The rarest Math question type (0.7% of the bank) and the highest return per hour of study, because it reduces to a single distinction. Covers random assignment as the licence for causal claims and random selection as the licence for generalisation, how to tell an observational study from an experiment, why confounding variables defeat observational data, and how to match a conclusion to the study design that supports it — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
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Title
SAT Math · Type 19 of 19
0.7% of the question bank — 11 of 1675 questions
Objectives
A study is described in three or four sentences, and you are asked which conclusion it supports. There is no arithmetic at all. The entire type rests on one distinction, and once you hold it the questions take about fifteen seconds each — which is why the rarest type on the section is also the one most worth an hour of your time.
Two sentences are the whole deck: random assignment licenses cause; random selection licenses generalisation. Neither one gives you the other.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 11 tagged questions of this type in the site's bank
Section
Section 1
Concept
A study is described in three or four sentences, and you are asked which conclusion it supports. There is no arithmetic at all. The entire type rests on one distinction, and once you hold it the questions take about fifteen seconds each — which is why the rarest type on the section is also the one most worth an hour of your time.
You will see it phrased in these ways:
Every one of these is answered by locating two words in the stem: whether subjects were randomly assigned, and whether they were randomly selected.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Evaluating statistical claims: the tell, the move, and the trap
The green panel is the complete method. There is nothing else to learn on this type, which is why eleven questions in a bank of 1,675 still justify a deck.
Prediction
The distinction is about how subjects reached their groups.
Predict first
Which description is an experiment?
Correct: Researchers randomly assigned 200 volunteers to receive either the drug or a placebo
Why: An experiment is defined by the researcher ASSIGNING subjects to groups. In the other three, subjects arrived in their groups by their own choices or circumstances, and the researcher merely observed — which makes all three observational studies, however carefully conducted. The word assigned is the tell, and it is the single most important word in this type.
Concept
Three shapes, all decided by the same two questions.
| variant | what it wants | the two questions |
|---|---|---|
| Which conclusion? | the strongest supported claim | assignment for cause, selection for scope |
| Compare two designs | which supports more | the one with assignment supports cause |
| Fix the study | what would license the claim | add whichever form of randomness is missing |
The third variant is the most instructive, because it forces you to say which kind of randomness was absent — which is exactly what the first two variants are testing implicitly.
Definition probe
Two different words doing two different jobs.
Sort into buckets
What does each form of randomness license?
Discrimination
Six studies, three kinds of conclusion.
Sort into buckets
What can this study support?
Warm-up
Try it before the rules.
Discussion prompt
Researchers randomly assigned 300 volunteers to a new exercise programme or a control group, and found the programme group had lower blood pressure. Can they conclude the programme lowers blood pressure? Can they conclude it would work for the general public?
Hint: There are two questions, and the answers differ.
Answer:
Yes to the first. Subjects were randomly ASSIGNED, which balances every other factor between the groups, so the difference in outcome can be attributed to the programme.
No to the second. The subjects were volunteers, not randomly SELECTED from the public. Volunteers differ — they tend to be more motivated and often healthier.
So the correct conclusion: the programme caused lower blood pressure among these volunteers.
What would license the wider claim: random selection from the general public, in addition to the random assignment already present.
That pair of answers — yes to cause, no to generalisation — is the most common configuration on this type.
Pattern
Two questions and a match. There is no third step and no arithmetic.
Ask: were subjects randomly ASSIGNED to groups by the researcher?
Why: If yes, it is an experiment and a causal claim is available. If no, it is observational and only association is available, however carefully it was run.
Ask: were subjects randomly SELECTED from a wider population?
Why: If yes, the finding extends to that population. If no, it applies only to the subjects studied.
Match the conclusion to what those two answers permit.
Why: Reject any choice claiming cause without assignment, or claiming a wider population without selection.
Both questions are answered by single words in the stem. Underlining assigned and selected is genuinely the whole technique.
