test positive anyway, and 990 − 99 = 891 test negative.
All the positives. 9 + 99 = 108 people test positive, and 9 of the 108 people who test positive have the condition. Since 108 ÷ 9 = 12, that is 1 in 12.
All the negatives. Of the 1 + 891 = 892 who test negative, 1 has the condition.

In words: only 9 of the 108 people who test positive have the condition, so a positive result is far from certain. The 99 false positives outnumber the 9 true ones, because the 990 people without the condition far outnumber the 10 with it.

Here is the same count as a table.

The illustrative screening table: 1,000 people sorted by whether they have the condition and how they test
Group	Test positive	Test negative	Total
Have the condition	9	1	10
Do not have the condition	99	891	990
Total	108	892	1,000

Read the table by rows first. The top row says the test catches most people who have the condition: 9 of 10. The second row says it wrongly flags some people who do not: 99 of 990. Then read the "Test positive" column. That column is what a person with a positive result belongs to, and only 9 of its 108 people have the condition.

The mistake this table cures has a name. Base-rate neglect is the mistake of judging what a positive result means from a test's accuracy alone while ignoring how rare or common the condition is in the group tested.

In practice (your tools). In a free spreadsheet, type 1,000 in one cell for the people tested. In the cells below it, work each count in words, one step per cell: the number with the condition, the number detected, the number without the condition, and the false positives. Then add the two positive counts. Label every cell in the column beside it, so each number says what it counts.

Misconception check — a positive result is near-certain. Students often reason, "the test catches 9 of every 10 cases, so a positive result means I almost surely have it." That is tempting, because the test really does catch most cases. But that sentence is about the top row of the table, and a person with a positive result belongs to the positive column. When the condition is rare, the false positives from the large healthy group crowd that column. Count the people before you trust the percent.

An honesty note: this table is illustrative, and real tests differ in how often they miss and how often they flag. That is why real screening programs often follow a positive result with a second, different test. The habit you take away is the count, not these exact figures.

Memory hook: few real cases, many false alarms — count the whole column.

6. Check the axis: misleading graphs

A friend shows you two bars on a phone screen. One is twice as tall as the other. "See? Ours is twice as good." Before you agree, look at the left edge of the graph. Where does the axis start?

A graph can be honest in every number and still mislead in its shape. Three tricks do most of the work.

A truncated axis is a graph axis that starts above zero, which makes bar heights stop matching the values they stand for and so exaggerates differences.
A stretched scale is an axis whose range is drawn so tall or so narrow that a small change spreads over most of the graph and looks steep.
A cherry-picked time window is a stretch of time chosen for a graph or a claim because it shows the pattern the speaker wants, while leaving out the periods that would change the picture.

Worked example — the ad's bar graph (illustrative). An ad's bar graph shows a rival brand's satisfaction rating of 50 and its own rating of 55. The vertical axis starts at 45.

The drawn heights. The rival's rating is 50 and the ad's own is 55, on an axis that starts at 45. The bars are drawn 50 − 45 = 5 units and 55 − 45 = 10 units tall.
What the eye sees. 10 ÷ 5 = 2, so one bar looks twice the other.
The real gap. The real gap is 55 − 50 = 5, and 5 ÷ 50 × 100 = 10 percent.

In words: the ad's taller bar shows a 10 percent gap, not a doubling. Had the axis started at zero, the bars would be drawn 50 and 55 units tall, and they would look nearly the same.

The other two tricks need no arithmetic to spot.

A stretched scale (illustrative). A city's monthly bus ridership rises a little over a year. Drawn on an axis that runs only from the lowest month to the highest, the line climbs from the bottom of the graph to the top. The rise is real, but its steepness is a choice about the axis.
A cherry-picked time window (illustrative). A city crime map shows one unusual month with a spike in reports. The months before and after, which would show the spike as a blip, are left out. The window decides the story.

Misconception check — the bar heights show the size of the difference. Students often compare bars by their drawn heights, since that is what a bar graph is for. That is tempting, because on an honest bar graph with an axis starting at zero, height does match value. But once the axis starts above zero, the heights no longer match the values. Read the axis numbers, compute the real gap, and only then look at the shape.

A limitation worth saying: an axis that starts above zero is not always a trick. On a line graph of body temperature, starting at zero would flatten every change that matters. The question is whether the drawing makes a difference look bigger than the numbers say.

Memory hook: read the axis first; then read the shape.

7. Put it together: one numbers check

You now have four questions, and they fit on an index card. Run them on any number in a claim, in this order.

Points or percent? If a percent changed, give the percentage points and the percent change.
Relative or absolute, and out of what base? If a risk changed, give "N in 1,000" before and after, and the change both ways.
How many of the positives? If a test or a screen is involved, count the people, starting from the base rate.
Where does the axis start, and what does the window leave out? If there is a graph, read its axis and its time range first.

Worked example — the whole check on this week's claims (illustrative). Run the four questions on the claims you met in this chapter.

The campus post. Support rose 10 percentage points, a 20 percent increase. "Jumped 20 percent" is true but tells only half.
The risk headline. A 50 percent relative rise from 2 in 1,000 to 3 in 1,000 is 1 more case per 1,000 people.
The screening result. Only 9 of the 108 people who test positive have the condition.
The ad's graph. The taller bar shows a 10 percent gap, not a doubling.

In words: each claim was true, and each told only part of the story. Support rose 10 percentage points, a 20 percent increase. The risk rose by 1 case per 1,000 people. A positive result is far from certain. The bars differ by 10 percent, not by double.

