Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- The dots on a scatter graph fall from left to right. Is the direction positive or negative?
Answer
Negative. - Which variable goes on the x axis, the independent or the dependent one?
Answer
The independent variable. - Find the mean of 10, 14, 18 and 22.
Answer
(10 + 14 + 18 + 22) ÷ 4 = 16.
Words used in this lesson
- correlation
- a link between two variables seen in a scatter graph. For example: positive, linear, strong.
- linear
- following a straight line. For example: dots in a band.
- strength
- how closely the dots follow the pattern. For example: strong or weak.
- outlier
- a dot that sits far from the pattern. For example: one dot far below the band.
- non-linear
- following a curve, not a straight line. For example: an arch.
Three words
A reporter has ten seconds to describe a scatter graph, so she uses three words: one for the direction, one for the form and one for the strength. Before you read on, guess what the three words might be for dots in a tight band running from the bottom left to the top right.
Direction: positive (the dots rise), negative (they fall) or none. Form: linear (the dots cluster about a straight line) or non-linear (a curve). Strength: strong, moderate or weak, by how closely the dots follow the pattern.
A dot that sits far from the pattern is an outlier. Mention it separately; it does not change the three words by itself.
Pick a pattern and slide the spread. Say the three words before you read them.
Rises, spread 1: positive, linear, strong. The largest gap from the trend line is 10.
Show me the steps
- Rises, spread 0: every dot sits on one straight line going up. Positive, linear, strong.
- Rises, spread 4: the dots stay in a band that rises, but the band is wide. Positive, linear, weak.
- Falls, spread 1: the dots go down to the right in a narrow band. Negative, linear, strong.
- No link: the dots have no direction. Changing the spread changes nothing, because there is no pattern to spread from.
- Curve: the dots rise and then fall. The relationship is clear but not linear, so it has no single direction.
Two slips to watch for. Strong does not mean steep: a shallow line of tightly packed dots is a strong correlation. And “no linear correlation” is not “no relationship”: dots in a clear arch follow a pattern, it is just not a straight line.
Worked examples
Puppies' ages (weeks) against weight (kg): the dots rise in a narrow band, and one dot sits far below the band. Describe it.
- The dots rise from left to right: the direction is positive.
- They cluster about a straight line: the form is linear.
- The band is narrow: the strength is strong. One dot far below the band is an outlier.
Graph A: dots within 2 units of the line y = 3x + 5. Graph B: dots within 12 units of the same line. Which correlation is stronger?
- The direction and steepness are the same: both lines have gradient 3.
- A's dots hug the line more closely (within 2 units, not 12).
- So A has the stronger correlation: strength is about closeness, not steepness.
Your go. The first step is given. Shoe length and height for 30 pupils: the dots rise in a band of medium width with no sign of a curve. Describe the correlation.
- As shoe length rises, height rises, so the direction is positive.
- Give the form and the strength.
Check the answer
Positive, linear and moderate.
Check before you go on
1. The dots on a scatter graph are spread widely about a rising pattern. Which description fits?
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. Five pairs (x, y): (1, 18), (2, 29), (3, 33), (4, 47), (5, 52). By how much does y rise from the first pair to the last?
3. A trend line is y = 8x + 10. At x = 7 a dot sits at y = 71. How far above the trend line is the dot?
Level 2 · Building
4. Graph A's dots lie within 3 units of its line. Graph B's dots lie within 9 units of the same line. B's band is how many times as wide as A's?
5. Dots on a scatter graph rise steadily and then fall steadily in a clear arch. Which statement is best?
6. A scatter graph has 20 dots. Five of them lie far from the band that holds the rest. What percentage of the dots are inside the band?
Level 3 · Stretch
7. The trend for x from 1 to 8 is y = 3x + 20, and no dot is more than 4 units from the line. What is the largest value of y that a dot could have?
8. Graph C: every dot lies within 3 units of a steep line that falls. Graph D: every dot lies within 3 units of a shallow line that rises. Which statement is correct?
Something to think about
Two graphs show a rising pattern. In graph A the dots are tightly packed about a very shallow line. In graph B the dots are widely scattered about a steep line. A pupil says B has the stronger correlation because it is steeper. What would you say?
One way to think about it
Steepness is the gradient of the pattern; strength is how closely the dots follow it. A is the stronger correlation, even though it is shallow, because its dots lie close to the line. B is steeper but its dots are spread out.
Exit ticket
9. The dots fall in a band of medium width about a straight line. Two dots sit far below the band. Which is the best description?