Section
Section 2
Concept
The defining question is whether the researcher decided which group each subject went into.
| experiment | observational study | |
|---|---|---|
| How groups form | the researcher assigns | subjects arrive by choice or circumstance |
| Typical wording | randomly assigned, allocated to | surveyed, compared, observed, recorded |
| Supports | a causal claim | an association only |
| Main weakness | subjects may not represent anyone | confounding variables |
The distinction is entirely about the mechanism of group formation, never about the size or the care of the study.
Concept
Assigning subjects randomly balances every other factor, known and unknown, between the groups.
This is why the phrase randomly assigned is worth circling the moment you read it. It is the only thing that licenses the word cause.
Prediction
One word decides whether cause is available.
Predict first
Which phrase in a study description licenses a causal conclusion?
Correct: participants were randomly assigned to two groups
Why: Random assignment is what balances other factors between groups and therefore licenses a causal claim. Random selection licenses generalisation instead, which is a different thing. Careful surveying and a large sample both improve the study without changing what kind of claim it can support.
Concept
Selecting subjects randomly from a population makes the sample representative of that population.
The two randomnesses answer different questions, and a study can have either, both, or neither.
Prediction
One randomness, one licence.
Predict first
A study randomly assigns 200 volunteers to two groups and finds a real difference. What can be concluded?
Correct: The treatment caused the difference among these volunteers
Why: Random assignment licenses a causal claim about the subjects studied. But the subjects were volunteers rather than a random sample of any wider group, so the finding cannot be extended to the general population. The third choice understates: assignment does license cause, not merely association.
Concept
A third factor affecting both quantities can produce an association with no causal link between them.
When a question asks why a causal conclusion is unjustified, naming a plausible confounder is usually the expected answer.
Concept
When subjects choose their own group, the groups differ before the study begins.
Self-selection defeats both randomnesses at once: the groups are not comparable, and the respondents represent nobody in particular.
Prediction
The other single randomness.
Predict first
A study randomly selects 1,000 residents of a city and finds that those who cycle have better health. What can be concluded?
Correct: Cycling is associated with better health among the city's residents
Why: Random selection licenses generalisation to the city's residents, so the association extends to that population. But nobody was assigned to cycle — residents chose for themselves — so healthier people may simply be more likely to cycle, or a third factor such as income may drive both. Only association is available.
Concept
No amount of data converts an observational study into evidence of cause.
This mirrors the bias point in type 18: size buys precision and never buys validity.
Concept
A study has assignment or not, and selection or not, giving four cases with four different scopes.
| assignment? | selection? | what it supports |
|---|---|---|
| yes | yes | cause, in the wider population |
| yes | no | cause, among the subjects studied |
| no | yes | association, in the wider population |
| no | no | association, among the subjects studied |
Memorise this table and the type is finished. Every question on it is asking which of these four rows the described study occupies.
Prediction
What could explain the association without causing it?
Predict first
Towns with more parks have lower obesity rates. Which is the most plausible confounding variable?
Correct: Town wealth, which affects both park provision and diet
Why: A confounder is a third factor influencing both of the quantities being compared. Wealthier towns can afford more parks and their residents tend to have better access to healthy food and healthcare — so wealth could produce the association with no causal link between parks and obesity at all. The other choices name the two variables themselves and a feature of the study rather than a third factor.
Two truths and a lie
Three of these statements about study design are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: No sample size converts an observational study into evidence of cause. A larger sample narrows the margin of error, which is about precision, and does nothing about confounding. A study of a million self-selected exercisers still cannot separate the exercise from the income, diet and health that travel with it. Only random assignment can do that, because only assignment breaks the link between the treatment and everything else about the subject.
Check
Two questions, then match.
Check your understanding
Researchers randomly assigned 400 students at one university to either a new study technique or their usual method. The new-technique group scored higher. Which conclusion is best supported?
Answer: A
Why: Random assignment licenses a causal claim, so improved is justified. But the subjects were students at one university, not a random sample of university students generally, so the conclusion cannot extend beyond that institution.
