Week 9 · Chapter

Chapter 9 — Inductive Generalization and Analogy

Critical Thinking Free sample edition

The module's primary reading · ~35–45 minutes · Objective 5 (with Objective 3 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 claims that reach from a few cases to a whole group. You will learn to ask two questions of any such claim: who was actually checked, and how did they get picked? The whole group is the population (everyone the claim is about), and the part that was checked is the sample (the people or things actually observed). You will see why a huge sample can still mislead, why one vivid story is weak evidence, and how to test an analogy (a comparison that says two things alike in some ways are alike in another). These tests help you decide how far a result can travel before you repeat it.

Why you already care about this

A new taco truck parks outside the student center. Your friend goes once, orders 1 taco, waits 20 minutes, and gets a cold tortilla. That night the group chat reads: "That truck is terrible. Don't bother." By morning 4 people have agreed, and none of them has eaten there.

You have done this too. A phone you bought broke, so you swore off the brand. Three friends loved a show, so you decided everyone would. Each time, a few cases stood in for a whole group. Sometimes that works well. Sometimes it badly misleads you.

For eight weeks you tested an argument's form, its words, and whether its reasons bear on the conclusion. This week you test the evidence behind a claim about a group.

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

tell a population from a sample and read an inductive generalization as an argument from one to the other;
name three ways a sample can mislead: too small, biased, or skewed by nonresponse;
explain why how a sample is chosen matters more than how big it is, using the 1936 Literary Digest poll;
run the sampling check on the running question's two surveys and say which is the better guide;
identify a hasty generalization and a generalization from one vivid anecdote;
test an argument from analogy by listing relevant similarities and looking for disanalogies.

Carry this line with you: how the sample was chosen beats how big it is — check the method first.

Where we are

Already covered: Week 1 taught you to find an argument, and Week 2 to put it in standard form and map it. Week 3 taught you validity and soundness for deductive arguments and strength and cogency for inductive ones. Weeks 4 and 5 taught the deductive forms. Weeks 6 to 8 taught how words, irrelevant reasons, and hidden assumptions can lead an argument astray.
This week: the second half of the course begins, turning from the passage on the page to the evidence in the world. You test generalizations from samples and arguments from analogy.
How they connect: a generalization and an analogy are both inductive arguments, so Week 3's strength and cogency are the standards you apply. Week 7's fallacy test returns with two new fallacies to key.
Where we're going next: Week 10 turns from samples to causes, testing claims that one thing brings about another.

Week 3 drew the line you lean on most this week. A deductive argument is meant to guarantee its conclusion and is judged valid or invalid. An inductive argument is meant to make its conclusion probable. It is strong when its premises, if true, make the conclusion likely, and cogent when it is strong and every premise is true. Even a cogent argument can have a false conclusion. Every argument in this chapter is inductive, so none of them can be valid; the question is always how strong.

1. Name the groups: population and sample

Think about tasting soup. You stir the pot, take one spoonful, and decide the whole pot needs salt. You never drank the whole pot. You reasoned from a small part to the whole, and you trusted the spoonful because you stirred first.

That move has names.

A population is the whole group a claim is about, such as every student enrolled on a campus, whether or not anyone has observed all of its members.

A sample is the part of a population that is actually observed or asked, such as the 400 students a survey reaches, used as evidence about the whole group.

An inductive generalization is an inductive argument that reasons from what is observed in a sample to a conclusion about the whole population, so it can be strong or weak but never valid.

In the soup, the pot is the population and the spoonful is your sample. Stirring is what makes your spoonful a fair stand-in. Every generalization you meet this week has three parts: a population, a sample, and a step from one to the other.

Worked example — the running question's population and samples (illustrative). The running question's surveys take place on an imagined campus.

The population. The claim is about the students on the campus, all 12,000 of them.
The samples. Each of two surveys asks 400 students.
How big a slice. 12,000 ÷ 400 = 30, so each sample is 1 student in 30.

In words: each survey checks 1 student in 30 and then reasons to all 12,000, so each is an inductive generalization, and the question is how strong that step is.

You can write a generalization in standard form, the way Week 2 taught.

P1: 240 of 400 students drawn at random from the enrollment list favor overnight hours.

So,

C: About 60 percent of the campus's students favor overnight hours.

If P1 is true, C is likely, but C could still be off. So you judge it with Week 3's inductive standards: strong or weak, cogent or not.

