Decide when a study supports causation

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Difficulty: Beginner
Estimated time: 26 minutes
Domains: Problem Solving and Data Analysis
Techniques: Random-assignmentObservational-studiesComparison-groupsCausationSelf-selection

What you’ll learn

Spot questions that are really about cause and effect.
Tell an experiment from an observational study.
Explain why random assignment lets a study show cause.
See why a comparison group alone doesn’t prove cause.
See how volunteering and self-selection limit what a study can show.
Keep random assignment and random sampling apart.

Prerequisites

You’re ready. No earlier Aniko lesson is required.

Why this matters on the SAT

Look at how the groups were formed

A study can find a real difference between two groups and still not prove what caused it. On the SAT, two words can flip the answer: randomly assigned.

Solution to the example

Start with how the groups were formed. The volunteers were randomly assigned, so chance decided who got the routine. That’s what lets you say the routine caused the difference, which rules out D.

Now check how far the conclusion can reach. The result is an average, so it doesn’t promise that every reader improved. That rules out A. And the people were volunteers, not a random pick from the community, so the study can’t speak for everyone there. That rules out C.

Choice B is correct. Its wording, “assignment to the routine caused,” can sound stiff. It means that in this experiment, being put in the routine group led to more pages read on average.

SAT example

Sixty volunteers joined a study of a new reading routine. The researchers randomly assigned $30$ volunteers to use the routine for $4$ weeks and the other $30$ to continue their usual reading habits. At the end of the study, the routine group read more pages per week on average.

Which conclusion is best supported?

A
The routine causes every reader to read more pages per week.
B
For the volunteers in the study, assignment to the routine caused a higher average number of pages read per week.
C
The routine causes all people in the volunteers’ community to read more pages per week.
D
The study shows an association, but no causal conclusion is justified.

Spot a study-design question

These questions describe a study, then ask something like:

Which conclusion is best supported?
Did the treatment cause the outcome?
Is the relationship causal, or only an association, meaning the two things tend to go together?
How could the study be changed to test cause and effect?

Some questions only look like these. If a question just asks you to describe an upward or downward trend in a scatterplot, you don’t need to judge cause. A trend can exist without one thing causing the other.

Others ask which larger group a sample represents. That’s about random sampling, which has its own lesson. For now, you only need one contrast:

\boxed{\text{Random assignment is about cause. Random sampling is about population.}}

In other words, random assignment decides whether you can say “caused.” Random sampling decides who the result can speak for.

Check your understanding:

A scatterplot shows that students who sleep longer tend to report better concentration. The question asks whether the longer sleep caused the better concentration. Is the upward trend enough to say yes?

What random assignment changes

In an experiment, the researchers decide which treatment each person gets, then compare the groups’ results. In an observational study, they don’t decide who gets what. People have already made their choices, or already have their traits, and the researchers record what happens.

A comparison group helps in both designs. Random assignment is what makes the causal difference.

Both designs have two groups. What differs is who formed them.

Why a comparison alone can’t prove cause

Picture a free Saturday SAT workshop. At the end of the term, students who signed up score higher than students who didn’t. Did the workshop help?

Maybe. But think about who signs up for a Saturday workshop. Those students may be more motivated, or have more free time to study. They might have scored higher even without the workshop. So the study can’t separate the workshop’s effect from the differences that came with the people who chose it.

An outside difference like motivation, one that could also explain the result, is called a confounding variable. When people pick their own groups, you can’t rule one out. So the most you can say is that the two things go together. SAT answer choices usually put it like this:

\boxed{\text{The variables are associated, but the study does not establish causation.}}

Why random assignment supports causation

Now run the study differently. Take the students who want to join, and let a coin flip decide who goes to the workshop. That’s random assignment: chance, not the students, picks each person’s group.

The motivated students now land in both groups, roughly half in each. So do the students with busy schedules, the strong test takers and the ones who sleep too little. The groups won’t match perfectly, but chance doesn’t favor either side, so no outside difference is likely to pile up in one group.

That leaves the workshop as the only difference the study builds in. If the workshop group then scores higher on average, that supports a causal conclusion for the students in the study.

It all comes down to one question: who picked the groups? If chance picked, the study can show cause. If people picked for themselves, it shows only an association.

Common mistake:

It’s tempting to see a treatment group and a comparison group and decide the study proves cause. But if people chose their own group, the groups may have differed before the treatment even started. Find the sentence that says how people got into the groups. If chance didn’t decide, stop at association.

What a comparison group can and can’t do

A comparison group is the group you measure the treatment group against. It might get no treatment, the usual treatment or a different treatment.

It might also get a placebo. That’s something that looks like the treatment but leaves out the part that’s supposed to work, like a sugar pill. When the comparison group gets no treatment, the usual treatment or a placebo, it’s often called a control group.

Why do you need one? Say students use a new study app for a month, and their scores go up. Maybe the app worked. Or maybe they’d have improved anyway, from a month of classes and practice. Without a group that skipped the app, you can’t tell.

So a strong experiment needs both pieces, and each does a different job:

The comparison group gives you a difference to measure.
Random assignment lets you say the treatment caused it.

Check your understanding:

A fitness center compares members who chose a new class with members who chose their usual workouts. The class group improved more. Does having the usual-workout group make the causal conclusion valid?

Example: Causation, but only for the volunteers

Put everything together, and a question like this comes down to three checks, in this order:

Who was in the study? The conclusion covers those people. It reaches a larger group only if they were randomly sampled from it.
Who picked the groups? If chance did, the conclusion can say “caused,” “effect” or “because of.” If people chose, or the researchers only observed, stop at association.
Average or every? A study that compares averages can’t promise the result for every person.