Choices B and C are the two directions of error — too broad and too weak — and choice D is the over-correction. Locating the right row of the four-combination table avoids all three.
Section
Section 3
Worked example
Researchers surveyed 2,000 randomly selected adults in a region and found that those who eat breakfast weigh less on average. What can be concluded?
Figure (svg): A table applying the two questions to an observational study
Ask about assignment: nobody was assigned to eat breakfast. Subjects chose for themselves.
Why: This makes it an observational study, so only association is available.
Ask about selection: 2,000 adults were randomly selected from the region.
Why: This licenses generalisation to the adults of that region.
Combine: there is an association between eating breakfast and lower weight among adults in that region.
Why: Selection gives the scope; the absence of assignment limits the claim to association.
This is the most common configuration on the test: a well-conducted survey supporting a generalisable association and nothing stronger.
Verify: name a confounder: people who eat breakfast may also have more regular schedules and more active jobs.
Why: A plausible third factor confirms that the causal claim is not available.
Answer: an association among adults in that region, not a causal claim
Worked example
150 volunteers were randomly assigned to a tutoring programme or a control group. The tutored group scored higher. What can be concluded?
Figure (svg): A table applying the two questions to an experiment on volunteers
Ask about assignment: yes, the researchers randomly assigned the groups.
Why: This balances other factors, so the tutoring is the remaining difference between the groups.
Ask about selection: no, the subjects volunteered.
Why: Volunteers differ from non-volunteers in motivation, so they represent no wider population.
Combine: the tutoring caused higher scores among these volunteers.
Why: Cause is available; the wider generalisation is not.
This configuration is extremely common in real medical research, where volunteers are recruited and then randomised. The causal finding is solid; the generalisation is the contested part.
Verify: check what is missing for the wider claim: random selection from all students.
Why: Naming what is absent confirms exactly which half of the conclusion is unavailable.
Answer: the tutoring caused higher scores, among these volunteers
Worked example
A company compared employees who chose to use its wellness app with those who did not, and found app users took fewer sick days. What can be concluded?
Figure (svg): Bars showing only the narrow association claim is supported
Assignment: no. Employees chose whether to use the app.
Why: Self-selection means the two groups differed before the study began.
Selection: no. These are the employees of one company, not a random sample of anyone.
Why: The findings apply to this workforce and no wider group.
So only an association among these employees is supported.
Why: Both randomnesses are absent, so both kinds of claim are unavailable.
This is the weakest of the four configurations and it is very common in workplace and marketing studies, which routinely report it as though it were causal.
Verify: name the confounder: health-conscious employees are more likely both to use the app and to be well anyway.
Why: A single plausible confounder is enough to defeat the causal reading.
Answer: an association among this company's employees, nothing more
Step zero
Before reading any answer choice.
Discussion prompt
A question describes a study and asks which conclusion is appropriate. What two words do you search the stem for?
Hint: Two phrases, doing two different jobs.
Answer:
Randomly ASSIGNED — if present, a causal claim is available.
Randomly SELECTED — if present, the finding generalises to the population named.
Underline whichever appears, and note which is absent. The absent one tells you what the correct answer will NOT claim.
If neither appears, the study supports only an association among the subjects actually studied — the weakest of the four conclusions.
Only then read the choices, and eliminate any that claim more than the two words permit. This usually leaves one standing.
Worked example
Researchers randomly selected 800 adults from a national register and then randomly assigned each to one of two diets. One diet produced greater weight loss. What can be concluded?
Figure (svg): A table showing both randomnesses present
Assignment: yes, subjects were randomly assigned to the diets.
Why: Other factors are balanced between the groups, licensing a causal claim.
Selection: yes, subjects were randomly selected from a national register.
Why: The sample represents the national adult population.
So the diet caused greater weight loss, and the finding generalises nationally.
Why: Both randomnesses are present, so both claims are available.
