Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- The dots rise in a tight band. Is the correlation positive or negative, strong or weak?
Answer
Positive and strong. - A line is y = 2x + 10. Find y when x = 9.
Answer
28, because 2 × 9 + 10 = 28. - Which is larger, 0.85 or 0.8?
Answer
0.85.
Words used in this lesson
- correlation coefficient
- a number r between −1 and 1 for how closely dots follow a straight line. For example: r = 0.97.
- strong
- r close to 1 or −1 (0.8 or more in size). For example: r = −0.9.
- weak
- r close to 0 (below 0.5 in size). For example: r = 0.2.
- outlier
- a dot far from the pattern that can change r and the line. For example: the ninth dot at height 10.
- linear
- following a straight line. For example: dots in a band.
One extra dot
Eight pupils' dots rise in a narrow band, and the spreadsheet says r = 0.97. A ninth pupil is added, and her dot sits well below the others. Before you read on: how much do you think one dot out of nine can change r? Make a guess as a number.
The correlation coefficient r lies between −1 and 1. The sign is the direction. The size measures how closely the dots follow a straight line: r = 1 or −1 is a perfect line, and r near 0 is no linear relationship. As a rough guide, 0.8 or more is strong, 0.5 to 0.8 is moderate, below 0.5 is weak.
We read r from a spreadsheet or a calculator's statistics mode, and we always look at the graph as well. An outlier can change r a lot.
Slide the ninth dot, then hide it. Say what r will do before you read it.
Ninth dot at (9, 30): r = 0.98 and the slope of the line is 2.18.
Show me the steps
- Ninth dot at (9, 30), on the trend: r = 0.98 and the slope is 2.18. The eight dots alone give r = 0.97.
- Ninth dot at (9, 10), far below the trend: r falls to 0.40 and the slope to 0.85. One dot has dragged the line down.
- Ninth dot at (9, 60), far above the trend: r is 0.79 and the slope is 4.18. A dot too high also hurts r.
- Hide the ninth dot: r = 0.97 and the slope is 2.08, whatever height the slider shows.
Two slips to watch for. r is not the gradient: a shallow line of tightly packed dots has r close to 1. And r is not a percentage: r = 0.5 is not “half as strong” as r = 1, so describe it with the words moderate and strong, and look at the graph.
Worked examples
A spreadsheet gives r = −0.83 for the hours of screen time and the hours of sleep of pupils. Describe the correlation.
- The sign is negative: as screen time goes up, sleep goes down.
- The size is 0.83, which is 0.8 or more: the correlation is strong.
- It is a strong negative linear correlation. It does not say that screen time causes the shorter sleep.
Graph A has r = 0.85 and graph B has r = −0.92. Which correlation is stronger?
- Strength is the size of r, so ignore the sign: 0.85 and 0.92.
- 0.92 is larger than 0.85.
- Graph B is stronger, even though it is negative.
Your go. The first step is given. A spreadsheet gives r = 0.04 for shoe size and test score. Describe the correlation.
- 0.04 is very close to 0.
- Say whether there is a linear relationship, and how strong.
Check the answer
Almost no linear relationship: a very weak correlation, with no useful direction.
Check before you go on
1. A spreadsheet gives r = −0.9 for two variables. Which description is correct?
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. Graph A has r = −0.74 and graph B has r = 0.68. What is the size of r for the graph with the stronger correlation (ignoring the sign)?
3. Five graphs have r = −0.9, 0.2, 0.75, −0.3 and 0.95. How many of them show a positive correlation?
Level 2 · Building
4. Six graphs have r = 0.92, −0.85, 0.5, −0.79, 0.81 and 0.1. Using “strong means 0.8 or more in size”, how many are strong?
5. A survey of twelve pupils gives r = 0.94 for revision hours and marks. One more pupil, who revised a lot and still scored very low, is added and r becomes 0.55. By how much did r fall?
6. Dots lie exactly on a symmetric arch and a spreadsheet gives r = 0. What does this tell you?
Level 3 · Stretch
7. A graph of height (cm) against arm span (cm) gives r = 0.6. Every arm span is converted to millimetres, so each y value is multiplied by 10. What is r for the new graph?
8. For the eight dots the line of best fit is y = 2.08x + 9.75. A ninth dot at x = 9 sits exactly on the line. What is its y, to one decimal place?
Something to think about
Every x value in a data set is increased by 10, so the whole cloud slides to the right. What happens to r, and why?
One way to think about it
r stays the same. Sliding the cloud does not change how closely the dots follow a straight line, so r measures the same pattern. The line moves with the dots, and its slope is unchanged.
Exit ticket
9. A scatter graph's r is 0.62. After an outlier is removed, r becomes 0.91. Which conclusion is best?