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L1-06 · Look for a common cause

Two things can go up and down together even when neither one causes the other. Often a third thing causes both. Looking only at the pattern cannot tell you which way the causes run.

Level 1 icon Level 1 · Grades 4–6 · about 40 minutes
Big idea. Two things can go up and down together even when neither one causes the other. Often a third thing causes both. Looking only at the pattern cannot tell you which way the causes run.
Before this lesson. L0-09, L1-04
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: Does ice cream make people swim?

Maya’s cousin Sam runs an ice cream stand next to the town pool. Sam keeps a notebook. Each day, Sam writes how many ice creams were sold and how many people went swimming. The numbers in this story are made up, but they look like real ones.

Leo reads the notebook for 10 made-up days. On days with lots of ice cream sold, lots of people swam. On days with little ice cream sold, few people swam. “Ice cream makes people want to swim!” says Leo.

Maya points to another column in the notebook: hot or cool. There were 5 hot days and 5 cool days. On the hot days, Sam sold 40 to 56 ice creams, and 38 to 49 people swam. On the cool days, Sam sold 12 to 26 ice creams, and only 10 to 16 people swam.

“Look only at the hot days,” says Maya. “Selling more ice cream does not go with more swimmers. The cool days are the same. The big pattern comes from the weather.”

Leo draws arrows. Hot weather points to ice cream sold. Hot weather also points to people swimming. “So hot weather is a common cause,” he says. “Giving out free ice cream on a cold day would not fill the pool.”

Ice creams sold and people swimming on 10 made-up days. Coloring the hot and cool days shows where the pattern comes from.

What the animation shows. Ten dots, one for each of 10 made-up days, show ice creams sold and people swimming. Together they rise from the lower left to the upper right. Then the dots turn orange for 5 hot days and blue for 5 cool days, and a flat line runs through each color. Last, a diagram appears: hot weather has one arrow to ice cream sold and one arrow to people swimming.

Hands-on: Find the hidden cause

About 20 minutes.

You need:

Steps:

  1. These numbers are made up for practice. Each pair is ice creams sold, then people swimming.

  2. Hot days: 40 and 46. 44 and 38. 48 and 49. 52 and 40. 56 and 45.

  3. Cool days: 12 and 13. 16 and 10. 19 and 16. 22 and 10. 26 and 13.

  4. Draw two lines for a graph. Label the bottom line ice creams sold, from 0 to 60. Label the side line people swimming, from 0 to 60.

  5. Make a pencil dot for each of the 10 days. Do not color them yet. Do the dots go up from left to right?

  6. Now color the dots for hot days orange and the dots for cool days blue.

  7. Look at the orange dots alone. Do more ice creams go with more swimmers? Then look at the blue dots alone.

  8. Draw a common-cause diagram. Write hot weather at the top. Draw one arrow to ice cream sold and one arrow to people swimming.

Practice together

On rainy days, more people carry umbrellas, and the roads are wet. Do umbrellas make roads wet? What is the common cause?

Check your answer

No. Rain is the common cause. Rain makes people carry umbrellas, and rain makes the roads wet. The diagram has rain at the top, with one arrow to umbrellas and one arrow to wet roads.

Try it yourself

1. In a made-up town, kids with bigger feet can read longer words. Does having big feet help you read? What third thing could cause both?

Check your answer

No. Age is a likely common cause. Older kids have bigger feet, and older kids have had more years to learn to read.

2. Leo notices that on days when many people wear shorts, many people buy cold drinks. Draw a common-cause diagram for this.

Check your answer

Hot weather goes at the top. One arrow goes from hot weather to wearing shorts. Another arrow goes from hot weather to buying cold drinks.

Talk about it

Check yourself

1. On hot days, more ice cream is sold and more people swim. What is the best explanation?

Show the answer

B. Hot weather is a common cause of both.

Hot weather makes people want ice cream, and it makes people want to swim. So the two go up together, even if neither causes the other.

2. Which diagram shows hot weather as a common cause?

Show the answer

B. Hot weather points to ice cream, and hot weather points to swimming.

In a common-cause diagram, both arrows start at the cause. Here they start at hot weather and point to the two things it changes.

3. Remember Lesson 4. Of 10 made-up days, 5 were hot. The pool was full on 4 of the hot days. Of the hot days, what fraction had a full pool? (Review of L1-04)

Show the answer

B. 4 out of 5

The question says of the hot days, so the total is the 5 hot days. The pool was full on 4 of them, so it is 4 out of 5.

Words to know

common cause
One thing that causes two other things. It can make those two things go up and down together.
scatter plot
A graph with one dot for each thing you measured, like one dot for each day.
diagram
A simple drawing that shows how things are connected. Here, an arrow points from a cause to what it causes.
For grown-ups: about this lesson

Why it matters. Pearl’s overview separates association from causation and uses causal diagrams in which a common cause, a confounder, opens a path that makes two variables associated even when neither affects the other Pearl, 2009. OpenStax calls such a hidden third variable a lurking variable and explains why random assignment in an experiment guards against it Barbara Illowsky and Susan Dean, 2023. Gopnik and colleagues give evidence that even young children learn causal structure from patterns in what they observe and do, which this lesson builds on with simple arrow diagrams Gopnik et al., 2004. The 10 days in the lesson are made up and were built so that, within hot days and within cool days, ice cream sold and swimmers are not related.

Watch for these mix-ups.

  • If two things rise together, one must cause the other. (A third thing may cause both; the pattern alone cannot say.)

  • A common cause means the two things are not really related. (They really do go together; they are just not cause and effect.)

  • The arrow in a diagram is a proof. (An arrow is a guess about causes that still needs evidence, such as a fair test.)

  • Splitting the data into hot and cool days always removes a pattern. (It does here because the example was built that way; real data can keep part of the pattern.)

To make it easier. Plot only 6 days, 3 hot and 3 cool, and talk about the diagram before drawing it.

To make it harder. Ask the child to invent a made-up table where splitting by a third thing does not remove the pattern, and to explain what that might mean. Then look at a real weather record and a pool or park attendance record if one is available.

Limits. The data are made up to make the idea clear. Real data are messier, and spotting a likely common cause is a guess that still needs testing.

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, 2009Barbara Illowsky and Susan Dean, 2023Gopnik et al., 2004

Next: L1-07 · Track a wave

References
  1. Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(none), 96–146. 10.1214/09-ss057
  2. Barbara Illowsky and Susan Dean. (2023). 1.4 Experimental Design and Ethics. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/1-4-experimental-design-and-ethics
  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