This configuration is rare in practice because it is expensive, and it is rare on the SAT too. When it appears, the strongest conclusion offered is usually correct.
Verify: check that both words appear in the stem: randomly selected and randomly assigned.
Why: Both are explicitly stated, which is what distinguishes this from the previous examples.
Answer: the diet caused greater weight loss in the national adult population
Worked example
A study finds that students who take music lessons have higher mathematics scores. Why is the conclusion that music lessons improve mathematics unjustified?
Figure (svg): Two dot plots showing music students scoring higher on average
Identify the design: nobody was assigned to take music lessons — families chose.
Why: This is observational, so a causal claim is unavailable from the outset.
Name a confounder: family income affects both the ability to pay for lessons and access to tutoring and resources.
Why: A third factor influencing both quantities produces the association without any causal link.
So the association is real and the causal explanation is not established.
Why: The data cannot distinguish music lessons from everything that travels with them.
Naming a specific plausible confounder is usually what these questions want, rather than the general statement that correlation is not causation.
Verify: check what would settle it: randomly assigning students to receive lessons or not.
Why: Assignment would break the link between lessons and family circumstances, isolating the effect.
Answer: because the study is observational, and confounders such as family income could explain the association
Faded example
From memory. These two sentences are the entire type.
Fill in the blanks
Random assignment licenses a claim about cause, because it balances other factors between the groups. Random selection licenses generalisation to the population sampled. A study is an experiment when the researcher assigns subjects to groups. And a confounding variable can explain an association without any causal link.
Why: The first two blanks are the whole deck. They are easy to confuse because both contain the word random and both sound like good practice — but they license completely different claims, and a study can have either without the other.
Worked example
A researcher wants to conclude that a new fertiliser increases crop yield nationally. She currently compares farms that already use it with farms that do not. What must change?
Figure (svg): A line showing the two additions needed to reach the desired conclusion
The desired conclusion has two parts: a causal claim, and a national scope.
Why: Each part requires its own form of randomness.
For the causal part, she must randomly ASSIGN farms to use the fertiliser or not.
Why: Currently farms chose for themselves, so the two groups may differ in soil, climate and management.
For the national part, she must randomly SELECT the farms from a national list.
Why: Currently the farms are whichever ones she happened to compare.
The fix-the-study variant is the clearest test of whether you hold the distinction, because it requires naming which randomness serves which purpose.
Verify: check both changes are needed: assignment alone would give cause among those farms only.
Why: Neither change substitutes for the other, which is the central point of the type.
Answer: she must randomly assign farms to the fertiliser AND randomly select them nationally
Fill the middle
Two questions, four answers.
Fill in the blanks
Assignment yes and selection yes supports cause in the wider population. Assignment yes and selection no supports cause among the subjects studied. Assignment no and selection yes supports association in the wider population. And neither supports association among the subjects studied.
Why: This four-row table is the complete content of the type. Every question is asking which row the described study occupies, and the correct answer is the strongest conclusion that row permits — no stronger, and no weaker.
Estimation
Rank the conclusions before looking at the study.
Predict first
Which conclusion is the STRONGEST claim, and therefore needs the most from a study design?
Correct: The treatment causes the effect in the general population
Why: Causation is stronger than association, and a claim about a general population is stronger than one about the study group. So the strongest claim needs both randomnesses — assignment for the causal part and selection for the population part. Ranking the choices by strength before examining the study is a useful habit, because the correct answer is the strongest claim the design actually supports.
Check
Look for who chose.
Check your understanding
A survey of 1,500 randomly selected town residents found that those who own dogs walk more each day. Which conclusion is best supported?
Answer: A
Why: Random selection from the town licenses generalisation to that town's residents. Nobody was assigned a dog, so only an association is available, and the scope stops at the town.
The two causal choices point in opposite directions, which is itself the argument: observational data cannot distinguish between them, so neither is available.