Notice what the check does not do. It does not tell you whether the library should stay open, whether a health choice is wise, or which brand to buy. Go back to the campus post. Once you know that support rose 10 percentage points to 60 percent, you can still argue for or against overnight hours. The check tells you how much the number can carry, and the course takes no side on the running question.

Misconception check — a number that fails the check is false. Students often decide that a number reported in a misleading way must be made up. That is tempting, because the misleading version felt like a trick. But nearly every number in this chapter was true: support did rise 20 percent, the risk did rise 50 percent, the bars did show 50 and 55. The flaw is in what the number leaves out or how it is drawn. Correct the picture; do not throw out the number.

Memory hook: points, base, positives, axis — four questions, one card.

Key terms

percentage points — The unit for the plain difference between two percents, found by subtracting one from the other; a share that rises from 50 percent to 60 percent rises 10 percentage points. (first used: 2. Points versus percent: report both)
percent change — The difference between a new value and a starting value divided by the starting value and multiplied by 100, so a rise from 50 percent to 60 percent is a 20 percent increase. (first used: 2. Points versus percent: report both)
absolute risk — The chance of an outcome stated as a count out of a named base, such as 2 in 1,000 people; a change in absolute risk is the number of extra or fewer cases per that base. (first used: 3. Relative versus absolute risk: name the base)
relative risk — A comparison of one risk with another stated as a ratio or a percent change, such as a 50 percent increase; without the base it leaves out how many people are affected. (first used: 3. Relative versus absolute risk: name the base)
base rate — How common a condition or outcome is in the whole group before any test or new evidence, such as 1 in 100 people. (first used: 5. Count the people: base rates and false positives)
false positive — A positive test result in a person who does not have the condition the test looks for. (first used: 5. Count the people: base rates and false positives)
false negative — A negative test result in a person who does have the condition the test looks for, a case the test misses. (first used: 5. Count the people: base rates and false positives)
natural frequencies — A way of setting out a probability problem as counts of people in a group of round size, such as 9 of the 108 people who test positive, instead of as percents or probabilities. (first used: 5. Count the people: base rates and false positives)
base-rate neglect — The mistake of judging what a positive result means from a test's accuracy alone while ignoring how rare or common the condition is in the group tested. (first used: 5. Count the people: base rates and false positives)
truncated axis — A graph axis that starts above zero, which makes bar heights stop matching the values they stand for and so exaggerates differences. (first used: 6. Check the axis: misleading graphs)
stretched scale — An axis whose range is drawn so tall or so narrow that a small change spreads over most of the graph and looks steep. (first used: 6. Check the axis: misleading graphs)
cherry-picked time window — A stretch of time chosen for a graph or a claim because it shows the pattern the speaker wants, while leaving out the periods that would change the picture. (first used: 6. Check the axis: misleading graphs)

Summary

A number in a passage is often a premise; find the conclusion before you weigh it.
Percentage points measure the plain gap between two percents; a percent change divides by the start.
From 50 percent to 60 percent is 10 percentage points and a 20 percent increase.
Write every risk as N in 1,000 and give every change both relative and absolute.
A 50 percent relative rise from 2 in 1,000 is 1 more case per 1,000 people.
When a condition is rare, false positives can outnumber true positives.
Set out a screening problem as counts of people and read the positive column.
Check the axis, the scale, and the time window before you read a graph's shape.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A gym reports that the share of members who use its app rose from 30 percent to 45 percent over a year (illustrative). Give the change in percentage points and as a percent change.
A headline says a new allergy spray "triples the risk" of a rare side effect, which goes from 1 in 10,000 to 3 in 10,000 (illustrative). Give the relative change and the absolute change.
A screening test for a food allergy is given to 10,000 people (illustrative). The allergy affects 1 in 50 of them. The test detects 190 of the people who have it, and it wrongly flags 5 percent of the people who do not. How many people test positive, and how many of them have the allergy?
A transit agency's bar graph shows weekday riders on two routes, 80 thousand and 84 thousand, on an axis that starts at 78 thousand (illustrative). How many times taller does the second bar look, and what is the real gap as a percent of the smaller route?

Answers (worked reasoning — read after attempting)

The change in percentage points is 45 − 30 = 15 percentage points. The percent change is (45 − 30) ÷ 30 × 100 = 50 percent. So app use rose 15 percentage points, which is a 50 percent increase, and a fair report gives both.
The relative change is (3 − 1) ÷ 1 × 100 = 200 percent, which is what "triples" means. The absolute change is 3 − 1 = 2 more cases per 10,000 people. So the risk tripled, and 9,997 of every 10,000 people still see no side effect.
Start with the base rate: 10,000 ÷ 50 = 200 people have the allergy, and 10,000 − 200 = 9,800 do not. The test detects 190 of the 200. It wrongly flags 9,800 × 5 ÷ 100 = 490 people without the allergy. In all, 190 + 490 = 680 people test positive, and 190 of the 680 people who test positive have the allergy. The false positives outnumber the true ones, which is base-rate neglect's trap.
The bars are drawn 80 − 78 = 2 and 84 − 78 = 6 units tall, and 6 ÷ 2 = 3, so the second bar looks 3 times as tall. The real gap is 84 − 80 = 4 thousand riders, and 4 ÷ 80 × 100 = 5 percent. The truncated axis turns a 5 percent gap into a bar that looks three times as big.

Next up in this module: the lecture, the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Evaluating Sources and Evidence, the numbers check becomes a source check, a method for tracing a claim back to where it started.

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