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L2-07 · Read a causal graph

A causal graph is a map of what we think causes what. Three small shapes appear again and again: a chain, a fork, and a collider. Looking only inside one group can hide a link in a chain or a fork, but it can create a fake link at a collider.

Level 2 icon Level 2 · Grades 7–9 · about 45 minutes
Big idea. A causal graph is a map of what we think causes what. Three small shapes appear again and again: a chain, a fork, and a collider. Looking only inside one group can hide a link in a chain or a fork, but it can create a fake link at a collider.
Before this lesson. L1-06, L2-04
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: Three shapes, one surprise

Leo is late for school on a rainy morning. Rain makes the road wet, and a wet road slows the traffic. “That is a chain,” says Maya: rain -> wet road -> slow traffic. In this simple map, if you already know the road is wet, hearing that it rained tells you nothing new about the traffic.

Next, Maya points at a news chart. In summer, ice cream sales rise, and so do sunburns. Neither one causes the other. Summer is a common cause, so the map is a fork: ice cream <- summer -> sunburn. If you compare only July days, most of the link between ice cream and sunburn fades away.

Then their friend Priya talks about an imagined arts school that is hard to get into. Maya invents a test with made-up numbers. There are 150 made-up applicants, each with a talent score and a hard work score from 1 to 10. The two scores are random and unrelated. The school admits a student only if talent plus hard work is above 13. That rule lets in 46 admitted students.

Among those 46 admitted students, a pattern appears. The ones with less talent tend to have more hard work. Why? A student with low talent only gets in by working very hard. A very talented student can get in with less work. The rule picks students by a collider: talent -> admitted <- hard work.

“So the pattern comes from the choosing, not from the students,” says Leo. Maya nods. “And every map we drew today is a guess. Data can support a map or argue against it. The arrows never appear on their own.”

A chain, a fork, and a collider, then 150 made-up applicants chosen by a collider.

What the animation shows. First, three maps appear with dots moving along the arrows. The chain runs rain -> wet road -> slow traffic. The fork has summer pointing to ice cream and to sunburn. The collider has talent and hard work both pointing into admitted. Next comes a scatter plot of 150 made-up applicants. Talent is across and hard work is up, and the dots show no trend. A dashed line marks where talent plus hard work is above 13. The 46 admitted students stay bright and the rest fade. A line through the admitted students slopes down: among them, less talent goes with more hard work.

Hands-on: Sort the stories, then play the admissions game

About 20 minutes.

You need:

Steps:

  1. Copy one story onto each card. (a) A storm cuts the power, the fridge warms up, and the milk spoils. (b) Good grades or a sports award can each win a prize.

  2. (c) Hot weather raises fan sales and also raises pool visits. (d) In children, age affects both shoe size and reading level. (e) Missing the bus makes you late, and being late means missing the warm-up.

  3. (f) A dead battery or a broken bulb can each make a flashlight stay dark. (g) Great food or a great view can each make a cafe busy. (h) Practice builds skill, and skill wins games. (i) A festival fills the hotels and jams the roads.

  4. Sort the cards into three piles: chain, fork, and collider. Draw each story as boxes and arrows.

  5. Now the admissions game. Flip the coin for talent: heads scores 3 and tails scores 0. Roll the die for hard work, from 1 to 6.

  6. Add the two scores. A total of 6 or more means the student is admitted. Play 30 students and record talent, hard work, and admitted or not.

  7. For all 30 students, find the average die roll for heads and for tails. Then do the same using only the admitted students.

  8. Compare the two pairs of averages. Among the admitted, which talent group needed more hard work?

Practice together

In the admissions game, work out exactly which rolls get a student admitted. What is the average hard work score of admitted students for tails and for heads?

Check your answer

Tails gives 0, so the student needs a 6 on the die. Heads gives 3, so the student needs 3, 4, 5, or 6. Among admitted students, tails students average 6 and heads students average 4.5. Among everyone, both groups average 3.5, because the coin and die are unrelated. The admission rule made the link.

Try it yourself

1. Exercise improves fitness, and fitness lowers resting heart rate. Which shape is this? In this map, if you compare only people with the same fitness, does exercise tell you more about heart rate?