Section
Section 4
Trap
The trap. A careful, large, well-funded study finds that people who take a supplement live longer, and a choice concludes the supplement extends life.
Nobody was assigned to take it. People who choose supplements also tend to exercise more, eat better and visit doctors more often.
The study's quality and size are irrelevant to this: neither can separate the supplement from everything that travels with it.
The fix. Search the stem for randomly assigned before considering any causal choice.
If subjects arrived in their groups by their own choices, only association is available.
Trap
The trap. A study randomly SELECTS participants and a choice concludes the treatment caused the outcome.
Random selection makes the sample representative; it does nothing about which subjects received the treatment.
The word random appears in the stem, which makes the causal choice feel licensed when it is not.
The fix. Read the word after random: selected or assigned.
Selected licenses generalisation. Assigned licenses cause. They are not interchangeable.
Error analysis
A researcher's conclusion. The observation is sound and the inference is not.
Annotate
On: \( \text{observational: coffee drinkers report less fatigue} \;\Rightarrow\; \text{coffee reduces fatigue} \)
Note that the causal claim may well be true. The question is never whether a conclusion is plausible, but whether this study establishes it — and those are different questions.
Trap
The trap. A randomised experiment on 200 volunteers shows a real effect, and a choice concludes the treatment works for everyone.
Random assignment licenses the causal claim about those volunteers. It says nothing about who they represent.
Volunteers differ systematically from non-volunteers, often being healthier and more motivated.
The fix. Treat the two claims separately: cause is one question, scope is another.
Assignment answers the first; only selection answers the second.
Elimination
Researchers randomly selected 500 adults from a city and found those who volunteer report higher life satisfaction.
Eliminate the wrong options
Which conclusion is supported? Three can be eliminated from the design alone.
Survives elimination: a
Why: Random selection from the city licenses generalisation to the city's adults, and nothing wider. The absence of random assignment means only an association is available, in either direction. Notice how mechanical the eliminations are: two choices fail on the missing assignment and one on the scope of the selection, and none of it required reading the numbers.
Trap
The trap. Having learned that correlation is not causation, you reject every conclusion and choose nothing can be determined.
An observational study genuinely does support an association, and a randomised experiment genuinely does support a causal claim about its subjects.
Rejecting everything is as wrong as accepting too much, and the SAT offers a nothing-can-be-concluded choice precisely to catch this.
The fix. Identify the strongest claim the design DOES support, rather than the strongest it does not.
Every one of the four combinations supports something.
Counterexample
A claim about what large studies can show.
Discussion prompt
A student says: with enough data, an observational study proves causation. Give a concrete counterexample.
Hint: Think of an association everyone accepts is not causal.
Answer:
Counterexample: ice cream sales and drowning deaths. Across millions of records they rise and fall together with near-perfect consistency.
No amount of additional data would establish that ice cream causes drowning. The association is real and the explanation is elsewhere.
The confounder is hot weather, which independently increases both ice cream sales and swimming.
Why more data does not help: every additional record contains the same confounding. The sample gets more precise about an association that was never causal.
What would settle it: randomly assigning people to eat ice cream or not, and comparing drowning rates. Absurd here, and that absurdity is the point — the causal question requires an intervention the observational data never performed.
The general principle: size addresses precision, and only assignment addresses confounding.
Edge cases
Observational studies cannot establish cause. Is that the end of the matter?
Discussion prompt
Are observational studies useless for causal questions, and when is random assignment impossible?
Hint: Consider whether anyone could ethically run the experiment.
Answer:
They are not useless. Observational studies generate the hypotheses that experiments later test, and they are often the only evidence available.
Random assignment is frequently impossible or unethical. Nobody can randomly assign people to smoke for thirty years, or to grow up in poverty.
In such cases researchers strengthen observational evidence by controlling for known confounders, looking for dose-response relationships, and checking that the association holds across many different populations.
The smoking and lung cancer case is the classic example: the causal conclusion was eventually accepted on observational evidence, but it took decades, many independent studies, and a biological mechanism.