Misconception check — a big enough sample makes a generalization valid. Students often think that if you check enough cases, the conclusion becomes certain. That is tempting, because a larger, well-chosen sample does make a conclusion more likely. But a generalization always leaves part of the population unchecked, and the unchecked part could differ. However strong it gets, it stays inductive. The words for it are strong and cogent, never valid.

Memory hook: stir the pot first; then trust the spoonful.

2. Spot the three traps: how a sample misleads

Suppose your gym wants to know whether members like the new class schedule. It could ask 5 members, ask whoever is in the weight room at 6 a.m., or email every member and count the replies. Each method can go wrong in its own way.

A sample can mislead in three ways.

It is too small. A handful of cases can land on unusual members by luck.
It is a biased sample, a sample chosen in a way that favors some members of the population over others, such as asking only the people already standing where the question matters most to them.
It suffers from nonresponse, the problem that arises when the people who choose to reply to a survey differ from the people who do not, so the replies misstate the population even if the people asked were well chosen.

Run those three traps on the gym's methods (illustrative).

Ask 5 members. Too small. Five people can all happen to be morning regulars.
Ask whoever is in the weight room at 6 a.m. Biased. Early risers are the members most likely to like an early class, and the night-shift worker who trains at 10 p.m. never gets asked.
Email every member and count replies. Open to nonresponse. Members who are upset about the schedule may be the most eager to reply, so the replies can lean unhappy even though everyone was asked.

The cure for the first two traps has a name too. A random sample is a sample drawn so that every member of the population has the same chance of being picked, which is different from asking whoever happens to be nearby. The gym could draw member numbers by lottery from its full list and then ask exactly those members. That is random in the technical sense. Standing at the door and asking whoever walks by is not.

A random sample is the usual way to get a representative sample. That is a sample whose members resemble the population in the ways that matter to the question, so that what holds in the sample is likely to hold in the population.

Misconception check — random means whoever I happened to ask. Students often call a sample random when the people in it were not planned: "I asked random people in the hallway." That is tempting, because in everyday speech "random" means unplanned. But a hallway sample favors the people who use that hallway at that hour. In this course a random sample gives every member of the population the same chance of being picked, which takes a list and a fair draw, not a stroll.

An honesty note: even a random sample can suffer from nonresponse if many of the people drawn never reply. A fair draw fixes who gets asked, not who answers, so a careful survey also reports how many people it asked and how many replied.

Memory hook: too few, tilted, or self-selected — check all three.

3. Check the method: representativeness beats size

If a sample can be too small, you might guess that the cure is always more. A real poll from 1936 shows why that guess fails.

Worked example — the 1936 Literary Digest poll (real, taught in words). The source for the electoral result is the National Archives.

The sample. The Literary Digest, a popular magazine, mailed straw-poll ballots to about 10,000,000 people. The names came largely from telephone directories and automobile-registration lists.
The replies. About 2,400,000 ballots came back.
The prediction. The magazine predicted that Landon would defeat Roosevelt.
The result. Roosevelt won in a landslide, 523 electoral votes to 8, the figure the National Archives reports.

In words: a sample of about 2,400,000 replies still pointed the wrong way, so its size did not protect it; the two explanations below say how its method may have failed.

Compare that with the running question's random sample of 400. The Digest's replies outnumbered it by a factor of about 6,000, since 2,400,000 ÷ 400 = 6,000. That is 2,400,000 replies versus 400, a 6,000-fold difference, and size still did not rescue the poll. A small sample drawn fairly can tell you more than a giant sample drawn badly.

Why did the poll miss? Two explanations are taught, and accounts differ on how much each one did.

The lists. The mailing lists were built from directories of telephone and car owners, so they may have favored some kinds of voters over others. That is the biased-sample trap.
The replies. The people who chose to mail their ballots back may have differed from the people who did not. That is the nonresponse trap.

The course teaches both explanations and grades neither as the cause. The one figure you are asked to remember is the electoral result, 523 to 8.

Misconception check — a large sample is automatically representative. Students often trust a result because the sample is huge: "they asked millions of people." That is tempting, because small samples really are risky, and more data feels like more truth. But size cannot fix a tilted method. A biased list stays biased however long it gets, and a pile of self-selected replies stays self-selected. Check how the sample was chosen first, then check its size.

A caveat: this is a real case from long ago, and historians still weigh its causes. Use it as an example of the two traps, not as a settled story about one cause.

