Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- r = −0.8 for two variables. Describe the correlation.
Answer
A strong negative linear correlation. - Find 20% of 450.
Answer
90. - What does the mean point of a data set tell you?
Answer
The mean of the x values and the mean of the y values: the centre of the data.
Words used in this lesson
- causation
- one thing makes another happen. For example: heating water makes it boil.
- lurking variable
- a hidden variable that affects both of the variables you are studying. For example: age affecting foot length and reading score.
- bias
- a sample or question that leans one way so that it does not represent everyone. For example: asking only pupils leaving the sports hall.
- reliable
- likely to give the same answer if repeated, from a fair sample. For example: a random sample.
- sample
- the part of a group that is actually asked or measured. For example: 50 pupils.
Bigger feet, better readers?
A newspaper headline reads “Children with bigger feet read better: study of ten children”. The dots really do rise together. Before you read on: could making a child's feet bigger raise their reading score? What else might explain why the two go together?
Correlation is a link in the data, not proof of cause. Two variables can move together because x causes y, because y causes x, because of a lurking variable that affects both, or by coincidence. The data alone cannot say which.
Bias is a second danger: if the sample is unrepresentative (self-chosen, convenient, too small, or asked a leading question) the data can mislead whatever r is. Ask how the data were collected, from whom and how many.
Press the button to colour the dots by age. Predict what r does inside each age group.
All ten children together: r = 0.93. Longer feet go with higher reading scores.
Show me the steps
- All ten children together: the dots rise in a clear band and r = 0.93. Longer feet go with higher reading scores.
- Show the ages: the six-year-olds (feet 16 to 19 cm) are the lower cluster and the eleven-year-olds (feet 22 to 25 cm) are the upper cluster.
- Inside each age group the dots show no trend: r = 0.00 for the six-year-olds and r = 0.00 for the eleven-year-olds.
- Age is the lurking variable. Older children have longer feet and have also had more years of reading lessons. Making a child's feet longer would not teach reading.
Two slips to watch for. A strong r does not prove a cause, and the graph alone cannot tell you which way round a cause would run. And a lurking variable has to be a sensible reason for both variables to change; name what it is, and do not just say “something else”.
Worked examples
Monthly ice-cream sales and monthly sunburn cases rise and fall together (r = 0.9). Does ice cream cause sunburn?
- Both numbers are higher in hot, sunny months.
- Hot sunny weather is the lurking variable: more people buy ice cream, and more people get sunburn.
- So the link is real but it is not cause and effect.
A survey asks 50 pupils as they leave the sports hall how much they enjoy sport. 45 say “a lot”. Can this be used for the whole school?
- Pupils leaving the sports hall are more likely to enjoy sport than the school as a whole.
- The sample is biased: it does not represent everyone.
- A fairer sample would pick 50 pupils at random from the school register.
Your go. The first step is given. Towns with more fire engines have more fire damage. Name a lurking variable.
- Ask what could make both the number of fire engines and the damage larger.
- Name it and say how it affects both.
Check the answer
The size of the town (or of the fires): bigger towns have more engines and more damage.
Check before you go on
1. A study finds r = 0.8 between hours of television watched and weight for a group of children. Which is the best interpretation?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A survey asks the 60 members of the school chess club whether they like board games. 51 say yes. What percentage say yes?
3. A careless report applies the 85% to the whole school of 900 pupils. How many pupils does it claim like board games?
Level 2 · Building
4. Cities with more libraries tend to have more crime. Which is the most sensible lurking variable?
5. A survey stands at a train station on a weekday morning and asks 120 people whether they cycle to work. 15 say yes. What percentage of the sample cycle?
6. You want to find out how Year 9 pupils travel to school. Which sample of 20 pupils is the fairest?
Level 3 · Stretch
7. In a reading study, the six-year-olds' mean score is 36 and the eleven-year-olds' mean score is 68. By how many points is the older group's mean higher?
8. A scatter graph of X against Y has r = 0.9 and the sample was chosen fairly. Which headline does the graph alone justify?
Something to think about
Cities with more fire engines tend to have more fire damage. A councillor says buying fewer fire engines would reduce the damage. Name a lurking variable and say what is wrong with the councillor's plan.
One way to think about it
The lurking variable is the size of the fire, or of the city: bigger cities and bigger fires need more engines and also suffer more damage. Fire engines do not cause the damage, so fewer of them would not reduce it and might make it worse.
Exit ticket
9. Ski helmet sales and ski injuries both peak in winter, with r = 0.85 over twelve months. A councillor says banning helmets would cut injuries. What is the flaw?