GCSE Maths: Scatter graphs (Higher)
Part of the GCSE Maths statistics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Scatter graphs
Bivariate data is data with two measurements for each person or object, such as the hours someone revised and their test mark. A scatter graph shows each pair as a point (a cross), with one measurement on the horizontal axis and the other on the vertical axis. Scatter graphs let us see whether the two things are related.
Correlation describes how the two variables are related. With positive correlation the points slope upwards: when one variable increases, so does the other (more hours revising, higher marks). With negative correlation the points slope downwards: when one increases, the other decreases (an older car is worth less). With no correlation there is no pattern. The correlation is strong if the points are close to a straight line and weak if they are widely scattered.
Correlation is not causation. Two things can be correlated without one causing the other. Ice cream sales and sunburn cases are positively correlated, but ice cream does not cause sunburn: both go up when the weather is hot. Always say there is a relationship or link unless you know the cause.
Line of best fit. A line of best fit is a straight line drawn through the points that follows the trend. It does not have to pass through any particular point. Draw it so that there are about equal numbers of points on each side of the line, and it passes close to the mean point (the mean of the x-values, the mean of the y-values). Use a ruler, and extend it across the data.
Using the line. Reading a value from the line within the range of the data is interpolation: it is usually reliable if the correlation is strong. Reading a value outside the range of the data is extrapolation: it is much less reliable, because the relationship may not continue (for example, marks cannot go above 100). Say that your answer is an estimate.
Key ideas
- Positive correlation: both increase. Negative: one increases as the other decreases. None: no pattern.
- Strong correlation: points close to a line. Weak correlation: points widely scattered.
- Correlation does not prove causation.
- A line of best fit follows the trend, with about equal numbers of points on each side, near the mean point.
- Interpolation (inside the data range) is more reliable than extrapolation (outside it).
- An outlier is a point that does not fit the pattern.
Method. Look at the direction of the points, and how close they are to a straight line.
- Look at the direction. The points go from the bottom left to the top right.As the hours increase, the marks increase: positive correlation.
- Look at the closeness. The points are close to a straight line.The correlation is strong.
- Write the answer in context. Use the words of the question.Strong positive correlation: students who revise for more hours tend to get higher marks.
Check. A line sloping up through the points would fit them closely. ✓
Watch out. Say positive or negative, and say strong or weak. Write it in context.
Method. Read across or up from the line of best fit. The answer is an estimate, so a small tolerance is allowed.
- Find 5.5 hours on the horizontal axis. We know the hours-value, so start on the hours-axis.hours = 5.5 hours
- Go straight up to the line of best fit. The line of best fit shows the trend, so it gives us the best estimate.Draw a vertical line up from the axis until it meets the line.
- Go across to the vertical axis and read the mark-value. Read the scale carefully.mark ≈ 70
Check. The point you read should lie on the line of best fit, not on one of the crosses. ✓
Watch out. Read from the line, not from the nearest cross.
Method. Check whether 12 hours is inside the range of the data, and whether the answer makes sense.
- Find the range of the data. The hours in the graph go from 1 to 9.1 to 9 hours
- Compare 12 hours with the range. 12 hours is outside the range of the data, so reading from the line is extrapolation.12 > 9, so this is extrapolation.
- Check the answer. The line gives about 110, which is more than 100.A test mark out of 100 cannot be bigger than 100.
- Conclude. The relationship probably does not continue in a straight line, so the prediction is unreliable.Not sensible.
Check. Using the line: 35.7 + 6.2 × 12 = 110.1. ✓
Watch out. Extrapolation is risky. Always say why the estimate may not be reliable.
| Adverts, x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Sales, y | 2 | 5 | 4 | 7 | 6 | 9 | 8 | 11 |
Method. Plot every point, find the mean point to guide the line, then draw a straight line that follows the trend.
