Lecture: Numbers in the News
Lecture script · about 15–20 minutes spoken · Week 11
Opening
Picture three numbers crossing 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 none of them tells you yet how big.
A nursing student meets this when explaining a screening result to a worried family. A business major meets it when reading a competitor's chart in a sales meeting. A club officer meets it when reporting how support for a proposal changed. In each case, you need to know what a number actually measures before you lean on it.
Here's where we are. Week 1 taught you to find an argument's premises and conclusion. Weeks 2 through 8 taught you to rebuild arguments, test their form, and catch fallacies. Week 9 taught you to check a sample, and Week 10 taught you to check a pattern for a hidden third factor.
In this lecture we check the numbers those claims are stated in. We'll take seven steps. First, finding the claim a number supports. Then percent against percentage points, relative risk against absolute risk, and one real case. Then screening tests, misleading graphs, and finally the whole check in one piece.
1. Find the claim first
Let's start with a friend holding up a phone. "Look, this ad says 9 out of 10 people love this blender. You should buy it." Before you ask whether that number 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's a premise, a claim offered as a reason to accept another claim. The other claim is the conclusion: "so you should buy this", or "so the city should act". Sometimes a passage only reports a number and argues for nothing at all. Week 1's test still works. What is the writer trying to get you to accept, and what is offered as the reason?
Here's the campus post in full. "Support for overnight hours jumped 20 percent in one year, so the library should open around the clock." The word "so" signals a conclusion. The conclusion is that the library should open around the clock. The premise is that support jumped 20 percent.
Now compare this sentence. "This year's random survey found 60 percent of students in favor of overnight hours." Nothing in that sentence is offered as a reason for anything else.
Pause and try this. Is that second sentence an argument or a report?
Look for a conclusion before you decide.
Here's the answer. It's a report. It states a number and offers it as a reason for nothing, so there is no conclusion to support yet.
So the habit is simple to say. Find the "so" first; then weigh the number.
2. Points versus percent
Think about your phone battery. It reads 50 percent at lunch, and after a short charge it reads 60 percent. How much did it go up? You could say 10, because 60 minus 50 is 10. You could also say it grew by a fifth of where it started. Both answers are true. 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 to give both whenever you give one.
Let's run it on the running question. Week 9's random sample asked 400 students from the enrollment list. Last year, 200 of the 400 were in favor, which is 50 percent. This year, 240 of the 400 were in favor, which is 60 percent.
Support rose from 50 percent to 60 percent. Subtract, and that's 10 percentage points. Now divide that 10 by the starting 50, and multiply by 100. That's a 20 percent increase. So the campus post's "jumped 20 percent" is true. But a reader who hears 20 percent may picture support going from 50 to 70, and that never happened.
The same tool measures the gap between Week 9's two surveys. The doorway sample read 75 percent, and the random sample read 60 percent. That's 15 percentage points apart, or 25 percent higher. But remember why. The doorway sample was biased, so a correct calculation doesn't rescue it.
One more case shows how a tiny base can inflate a percent. A store sold 2 units of a new item, then 4 units. That's 2 more units, and it's also up 100 percent.
Pause and try this. A friend says, "Support went from 50 percent to 60 percent, so it rose 10 percent." What would you correct?
Say the fix in one sentence before you go on.
Here's the answer. It rose 10 percentage points, which is a 20 percent increase. Calling the 10 a percent mixes up the plain gap with the growth.
Points for the gap, percent for the growth; always say both.
3. Relative versus absolute risk
A friend reads a headline aloud. "Doing this raises your risk by half!" Your stomach drops. Then you ask the question this section is about. Raises it from what?
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.
Here's the pair this course uses. Suppose a risk rises from 2 in 1,000 to 3 in 1,000. In absolute terms, that's 1 more case per 1,000 people. In relative terms, you divide that 1 extra case by the starting 2, and multiply by 100. That's a 50 percent increase.
Now suppose a different risk falls from 4 in 1,000 to 3 in 1,000. In absolute terms, that's 1 fewer case per 1,000 people. In relative terms, that 1 divided by the starting 4, times 100, is a 25 percent decrease.
Notice something odd. Both changes are 1 case per 1,000 people. Yet one sounds like 50 percent and the other like 25 percent. The relative number depends on where the risk started.
Pause and try this. A headline says a risk rose 50 percent, from 2 in 1,000 to 3 in 1,000. How many more people in every 1,000 are affected?
Picture a room of 1,000 people before you answer.
Here's the answer. One more person in every 1,000. The 50 percent is real, but it describes growth from a small start, not half of everyone.
A small absolute change can still matter across a big population. That's why you want both numbers, not one.
4. A real case, in words
So far every number in this lecture has been illustrative. Here's a real case, from a federal health agency, taught in words. No figure from it will be on a 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.
Let's hold both numbers up at once. The 20 percent compares the risk with screening against the risk without it. The 0.4 percent is the plain difference in the risk itself. They differ because the starting risk was small. A fifth of a small risk touches few people.