Here are all three on one problem.

Worked example

Researchers recruited $120$ volunteers from one community college to study a weekly digital planner. The volunteers were randomly assigned to use either the digital planner or the college’s usual planning guide for $8$ weeks. Students assigned to the digital planner completed more assignments on average.

Which conclusion is best supported?

A
Every student at the college would complete more assignments by using the digital planner.
B
The digital planner is associated with assignment completion, but no causal conclusion can be made for the volunteers.
C
For the $120$ volunteers, assignment to the digital planner caused a higher average number of completed assignments than assignment to the usual guide.
D
The digital planner causes all community-college students to complete more assignments.

Step 1

Check 1: Who was in the study?

The study used $120$ volunteers from one college. So the conclusion is about them. It can’t quietly grow to cover every student at that college, let alone all community-college students.

Whether the planner caused anything is a separate question. That’s the next check.

Step 2

Check 2: Who picked the groups?

Randomly assigned means chance put each volunteer into one of the two groups. So this is a randomized experiment, and the conclusion is allowed to say “caused,” for these volunteers.

Step 3

Check 2, continued: Compared with what?

The comparison group used the college’s usual planning guide. Both groups had the same $8$ weeks, and only the planning tool was different.

That tells you what “caused” means here: the planner beat the usual guide. It says nothing about how the planner compares with using no plan at all.

Step 4

Check 3: Average or every?

The result is about the average number of completed assignments. An average can go up even if some students didn’t improve. So the right conclusion says “on average,” not “every student.”

Step 5

Pick the choice that passes all three

Choice C passes all three:

Who was in the study? C stays with the $120$ volunteers.
Who picked the groups? Chance did, so C can say “caused.”
Average or every? C describes a higher average, not a promise for every student.

Each wrong choice fails at least one check. A and D reach past the volunteers, and both promise the result for every student. B turns down the causal claim that random assignment supports.

Check your understanding:

Suppose the college had first randomly selected the $120$ students from all enrolled students, and every selected student took part before the random assignment. What more could you conclude?

Common mistake:

Some students see the word volunteers and decide no causal conclusion is possible. But volunteering only limits who the result covers. It doesn’t undo the random assignment inside the study. Keep two checks apart: who was in the study, and who picked the groups.

Two jobs for chance: sampling and assignment

Chance can do two different jobs in a study. To see both, go back to the Saturday workshop. In each version below, suppose the workshop group scores higher on average.

Random sampling decides who is in the study. If the school picks students at random from its full list, the results can speak for the whole school, not only for the students who took part.

Random assignment decides who gets which treatment. If a coin flip decides which students attend, you can say the workshop caused the difference.

A study can use one job, both or neither:

Volunteers, then a coin flip: the workshop caused higher scores on average, but only for these volunteers.
A random sample, but students choose whether to attend: attending and higher scores go together across the whole school, but you can’t say the workshop caused it.
A random sample, then a coin flip: the workshop caused higher scores on average, and that average effect can reasonably extend to the whole school.
Volunteers who choose whether to attend: attending and higher scores went together for these students. That’s all you can say.

None of these limits means the workshop doesn’t work, or that an association is meaningless. They only mark how far this study’s evidence reaches.

Common mistake:

A huge sample can feel like proof of cause, but it isn’t. More people make an estimate more precise, but they can’t create random assignment. If thousands of people chose their own groups, the study still shows only an association. A bigger study gives a sharper picture of that association, not a cause.

Choose the conclusion by hand

You won’t need Desmos for these, even when the question shows a percent or an average. A calculator can’t tell whether chance picked the groups, what else might explain a difference, or which population the results cover.

These are reading questions, and the three checks are the whole method: who was in the study, who picked the groups, and average or every.

Practice problems

Each problem is a little harder than the last. Run the three checks on every answer choice.

People chose their own group

Practice problem

A library offered an optional weekly writing circle. At the end of $10$ weeks, members who chose to attend the circle had completed more pages of writing on average than members who did not attend.

Which conclusion is best supported?

Calculator loads as you approach

Look for the sentence that says how members ended up in each group.

Volunteers, randomly assigned

Practice problem

Ninety volunteers at a language center entered a study. Researchers randomly assigned half to receive a daily vocabulary reminder and half to receive no reminder. After $5$ weeks, the reminder group correctly answered more vocabulary questions on average.

Which conclusion is best supported?

Calculator loads as you approach

Keep two checks apart here: who was in the study, and who picked the groups.

Picked, then joined, then assigned

Practice problem

Researchers randomly selected $240$ residents from a city directory for a study of a meal-planning website. Of those selected, $168$ agreed to participate. The researchers randomly assigned $84$ participants to use the website for $6$ weeks and the other $84$ to use a printed planning sheet. The website group prepared more meals at home on average.

Which conclusion is best supported?

Calculator loads as you approach

Your work here is saved. Read the whole study before you look at the choices.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Run three checks on every answer choice: who was in the study, who picked the groups, and average or every.
If chance picked the groups, the study can show cause. If people picked for themselves, or the researchers only observed, it shows only an association.
A comparison group gives you a difference to measure. Without random assignment, it doesn’t prove cause.
Volunteers can still be randomly assigned. Volunteering limits who the result covers, not whether it shows cause.
Random assignment is about cause. Random sampling is about population. And a result about averages is never a promise for every person.

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