Check your answer

It is a chain: exercise -> fitness -> heart rate. In this simple map, once you know fitness, exercise adds no new information about heart rate. Real bodies may have other paths, so the map is a guess to test.

2. A website lists only cafes with at least 4 stars. Stars go up with tasty food and with low prices. Among the listed cafes, cheap ones seem to have worse food. What could explain this?

Check your answer

Stars are a collider: tasty food -> stars <- low prices. A cheap cafe can reach 4 stars with ordinary food, but an expensive one needs great food. Listing only 4-star cafes can create the pattern even if price and taste are unrelated.

Talk about it

Check yourself

1. Which map is a fork?

Show the answer

C. ice cream <- summer -> sunburn

In a fork, one cause points to two effects. Summer affects both ice cream sales and sunburns.

2. In the made-up admissions data, among the 46 admitted students, less talent went with more hard work. In all 150 applicants, the two were unrelated. What is the best explanation?

Show the answer

A. Choosing students by talent plus hard work created the pattern.

Admission is a collider. Picking only students above the cutoff means low talent must be balanced by high hard work, so a downward pattern appears.

3. Remember Lesson 3. In an imagined town, months with more sunscreen sold also have more sunburns. What does the scatter plot alone show? (Review of L2-03)

Show the answer

B. Sunscreen sales and sunburns rise together in this data.

A scatter plot shows an association. Here sunny weather is a likely common cause of both, and the plot cannot tell which way causes run.

Words to know

causal graph
A drawing of boxes and arrows. Each arrow says we think one thing directly affects another.
collider
A box that two arrows point into, as in talent -> admitted <- hard work.
condition on
To look only at the records in one group, such as only admitted students or only July days.
For grown-ups: about this lesson

Why it matters. The lesson introduces the three building blocks of causal diagrams: chains, forks (common causes) and colliders. Pearl’s overview explains how conditioning on the middle of a chain or fork blocks an association, while conditioning on a collider can open one Pearl, 2009. Bareinboim’s causal AI textbook builds on the same graphs Bareinboim, 2025. Gopnik and colleagues argue that young children learn causal maps of this kind from patterns in data and from their own interventions Gopnik et al., 2004. The lesson presents graphs as working guesses because data can support or challenge a graph but cannot draw it alone. The collider scatter uses the course’s seeded generator, so the 150 applicants and the 46 admitted match the text.

Watch for these mix-ups.

  • Looking inside a group always makes the picture clearer. (Conditioning on a collider can create a link that is not there in the whole population.)

  • If two things are linked among admitted students, one must cause the other. (The selection rule itself can produce the link.)

  • A causal graph is read straight off the data. (Arrows are assumptions; data can support or challenge a graph, but cannot draw it alone.)

  • In a fork, the two effects cause each other. (They share a cause; neither needs to affect the other.)

To make it easier. Skip the card sort and draw only the three story graphs. Play the admissions game with 12 students and compare only the admitted ones.

To make it harder. Pool the class’s admissions games, or list all 12 coin and die outcomes and work out the exact averages. Then change the cutoff to 4 or 5 and predict how the pattern changes before checking.

Limits. The chain, fork and collider stories are simplified maps. Real situations usually have more causes and more arrows, and the collider data are made up.

On the computer. Grown-ups can see where this idea leads later in Lab: Confounding and intervention, written for older learners.

Sources for this guide. Pearl, 2009Bareinboim, 2025Gopnik et al., 2004

Next: L2-08 · Observe or intervene

References
  1. Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(none), 96–146. 10.1214/09-ss057
  2. Bareinboim, E. (2025). Causal Artificial Intelligence: A Roadmap for Building Causally Intelligent Systems. https://causalai-book.net/
  3. Gopnik, A., Glymour, C., Sobel, D. M., Schulz, L. E., Kushnir, T., & Danks, D. (2004). A Theory of Causal Learning in Children: Causal Maps and Bayes Nets. Psychological Review, 111(1), 3–32. 10.1037/0033-295X.111.1.3