On the SAT, none of this applies. The test asks what a single described study supports, and the answer is governed strictly by the two randomnesses.
So the exam rule and the real-world picture differ, and it is worth knowing both — the exam rule is a simplification of genuine scientific caution, not an arbitrary convention.
Check
Name which randomness is missing.
Check your understanding
A researcher compares patients who chose acupuncture with those who did not, and wants to conclude that acupuncture reduces pain for the general population. What must be added?
Answer: A
Why: The desired conclusion has two parts. Reduces pain is causal and requires random assignment. For the general population is a scope claim and requires random selection. The current study has neither, so both must be added.
This variant is the clearest test of the distinction, because it forces you to name which randomness serves which half of the conclusion.
Section
Section 5
Matching
Six designs, six conclusions.
Match the pairs
Why: The last two rows both have large or convenient samples and neither supports generalisation, because neither used random selection. That is the same point as the bias rule in type 18, and it is why the two types are worth studying together.
Sorting
Each describes how subjects reached their groups.
Sort into buckets
Experiment or observational study?
The distinction is entirely about the mechanism of group formation. Size, care, cost and sophistication are all irrelevant to it.
Comparison
Fill the blanks from memory. This table is the whole type.
Comparison matrix
| random assignment | random selection | |
|---|---|---|
| What it decides | which group each subject goes into | who is in the study at all |
| What it licenses | a causal claim | generalisation to the population sampled |
| What it does NOT give | any claim about a wider population | any causal claim |
| Absent when | subjects chose their own group | subjects volunteered or were convenient |
The third row is the one worth rehearsing. Each randomness gives exactly one thing and explicitly does not give the other, which is why a study can be rigorous in one dimension and useless in the other.
Trade off
Fill in what each does and does not achieve.
Comparison matrix
| feature | what it buys | what it does not |
|---|---|---|
| Random assignment | a causal claim about the subjects | generalisation beyond them |
| Random selection | generalisation to the population | any causal claim |
| A larger sample | a narrower margin of error | validity, cause, or freedom from bias |
| A careful, expensive study | better data quality | any upgrade in what can be concluded |
The bottom two rows are what students most often get wrong. Size and quality are real virtues that change nothing about which KIND of conclusion a design supports.
Real world
One minute on why this distinction is worth more than 0.7 per cent.
Discussion prompt
Health headlines routinely report observational findings with causal language. How can a reader tell what a study actually showed?
Answer:
Look for the word assigned. A randomised controlled trial says participants were randomly assigned. An observational study says participants were followed, surveyed or compared.
The vocabulary of the report is a reliable tell. Observational findings are described as linked to or associated with; experimental findings say reduces or causes.
A famous reversal: observational studies found hormone replacement therapy was associated with lower heart disease. A later randomised trial found the opposite — the association came from healthier, wealthier women being more likely to receive it.
That case cost real lives, and it is the standard example of why the distinction is not academic pedantry.
The practical skill: when you read that something is linked to an outcome, ask who chose. If the participants chose, a confounder is always available as an explanation.
Which is exactly the SAT question, asked about a study described in three sentences.
Ranking
Order from weakest to strongest.
Put in order
Why: The weakest supports only an association among those studied. Adding random selection extends that association to a population. Random assignment instead gives a causal claim, which is a stronger kind of claim even though its scope is narrow. And both randomnesses together give a causal claim about a population, which is the strongest conclusion any study can support. Note the judgement in ranking (b) below (c): causation is treated as the stronger claim, since a narrow causal finding usually tells you more about mechanism than a broad association does.
