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L2-03 · Read plots without inventing causes

A scatter plot shows how two quantities vary together. You can describe its direction, form and strength, and spot points that do not fit. The plot alone cannot tell you what causes what, which way a cause runs, or what happens outside the range of the data.

Level 2 icon Level 2 · Grades 7–9 · about 45 minutes
Big idea. A scatter plot shows how two quantities vary together. You can describe its direction, form and strength, and spot points that do not fit. The plot alone cannot tell you what causes what, which way a cause runs, or what happens outside the range of the data.
Before this lesson. L2-01, L2-02
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: Does sunscreen cause sunburn?

Maya finds a table of data for an imagined town, with 12 made-up months. For each month it lists the number of sunscreen bottles sold and the number of sunburn cases at the town clinic. In January the town sold 20 bottles and saw 3 cases. In July it sold 160 bottles and saw 26 cases.

She makes a scatter plot with one dot per month. Then she describes it the way a statistician would. The direction: positive, because more bottles go with more cases. The form: linear, because the dots lie close to a straight line. The strength: strong, because they hug the line tightly. No point sits far from the rest, so there is no outlier.

Leo grins. “So sunscreen causes sunburn!” Maya draws a question mark between the two quantities. There are at least three stories. Maybe people with sunscreen stay out longer. Maybe people buy sunscreen after they get burned. Or maybe a common cause drives both: more sunshine means more sunscreen sold and more sunburns. The plot could look the same in all three stories.

To decide between stories you need more data, such as hours of sunshine each month. In this made-up town, sunshine is high in exactly the months when both numbers are high. That fits the common cause story. It still does not prove it, but it shows why the plot alone cannot settle the question.

Then Kai brings his own trap. His made-up growth record says he was 50 cm long at birth and 86 cm at age 2. That is 18 cm per year. A straight line says height = 50 + 18 times age. At age 14 the line gives 50 + 252 = 302 cm, a teenager over 3 meters tall! The line fits the first two years, but using it outside the data is extrapolation, and here it fails badly.

A scatter plot of sunscreen sold and sunburn cases in an imagined town, a hidden common cause, and a trend line pushed past the data.

What the animation shows. Twelve dots appear, one for each of 12 made-up months in an imagined town. Bottles of sunscreen sold go across and sunburn cases go up. A straight trend line is drawn through them. Three labels appear: direction: positive, form: linear, strength: strong. Then two boxes, sunscreen sold and sunburn cases, appear with a question mark between them: which causes which? A third box, sunshine, appears above with arrows to both. One cause can drive both. Last, the trend line is pushed far to the right, past the biggest month. That part turns red with the words outside the data: no evidence here.

Hands-on: Plot it, then doubt it

About 20 minutes.

You need:

Steps:

  1. Draw axes. Put sunscreen bottles sold across, from 0 to 200, and sunburn cases up, from 0 to 40. Label both axes.

  2. Plot the made-up months January to June as (bottles, cases): (20, 3), (30, 5), (50, 7), (80, 13), (110, 16), (140, 25).

  3. Plot July to December: (160, 26), (150, 24), (100, 13), (60, 10), (30, 4), (20, 2).

  4. Describe the plot in one sentence each for direction, form and strength. Is any dot far from the others?

  5. Sketch a straight line with your ruler that follows the dots, with about as many dots above it as below.

  6. Write three possible explanations: sunscreen leads to sunburn, sunburn leads to buying sunscreen, and a common cause. Draw each as boxes and arrows.

  7. For each explanation, write one piece of extra data that could help check it.

  8. Shade the range of bottles your data covers (20 to 160). Use your line to predict cases for 120 bottles. Then extend the line with your ruler to 200 bottles, the edge of your grid, and read off a prediction there.

  9. Say which of your two predictions you trust less, and why. What might change in a month with far more sun than any month in the data?

Practice together

A made-up plot shows 20 students. Across is hours of screen time in the evening, and up is hours of sleep. The dots fall from upper left to lower right, close to a line. Describe the plot, and give one reason it might not show that screens cut sleep.