- Plot the points. Each pair is one point: go along to x, then up to y.(1, 2), (2, 5), (3, 4), (4, 7), (5, 6), (6, 9), (7, 8), (8, 11)
- Work out the mean point. The line of best fit should pass close to the mean point (mean of x, mean of y).Mean of x = 36 ÷ 8 = 4.5
Mean of y = 52 ÷ 8 = 6.5
Mean point = (4.5, 6.5) - Draw a straight line with a ruler. It follows the trend of the points and passes close to the mean point. There should be about the same number of points above and below it: here 4 above and 4 below.
Check. There are 4 points above the line and 4 below it. ✓
Watch out. The line does not have to go through the first or last point, and does not need to go through any point at all.
Method. Think about whether there could be another reason for both to increase.
- Describe the correlation. When ice cream sales rise, so do sunburn cases.Strong positive correlation.
- Look for another cause. Hot, sunny weather makes people buy ice cream and also causes sunburn.A third factor (hot weather) explains both.
- Conclude. Correlation shows a link, not that one thing causes the other.No. The correlation does not show that ice cream causes sunburn.
Check. Ice cream is eaten indoors in winter as well, with no sunburn. ✓
Watch out. Never write “this causes that” from a scatter graph alone.
Watch out
A line of best fit is not found by joining the first and last points, and it does not have to pass through the origin. Also, never use a line of best fit far outside the data: that is extrapolation and it may not be reliable.Exercise 24J Scatter graphs
Non-calculatorMethod. Look at the direction and at how close the points are to a line.
- Direction. The points slope up from left to right.Positive
- Closeness. The points are close to a straight line.Strong
Check. A straight line sloping up would fit the points closely. ✓
Watch out. Name the direction and the strength.
Method. Look at the direction and at how close the points are to a line.
- Direction. The points slope down from left to right.Negative
- Closeness. The points are close to a straight line.Strong
Check. As age goes up, value goes down. ✓
Watch out. Negative means one goes up as the other goes down.
Method. Look for a pattern.
- Direction. The points do not slope up or down.No clear direction
- Conclude. The points are scattered with no pattern.No correlation
Check. No straight line fits the points well. ✓
Watch out. If there is no pattern, say “no correlation”.
Method. Look at the direction and at how close the points are to a line.
- Direction. The points slope slightly upwards, left to right.Positive
- Closeness. The points are widely scattered about any line.Weak
Check. The upward trend is there but it is not clear-cut. ✓
Watch out. Weak correlation still has a direction.
| Maths mark | 34 | 48 | 55 | 62 | 70 | 75 | 82 | 90 |
|---|---|---|---|---|---|---|---|---|
| Science mark | 40 | 52 | 50 | 66 | 68 | 80 | 79 | 92 |
Method. Plot each pair as a point, then describe the pattern.
- Plot the points. Each student gives one point: maths mark along, science mark up.(34, 40), (48, 52), (55, 50), (62, 66), (70, 68), (75, 80), (82, 79), (90, 92)
- Draw the graph. Use equal scales on both axes.
- Describe the correlation. The points slope upwards and lie close to a line.Strong positive correlation: students with high maths marks tend to have high science marks.
Check. The correlation is positive: the best students in maths are also the best in science. ✓
Watch out. Plot carefully: both axes need an equal scale.
Method. Add each column and divide by 8.
- Mean of the maths marks. Add them, then divide by 8.34 + 48 + 55 + 62 + 70 + 75 + 82 + 90 = 516; 516 ÷ 8 = 64.5
- Mean of the science marks. Do the same.40 + 52 + 50 + 66 + 68 + 80 + 79 + 92 = 527; 527 ÷ 8 = 65.875
Check. The line of best fit should pass close to this point. ✓
Watch out. Write the mean point as a coordinate pair (x, y).
Method. Read across or up from the line of best fit. The answer is an estimate, so a small tolerance is allowed.
- Find 3 hours on the horizontal axis. We know the hours-value, so start on the hours-axis.hours = 3 hours
- Go straight up to the line of best fit. The line of best fit shows the trend, so it gives us the best estimate.Draw a vertical line up from the axis until it meets the line.
