At a glance. One lesson of about 60 minutes. Correlation versus causation, lurking variables, biased and fair samples, and what a graph alone can justify.
Calculator: no calculator. No calculator is needed: the arithmetic is percentages of simple numbers and differences.
What students will learn
Know
- Correlation does not imply causation.
- A lurking variable can explain a link.
- A biased sample can mislead whatever r is.
Understand
- A model is only as good as its data.
- A strong correlation is evidence of a relationship, not proof of cause.
Do
- Name a lurking variable.
- Identify bias in how data were collected.
- Say what a scatter graph can and cannot justify.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: percentages and differences in the sample questions (questions 2, 3, 5 and 7).
- Criterion B. Investigating patterns: students use the split by age to test an explanation.
- Criterion C. Communicating: each claim is stated with its evidence and its limit.
- Criterion D. Applying mathematics in real-life contexts: headlines, surveys, public health claims.
The one thing to do with the figure
Ask for the explanation before pressing the button. Most students say age or growth. Then press it and let them see r = 0.00 inside each age.
Ask: if age is the explanation, what would a fair study need to hold fixed? This is the bridge to designing a fair study.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the headline. |
| 12 min | The figure; explain the link. |
| 12 min | Worked examples; fire engines in pairs. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | A strong negative linear correlation; 90; the centre of the data. |
|---|---|
| Worked examples | Hot weather is the lurking variable. The sports-hall sample is biased. |
| Faded example | The size of the town or fire. |
| 1. Interpreting a correlation | A. B jumps to a cause; C misreads r. |
| 2. Percentage in a sample | 85 per cent. |
| 3. Applying a biased sample | 765, an unreliable claim because the sample is biased. |
| 4. Naming a lurking variable | C. A is not linked to crime; B jumps to a cause. |
| 5. Percentage in a sample | 12.5 per cent. |
| 6. Choosing a fair sample | A. B and C pick pupils by how they travel. |
| 7. Comparing group means | 32 points. |
| 8. What a graph alone can justify | B. A and C both claim cause. |
| 9. Spotting the flaw | C. A treats correlation as cause; B misreads r. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B or C in question 1 | Jumps to cause or misreads r. | Ask: what else could make both bigger? |
| 15, 51 or 9 in question 2 | Reports the complement or a count. | Write part ÷ whole × 100. |
| 51, 85 or 7650 in question 3 | Reports the sample, the percentage or forgets 0.9. | Ask what 90% is as a decimal. |
| A or B in question 4 | Picks an irrelevant quantity or a cause. | Ask: could this affect both? |
| 8, 15 or 87.5 in question 5 | Inverts, or reports the count or the complement. | Ask what fraction of the 120 cycle. |
| B or C in question 6 | Chooses a convenient or self-selected sample. | Ask: does every pupil have the same chance? |
| 104, 16 or 1.9 in question 7 | Adds, halves or divides. | Ask: how many points higher means subtract. |
Questions worth asking
- What could make both of these numbers larger? Listen for a plausible third variable, named.
- Does a random sample guarantee the right answer? Listen for “no, but it removes the bias of who was chosen”.
- Is correlation useless then? Listen for “no: it is evidence of a link, and a reason to investigate further”.
Differentiation
A way in for anyone stuck
- Give a card with four explanations: x causes y, y causes x, a lurking variable, coincidence.
- Provide named lurking variables to choose from for the first examples.
- Use a sentence frame: Both go up because … affects both.
Stretch
- Design a study that could test whether sleep really causes better reaction times. What would you hold fixed?
- Find a news headline that treats a link as a cause, and rewrite it.
Students learning in English as an additional language
Pair “lurking” with “hiding in the background” and “bias” with “unfair in one direction”.
| Word | Plain meaning | Example |
|---|---|---|
| causation | one thing makes another happen | heating water makes it boil |
| lurking variable | a hidden variable that affects both of the variables you are studying | age affecting foot length and reading score |
| bias | a sample or question that leans one way so that it does not represent everyone | asking only pupils leaving the sports hall |
| reliable | likely to give the same answer if repeated, from a fair sample | a random sample |
| sample | the part of a group that is actually asked or measured | 50 pupils |
Sentence starters for explaining aloud or in writing:
- The link might be explained by … which affects both.
- The sample is biased because …
- The data show a … but they cannot show that …
Two things not to say
“Correlation never means anything.” It is evidence of a link.
“A large sample cannot be biased.” A large biased sample is still biased.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- PercentageA survey of 40 people finds 30 prefer tea. What percentage prefer tea?
75 per cent. - SpotSunburn and ice cream sales: name the lurking variable.
Hot sunny weather.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- SampleTo find out how pupils spend lunchtime, you ask pupils in the library. Is the sample fair?
No: library users are not typical, so it is biased. - Claimr = 0.7 between hours of homework and test score. Is it fair to say homework causes the higher scores?
Not from the data alone: ability or motivation could explain both.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA bigger sample from the same biased group: is it fairer?
No: it is more precise about the wrong group. - From Lesson 5Eight dots give r = 0.95. One outlier changes r to 0.5. Is the original or the new r a better summary of most dots?
The original r describes the eight dots; the outlier should be examined.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The button works with the keyboard and by touch; “Show me the steps” describes four states in words.
On paper: ten stickers in two colours on a pre-drawn graph reproduce the figure.