Check your answer

Direction: negative, since more screen time goes with less sleep. Form: roughly linear. Strength: fairly strong, since the dots stay near the line. It might not be cause, because a common cause such as homework or stress could keep students up and on screens. The cause could also run the other way: students who cannot sleep may pick up a screen.

Try it yourself

1. In a made-up survey of students aged 6 to 14, bigger shoe sizes go with higher reading scores. Name a common cause.

Check your answer

Age. Older students tend to have bigger feet and also read better. Shoe size does not cause reading skill, and reading does not grow feet.

2. A made-up record shows an ice cube’s mass dropping by about 2 grams a minute over its first 5 minutes in a warm room. It started at 30 grams. What does the line predict at 20 minutes? Why should you doubt it?

Check your answer

The line gives 30 - 2 times 20, which is -10 grams. A mass cannot be negative. The trend only covered 5 minutes, so using it at 20 minutes is extrapolation, and the cube will simply melt away.

Talk about it

Check yourself

1. In the imagined town, a strong positive scatter plot of sunscreen sold and sunburn cases, by itself, shows that:

Show the answer

B. The two numbers tend to rise and fall together.

A scatter plot shows how two quantities vary together. It cannot tell you which story about causes is true. Sunshine is a good guess, but checking it needs more data.

2. Kai’s made-up growth line fits ages 0 to 2. Using it to predict his height at age 14 is:

Show the answer

B. Extrapolation, which is risky

Age 14 is far outside the data, so it is extrapolation. Growth slows down, and the line gives a silly 302 cm.

3. Remember Lesson 1. Where does the line y = 2x + 3 meet the y axis? (Review of L2-01)

Show the answer

C. At y = 3

When x = 0, y = 2 times 0 plus 3, which is 3. The constant tells you where the line starts. The 2 is how much y goes up for each step of 1 across.

Words to know

scatter plot
A graph with one dot for each record, placed by two measured quantities, one on each axis.
common cause
A third thing that pushes two other quantities up or down together, so they move together even if neither affects the other.
extrapolation
Using a trend to predict far outside the range of the data you have. It is risky, because the pattern may change.
For grown-ups: about this lesson

Why it matters. OpenStax describes the relationship in a scatter plot by its direction (positive or negative), form (linear or not) and strength Barbara Illowsky and Susan Dean, 2023. Common Core asks learners to construct and interpret scatter plots and describe patterns, including outliers, in 8.SP.1, and to distinguish correlation from causation in S-ID.9 NGA Center and CCSSO, 2010. In Pearl’s framework, a shared cause (a confounder) can produce an association between two variables even when neither affects the other, and the data alone cannot choose among the causal stories Pearl, 2009. The extrapolation example (a growth line giving a 302 cm teenager) shows that a fitted line is supported only inside the range of the data.

Watch for these mix-ups.

  • A strong correlation shows cause. (Strength says how tightly dots follow a line; it says nothing about why.)

  • If A and B rise together, A must cause B. (B could cause A, or a common cause could drive both.)

  • A line that fits well works everywhere. (It is supported only inside the range of the data; outside it, the pattern may change.)

  • Positive means good. (Positive only means both quantities tend to increase together.)

To make it easier. Plot only 6 of the 12 months, describe the direction, and draw just the common cause diagram.

To make it harder. Make up monthly sunshine hours for the imagined town and plot sunscreen against sunshine and sunburn against sunshine. Then look up a real pair of quantities in a public data table and write three possible explanations for any pattern you see.

Limits. The town, its sales, its clinic cases and Kai’s growth record are all made up. Nothing here is health advice, and the lesson does not suggest that sunscreen causes sunburn; the point is that a plot alone cannot tell.

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. Barbara Illowsky and Susan Dean, 2023NGA Center and CCSSO, 2010Pearl, 2009

Next: L2-04 · Calculate conditional probability

References
  1. Barbara Illowsky and Susan Dean. (2023). 12.2 Scatter Plots. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/12-2-scatter-plots
  2. NGA Center and CCSSO. (2010). Common Core State Standards for Mathematics. Common Core State Standards Initiative. https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf
  3. Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(none), 96–146. 10.1214/09-ss057