- Go across to the vertical axis and read the mark-value. Read the scale carefully.mark ≈ 54
Check. The point you read should lie on the line of best fit, not on one of the crosses. ✓
Watch out. Read from the line, not from the nearest cross.
Method. Read across or up from the line of best fit. The answer is an estimate, so a small tolerance is allowed.
- Find 7 hours on the horizontal axis. We know the hours-value, so start on the hours-axis.hours = 7 hours
- Go straight up to the line of best fit. The line of best fit shows the trend, so it gives us the best estimate.Draw a vertical line up from the axis until it meets the line.
- Go across to the vertical axis and read the mark-value. Read the scale carefully.mark ≈ 79
Check. The point you read should lie on the line of best fit, not on one of the crosses. ✓
Watch out. Read from the line, not from the nearest cross.
Method. Read across or up from the line of best fit. The answer is an estimate, so a small tolerance is allowed.
- Find 80 on the vertical axis. We know the mark-value, so start on the mark-axis.mark = 80
- Go across to the line of best fit. The line of best fit shows the trend.Draw a horizontal line across until it meets the line.
- Go down to the horizontal axis and read the hours-value. Read the scale carefully.hours ≈ 7
Check. The point you read should lie on the line of best fit, not on one of the crosses. ✓
Watch out. Read from the line, not from the nearest cross.
Method. Compare 4.5 with the range of the data, 1 to 9 hours.
- Check the range. 4.5 is between 1 and 9.1 < 4.5 < 9
- Name it. Inside the range of the data is interpolation.Interpolation, so it is likely to be reliable.
Check. The graph has points either side of 4.5 hours. ✓
Watch out. Inside the data range is interpolation.
Method. Compare 15 with the range of the data, 1 to 9 hours.
- Check the range. 15 is bigger than 9.15 > 9
- Name it. Outside the range of the data is extrapolation.Extrapolation, so it is not reliable.
Check. The line would predict 129, which is more than 100. ✓
Watch out. Outside the data range is extrapolation.
Method. Compare 0.5 with the range of the data, 1 to 9 hours.
- Check the range. 0.5 is less than 1.0.5 < 1
- Name it. Outside the range, so extrapolation.Extrapolation, so it is not reliable.
Check. There are no points to the left of 1 hour. ✓
Watch out. Both ends count: below the lowest value is also extrapolation.
Method. Compare the age used with the ages in the data, and think about whether the pattern could continue.
- Compare with the data. The data only covers ages 3 to 12, and 40 is far outside it.Extrapolation
- Think about the real situation. Children stop growing at around 18, so height does not keep rising in a straight line.The trend does not continue.
- Conclude. The estimate is not reliable.Predictions far outside the range of the data are not reliable.
Check. A height of 3 m is impossible for an adult. ✓
Watch out. Extrapolation is only safe over a short distance, if at all.
Method. Look for another reason that explains both.
- Describe the link. More firefighters go with more damage.Positive correlation.
- Find another cause. A bigger fire causes more damage and also needs more firefighters.Size of the fire is the common cause.
- Conclude. A correlation does not mean one causes the other.No.
Check. Sending fewer firefighters to a big fire would not reduce the damage. ✓
Watch out. Correlation does not prove causation.
Method. Read across and up from the circled point.
- Read the horizontal value. Look down from the circled cross to the axis.x = 4.5
- Read the vertical value. Look across to the axis.y = 20
- Write the coordinates. (4.5, 20)
Check. It is the only point far below the others. ✓
Watch out. Write the horizontal value first.
Method. Think about what could make a result very different from the pattern.
- Think about the pattern. Students who revise 4.5 hours usually score about 60 to 65.This student scored 20.
- Suggest a reason. Anything sensible will do.For example, the student was ill during the test.
Check. Any sensible reason is accepted. ✓
Watch out. An outlier may be a real unusual result or a mistake.
Method. Decide whether the point follows the trend.