The relative number is real, and so is the absolute one. They answer different questions. A report that gives only the 20 percent sounds bigger than the result. A report that gives only the 0.4 percent can sound like nothing happened. A fair report gives both.
Pause and try this. Someone says, "Relative numbers are always tricks, so only the absolute number counts." What's wrong with that?
Think about what each number tells you.
Here's the answer. Both numbers describe the same result. The relative one compares options, and the absolute one shows how many people are touched, so leaving either out misleads.
And one honest caveat. This case teaches how the numbers relate. It isn't advice about anyone's own screening.
5. Count the people
Imagine a phone app that warns you about security problems. When a real problem shows up, it almost always catches it. 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 is in the whole group before any test, such as 1 in 100 people. A false positive is a positive test result in a person who does not have the condition. A false negative is a negative result in a person who does have it, a case the test misses.
The clearest way to work this is with natural frequencies: counts of people in a group of round size, instead of percents. Gigerenzer and Hoffrage, writing in 1995 in Psychological Review, found that people reason about these problems more easily in counts.
So let's count. A test is given to 1,000 people, and the condition affects 1 in 100. That means 10 people have it. The test catches 9 of them and misses 1. That leaves 990 people without the condition. The test wrongly flags 10 percent of them, which is 99 people. The other 891 test negative.
Now gather everyone who tested positive. That's 9 real cases plus 99 false alarms, 108 people in all. Only 9 of the 108 people who test positive have the condition. That's 1 in 12. Of the 892 who test negative, only 1 has the condition.
Pause and try this. A friend tests positive and says, "The test catches 9 of every 10 cases, so I almost surely have it." Using the 108 positives, what would you say?
Count the positive group before you answer.
Here's the answer. Only 9 of the 108 people who test positive have the condition. The 99 false positives from the large healthy group crowd out the 9 true ones.
That mistake has a name: base-rate neglect, judging a positive result from the test's accuracy alone while ignoring how rare the condition is. Rare condition, many false alarms; count the positive column.
6. Check the axis
A friend shows you an ad with two bars. One bar is twice as tall as the other. "See? Ours is twice as good." Before you agree, find the bottom of the vertical axis. Where does it start?
Here's the ad's graph, described in words. The rival brand's satisfaction rating is 50, and the ad's own rating is 55. The vertical axis doesn't start at zero. It starts at 45. So the rival's bar is drawn 5 units tall, and the ad's bar is drawn 10 units tall. The second bar looks twice as big.
But the real gap is 55 minus 50, which is 5. And 5 out of 50 is 10 percent. The ad's taller bar shows a 10 percent gap, not a doubling. A graph axis that starts above zero like this is a truncated axis, and it makes bar heights stop matching the values they stand for.
Two other tricks need no arithmetic. A stretched scale draws the axis so narrow that a small change climbs from the bottom of the graph to the top. Think of a city's bus ridership rising a little over a year, drawn so the line looks like a cliff. And a cherry-picked time window shows only the stretch of time that fits the story. Think of a crime map showing one unusual month and leaving out the months around it.
Pause and try this. If the ad's axis started at zero instead of 45, how tall would the two bars be drawn?
Picture the bars rising from the bottom edge.
Here's the answer. They'd be drawn 50 and 55 units tall, and they'd look nearly the same height. Read the axis first; then read the shape.
One limitation worth saying out loud. An axis that starts above zero isn't always a trick. The question is whether the drawing makes a difference look bigger than the numbers say.
7. One numbers check
You now have four questions, and they fit on an index card. Points or percent? Relative or absolute, and out of what base? How many of the positives? Where does the axis start, and what does the window leave out?
Run them on this week's claims. Support rose 10 percentage points, a 20 percent increase, so "jumped 20 percent" is true but tells only half. A 50 percent relative rise in a risk is 1 more case per 1,000 people. Only 9 of the 108 people who test positive have the condition. And the ad's taller bar shows a 10 percent gap, not a doubling.
Notice what the check doesn't do. It doesn't tell you whether the library should stay open. Once you know support rose 10 percentage points to 60 percent, you can still argue either way, and the course takes no side on that question. The check tells you how much a number can carry.
Pause and try this. A classmate says, "The campus post was misleading, so its number must be made up." Is that right?
Think back to the 50 and the 60.
Here's the answer. No. Support really did rise 20 percent. The flaw is in what the post left out, the 10 percentage points, not in the number itself.
Points, base, positives, axis: four questions, one card.
Closing
Let's pull it together. A number in a passage is often a premise, so find the conclusion first. A change between percents is 10 percentage points and a 20 percent increase, and you give both. A risk change from 2 in 1,000 to 3 in 1,000 is a 50 percent relative rise and 1 more case per 1,000 people. The lung-screening case shows a real relative change that sounds bigger than its absolute one. Counting people shows that only 9 of the 108 positives are real. And a truncated axis can make a 10 percent gap look like double.
Next in this module come the slide deck, the Readings and Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Evaluating Sources and Evidence, you turn from the number to its source, and you trace a claim back to where it started.
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