Memory hook: more of a tilted sample is still tilted; check the method.

4. Run the sampling check: the two surveys

Now run the check on the running question. Two illustrative surveys ask students whether the campus library should stay open around the clock during finals week.

Worked example — the doorway sample against the random sample (illustrative). Read both surveys, then run the three traps.

The doorway sample. 400 students were asked at the library entrance at 11 p.m., and 300 said yes. 300 ÷ 400 × 100 = 75 percent.
The bias check. The doorway sample asked only people already in the library late at night. They are the students most likely to want it open late, so the sample was chosen in a way that favors some members of the population. It is a biased sample.
The random sample. 400 students were drawn at random from the enrollment list, so every enrolled student had the same chance. 240 said yes. 240 ÷ 400 × 100 = 60 percent.
Size against method. Both samples hold 400 students, so size cannot explain why one reads 75 percent and the other 60 percent. The method of choosing does.
The verdict. The random sample is the better guide to the 12,000.

In words: the doorway survey's 75 percent comes from a biased sample, while the random sample's 60 percent is the better guide to the campus. The two samples are the same size and differ only in how they were chosen.

Here is the same comparison in a small table.

The running question's two illustrative surveys: who was asked, how many said yes, and the share in favor
Survey	Who was asked	Students asked	In favor	Share in favor
Doorway sample	students at the library entrance at 11 p.m.	400	300	75 percent
Random sample	students drawn at random from the enrollment list	400	240	60 percent

Now connect this to Week 3. Write each survey as a generalization and judge it.

The doorway argument. "300 of 400 doorway students favor overnight hours, so about 75 percent of the campus does." Its premise can be true, since those 300 really did say yes. The argument is still weak, because the sample is biased.
The random-sample argument. "240 of 400 randomly chosen students favor overnight hours, so about 60 percent of the campus does." This argument is strong. If its premise is true, it is cogent, and yet it can still miss.

This is a lesson in structure, not a verdict on the library. The sampling check tells you which number is the better guide to student opinion. It does not tell you whether the library should stay open, and the course takes no side on that question.

Misconception check — the doorway students must have lied. Students often assume that a misleading survey result comes from false answers. That is tempting, because 75 percent and 60 percent cannot both describe the campus. But nothing suggests anyone lied. The 300 doorway students may have answered honestly. The flaw is in the step from those 400 to all 12,000, which is a flaw in the reasoning, not in the replies.

Memory hook: same size, different door; the door decides.

5. Count the cases: hasty generalization and the vivid anecdote

Go back to the taco truck. One visit, one cold tortilla, one verdict about every meal the truck will ever serve. You now have a name for that move.

A hasty generalization is the fallacy of drawing a conclusion about a whole group from too few cases, or from cases that do not represent the group.

Its most common form leans on a story. A generalization from one vivid anecdote is a hasty generalization that rests a claim about a whole group on a single memorable story, letting how striking the case is stand in for how typical it is.

Worked example — the flu shot (illustrative). "My cousin caught the flu even though she had a flu shot, so flu shots don't work."

Find the population. The conclusion is about flu shots in general, given to many people.
Find the sample. The sample is 1 person, the cousin.
Run the traps. A sample of 1 is too small, and it was picked because it was memorable, not by a fair draw.

In words: the sample is 1 person and the conclusion is about every flu shot, so the passage commits the hasty generalization, here a generalization from one vivid anecdote. Whether flu shots work is a question about many people, and one case cannot settle it either way.

The verdict does not say the cousin's story is false; she may well have caught the flu. A fallacy is a flaw in the reasoning, as Week 7 taught you, and the flaw here is the step from 1 case to all of them.

The non-fallacious neighbor is a generalization from a large, representative sample. A health agency that tracks thousands of people chosen to represent the population can make a strong inductive generalization, and that kind of evidence is what the cousin's story cannot replace.

Misconception check — one vivid story is evidence of a trend. Students often find a single story more convincing than a table of numbers. That is tempting, because a story has a face and a feeling, and it is easy to remember. But how memorable a case is tells you nothing about how typical it is. Ask how many cases stand behind the claim and how they were chosen. One story is a reason to ask the question, not an answer to it.

Memory hook: one story is a sample of 1, not a trend.

6. Test the comparison: arguments from analogy

You are choosing a new phone. Your last phone, from the same maker, had a battery that lasted all day. The new model uses the same battery, the same screen size, and the same kind of chip. So you expect its battery to last all day too.