- Look at the point. It is far from all the others, so it is an outlier.Does not follow the pattern.
- Decide. The line of best fit should show the trend. An outlier would pull the line away from the other points.Ignore the outlier when drawing the line.
Check. The other 11 points follow a clear line. ✓
Watch out. Mention the outlier, but do not let it distort the line.
Method. Think about what a line of best fit is meant to do.
- Recall the rule. A line of best fit follows the trend, with about equal numbers of points above and below it.It should fit all the points.
- Check what Dan did. The end points may be unusual, so a line through them could have most points on one side.It uses only two points.
- Conclude. It does not fit the overall trend.
Check. A better line passes near the mean point of all the data. ✓
Watch out. Use all the points, not just two of them.
Method. Substitute x = 6 into the equation.
- Write the equation. y = 2x + 3
- Substitute x = 6. Replace x with 6.y = 2 × 6 + 3 = 12 + 3 = 15
Check. 6 km at £2 per km is £12, plus the £3 fixed charge, is £15. ✓
Watch out. Do the multiplication before the addition.
Method. Look at what changes when x changes.
- The number 3. When x = 0 (no distance), y = 3.A fixed charge of £3.
- The number 2. Each extra kilometre adds £2 to y.The cost per km is £2.
Check. Check: 1 km costs 2 + 3 = £5 and 2 km costs £7, which is £2 more. ✓
Watch out. The gradient is the change in y for each 1 more in x.
Method. Gradient = change in y ÷ change in x.
- Find the change in y. 97 − 35 = 62
- Find the change in x. 10 − 0 = 10
- Divide. 62 ÷ 10 = 6.2
Check. Moving across 10 raises the line by 62, so by 6.2 for each 1. ✓
Watch out. Both changes must be taken in the same order.
Method. The line crosses the y-axis at (0, 35), so c = 35.
- Write the form. m is the gradient and c is the y-intercept.y = mx + c
- Put in m = 6.2 and c = 35. The point (0, 35) is on the y-axis.y = 6.2x + 35
Check. When x = 10: 6.2 × 10 + 35 = 97. ✓
Watch out. The intercept is the value of y when x = 0.
Method. Substitute x = 7.5.
- Substitute. y = 6.2 × 7.5 + 35
- Work out. 6.2 × 7.5 = 46.546.5 + 35 = 81.5
Check. x = 7.5 is inside the data range 0 to 10, so this is interpolation. ✓
Watch out. Multiply before you add.
Method. Interpret the gradient as the change in y for each 1 increase in x, then check the extreme value.
- Gradient. The gradient 6 means y increases by 6 when x increases by 1.Each extra hour adds about 6 points.
- Substitute 30. 6 × 30 + 20 = 200
- Comment. 30 hours a week is likely to be outside the data, and the score is unlikely to rise in a straight line for ever.It is an extrapolation and probably unreliable.
Check. Using x = 5: 6 × 5 + 20 = 50, and x = 6 gives 56, which is 6 more. ✓
Watch out. Say what the gradient means in the context.
Method. Strength depends on how close the points are to a straight line.
- Graph A. The points are close to a line sloping down.Strong negative correlation
- Graph B. The points are widely scattered but slope down.Weak negative correlation
- Compare. The closer the points are to a line, the stronger the correlation.Graph A is stronger. Both are negative.
Check. Both slope downwards, so both are negative. ✓
Watch out. Strength and direction are separate things.
Method. Substitute the x-value of the mean point into the equation and see whether the y-value matches.
- Substitute x = 6. If the line passes through (6, 40), then y should be 40.3 × 6 + 22 = 18 + 22 = 40
- Compare. The value matches the y-value of the mean point.So the line passes through (6, 40).
- Estimate when x = 8. Substitute.3 × 8 + 22 = 24 + 22 = 46
Check. From (6, 40), going 2 to the right raises the line by 2 × 3 = 6: 40 + 6 = 46. ✓
Watch out. A good line of best fit passes through (or very near) the mean point.