Concept
This type is 0.7 per cent of the section — about one question — and it is a single distinction, which makes it the best return per hour on the entire test.
| session | what you do | why |
|---|---|---|
| 1 | Learn the two sentences: assignment licenses cause, selection licenses generalisation. | This is genuinely the whole type; twenty minutes is enough. |
| 2 | Classify fifteen described studies as experiment or observational, naming the deciding word. | Builds the reflex of searching for assigned. |
| 3 | For ten studies, write which of the four combinations it occupies and what it supports. | The four-row table is the answer to every question on this type. |
| 4 | Ten conclusion questions, eliminating choices on scope and on causal language before reading further. | Trains the mechanical elimination that makes these fifteen-second questions. |
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Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state what random assignment licenses and why, what random selection licenses and why, and what a study with neither can support.
Hint: Two sentences and a consequence.
Answer:
Random assignment licenses a causal claim, because it balances every other factor — known and unknown — between the groups, so the treatment is the only systematic difference left.
Random selection licenses generalisation to the population the sample was drawn from, because it makes the sample representative of that population.
Neither gives the other. An experiment on volunteers supports cause among volunteers; a random survey supports association in a population.
With neither, a study supports only an association among the subjects actually studied — the weakest of the four conclusions, and still a real one.
If you also said that size and care change nothing about which kind of claim is available, you have the version that resists every distractor on this type.
Explain it
Two minutes, out loud.
Discussion prompt
A friend sees the word random in a study description and assumes any conclusion is fine. How do you separate the two meanings for them?
Answer:
Tell them there are two different randomnesses, and they do completely different jobs.
Random SELECTION decides who is in the study. It makes the participants representative, so findings extend to the population they came from. It says nothing about cause.
Random ASSIGNMENT decides which group each participant goes into. It makes the groups comparable, so a difference in outcome can be blamed on the treatment. It says nothing about who they represent.
Give the two examples side by side. A perfect national survey with no assignment: generalisable, but only an association. A perfect experiment on volunteers: causal, but only about those volunteers.
Then the practical instruction: read the word immediately after randomly, and note which of the two is missing. The missing one tells you what the answer will not claim.
Commit first
Commit before you check.
Predict first
A study randomly selects 2,000 people nationally and observes that gym members have lower cholesterol. What is the strongest supported conclusion?
Correct: Gym membership is associated with lower cholesterol nationally
Why: Random selection is present, so the finding generalises to the national population. Random assignment is absent — nobody was assigned a gym membership — so only an association is available. The third choice understates by ignoring the random selection, and the second over-claims by ignoring the missing assignment. This is the selection-without-assignment row of the four-combination table.
Connect it up
Blank paper — and this one genuinely fits.
Draw it
Draw a two-by-two grid. Label the rows random assignment yes and no, and the columns random selection yes and no. Fill each of the four cells with what that combination supports: cause in the population, cause among the subjects, association in the population, association among the subjects. Beside the grid write the two sentences that generate it: assignment balances other factors so it licenses cause, and selection makes the sample representative so it licenses generalisation. Underneath, draw an example of a confounder — three circles, with an arrow from a third factor to each of the two observed quantities and no arrow between them. At the bottom write: size and care change nothing about which cell you are in.
Exit ticket
One question before you close the deck.
Predict first
A study reports that participants were randomly selected but not randomly assigned. Which claim is available?
Correct: An association, generalisable to the population sampled
Why: Random selection makes the sample representative, so whatever is found extends to the population it was drawn from. Random assignment is absent, so the groups formed themselves and confounders cannot be ruled out — meaning only an association is available. Both causal choices require assignment, and rejecting everything is the over-correction: a well-drawn observational study genuinely does establish a generalisable association.
Recap
One type, one distinction, two sentences.
| never do this | do this instead |
|---|---|
| Conclude cause from an observational study | Say associated with, and name a possible confounder |
| Treat random selection as licensing cause | Read the word after randomly: assigned or selected |
| Generalise an experiment on volunteers | Keep the causal claim and narrow the population |
| Argue a huge study proves causation | Size buys precision, not validity |
| Answer nothing can be concluded by default | Identify the strongest claim the design supports |
| Judge a study by how careful it sounds | Judge it by how subjects reached their groups |
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