That is a different kind of inductive argument.

An argument from analogy is an inductive argument that reasons that two things alike in several known ways are also alike in a further way. It is stronger when the similarities bear on the conclusion and no relevant difference outweighs them.

A disanalogy is a relevant difference between the two things an analogy compares, one that bears on the conclusion and so weakens the argument.

To test an analogy, take three steps:

List the relevant similarities, the ones that bear on the conclusion.
Look for relevant differences, the disanalogies.
Judge whether the similarities bear on the conclusion strongly enough to survive the differences.

Worked example — the phone and the brain (illustrative). Run the three steps on two comparisons.

The new phone and the last model. Similarities: the same battery, the same screen size, and the same kind of chip, and all three bear on battery life. Disanalogy check: if the new model adds a brighter screen or a feature that runs all the time, that difference would bear on battery life too. With no such difference, the similarities bear directly on the conclusion, so the analogy is fairly strong.
The brain and a computer. People sometimes say memory works like a computer's hard drive, storing each event as a file. Similarities: both take in information, keep some of it, and use it later. Disanalogy check: the comparison says nothing about how memories are actually kept, and it leaves out that people often remember an event differently each time they recall it. The comparison can help you picture memory, but it fails as proof that memory works like a hard drive.

In words: the phone analogy is fairly strong because its similarities bear on battery life, while the brain analogy is a useful picture that cannot prove how memory works, because its similarities do not bear on how memories are kept.

Misconception check — an analogy is a proof. Students often treat a vivid comparison as if it settled a question: "the brain is a computer, so memory is a file." That is tempting, because a good analogy makes a hard idea feel familiar. But an analogy is an inductive argument at best, and often it is only a picture. Ask which similarities bear on the conclusion and which differences might break it.

An honesty note: the number of similarities is not the whole test. Two things can share ten features that do not matter and differ in the one that does. Relevance comes first, then count.

Memory hook: alike in the ways that matter — not merely alike.

7. Key the weak analogy: the college and the store

Some analogies are not weak by degree; they miss. Their similarities are real, but none bears on the conclusion you are asked to accept.

A weak analogy is the fallacy of resting a conclusion on a comparison whose similarities do not bear on that conclusion, or are outweighed by relevant differences.

Worked example — the college as a store (illustrative). "A college is like a store, and in a store the customer is always right, so students should decide their own grades."

List the similarities. Students pay, as customers do, and both a college and a store serve the people who pay.
Look for the disanalogy. A grade is a judgment of whether work meets a standard, not a product the buyer chooses. In a store, you can pick the shirt you like. A grade reports how your work measured up.
Judge whether the similarity bears on the conclusion. Paying is a real similarity, but it does not bear on who should judge whether work meets a standard.

In words: the similarity is that students pay, as customers do, the disanalogy is that a grade is a judgment and not a purchase, and since the similarity does not bear on the conclusion, the passage commits the weak analogy.

That gives this week's tally from the course's fallacy set. 1 hasty generalization + 1 weak analogy = 2 Week 9 passages. Through Week 8 you keyed 15 passages, so 15 + 2 = 17 passages are keyed through Week 9. The set holds 20, and 20 − 17 = 3 passages are still to come in later weeks.

Keep structure apart from agreement, as you have all term. Calling the store analogy weak does not settle how grading should work; someone could build a better argument on other grounds. Your verdict is about this argument's link, not its topic.

Misconception check — spotting a disanalogy defeats any analogy. Students often learn to find a difference and then declare every analogy broken. That is tempting, because every comparison has differences. But only a relevant difference weakens an analogy, one that bears on the conclusion. The new phone and the old one differ in color, and that difference says nothing about the battery. Ask whether the difference bears, not merely whether it exists.

Memory hook: a real likeness can still be the wrong one; ask if it bears.

Key terms

population — The whole group a claim is about, such as every student enrolled on a campus, whether or not anyone has observed all of its members. (first used: 1. Name the groups: population and sample)
sample — The part of a population that is actually observed or asked, such as the 400 students a survey reaches, used as evidence about the whole group. (first used: 1. Name the groups: population and sample)
inductive generalization — An inductive argument that reasons from what is observed in a sample to a conclusion about the whole population, so it can be strong or weak but never valid. (first used: 1. Name the groups: population and sample)
biased sample — A sample chosen in a way that favors some members of the population over others, such as asking only the people already standing where the question matters most to them. (first used: 2. Spot the three traps: how a sample misleads)
nonresponse — The problem that arises when the people who choose to reply to a survey differ from the people who do not, so the replies misstate the population even if the people asked were well chosen. (first used: 2. Spot the three traps: how a sample misleads)
random sample — A sample drawn so that every member of the population has the same chance of being picked, which is different from asking whoever happens to be nearby. (first used: 2. Spot the three traps: how a sample misleads)
representative sample — A sample whose members resemble the population in the ways that matter to the question, so that what holds in the sample is likely to hold in the population. (first used: 2. Spot the three traps: how a sample misleads)
hasty generalization — The fallacy of drawing a conclusion about a whole group from too few cases, or from cases that do not represent the group. (first used: 5. Count the cases: hasty generalization and the vivid anecdote)
generalization from one vivid anecdote — A hasty generalization that rests a claim about a whole group on a single memorable story, letting how striking the case is stand in for how typical it is. (first used: 5. Count the cases: hasty generalization and the vivid anecdote)
argument from analogy — An inductive argument that reasons that two things alike in several known ways are also alike in a further way. It is stronger when the similarities bear on the conclusion and no relevant difference outweighs them. (first used: 6. Test the comparison: arguments from analogy)
disanalogy — A relevant difference between the two things an analogy compares, one that bears on the conclusion and so weakens the argument. (first used: 6. Test the comparison: arguments from analogy)
weak analogy — The fallacy of resting a conclusion on a comparison whose similarities do not bear on that conclusion, or are outweighed by relevant differences. (first used: 7. Key the weak analogy: the college and the store)

Summary

A population is the whole group a claim is about; a sample is the part observed.
An inductive generalization reasons from sample to population and is strong or weak, never valid.
A sample misleads when it is too small, biased, or skewed by nonresponse.
A random sample gives every member the same chance; asking whoever is nearby is not random.
How a sample is chosen matters more than how big it is, as the 1936 poll shows.
The doorway sample's 75 percent is biased; the random sample's 60 percent is the better guide.
A hasty generalization rests a claim about a group on too few or untypical cases.
Test an analogy by listing relevant similarities and looking for disanalogies.
A weak analogy rests on similarities that do not bear on its conclusion.

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 bakery owner wants to know whether customers would like the shop to open at 5 a.m. instead of 7 a.m. At 6:30 one morning, she asks the 30 customers in line, and 24 say yes (illustrative). Find the population, the sample, and the percent in favor, and name the trap.
A streaming service emails a survey to 5,000 subscribers, chosen at random from its full list, and 250 reply (illustrative). What percent replied, and which trap should you check before trusting the results?
A neighbor says (illustrative): "My coworker's electric scooter broke down twice in one month, so electric scooters are unreliable." Name the fallacy and say what kind of evidence would bear on the conclusion.
A team captain argues (illustrative): "A group project is like a relay race: in a relay, the slowest runner sets the pace for the whole team, so in our project the slowest member decides our grade." List one relevant similarity, one disanalogy, and say how strong the analogy is.

Answers (worked reasoning — read after attempting)

The population is all of the bakery's customers, and the sample is the 30 people in line at 6:30. The percent in favor is 24 ÷ 30 × 100 = 80 percent. The trap is a biased sample: people already in line before 7 a.m. are the customers most likely to want an earlier opening, and the customers who come at noon were never asked.
250 ÷ 5,000 × 100 = 5 percent replied. The draw was random, so the people asked were well chosen. Nonresponse is the trap to check: the 5 percent who replied may differ from the 95 percent who did not, for example by being the most pleased or the most annoyed.
A hasty generalization, here a generalization from one vivid anecdote. The sample is 1 scooter and the conclusion is about electric scooters in general. Evidence that would bear on it is a repair record for many scooters, chosen to represent the ones people actually ride.
One relevant similarity: in both, the result depends on several people's work, so one person's weak part can pull the whole result down. One disanalogy: a relay runner cannot run faster than they are able, but project members can help each other, divide parts, or revise a weak section before it is turned in. Because that difference bears directly on the conclusion, the analogy is weak as an argument, though it makes a fair point that every part matters.

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 Causal Reasoning, the sampling check becomes a test for causes, comparing groups to ask whether one thing brings about another.

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