Teach Me Maths

Scatter Graphs and Correlation

Question 11 mark

What type of correlation does this scatter graph show?

Hint

Look at the overall direction of the points as you move from left to right.

Worked solution

What type of correlation does this scatter graph show?

  1. Positive correlation
  2. Negative correlation
  3. No correlation
  1. The points tend to rise from left to right: pupils with greater heights tend to have greater arm spans. This is positive correlation.

Question 21 mark

The graph compares the age and value of used bicycles. What type of correlation is shown?

Hint

Describe the overall direction, not one individual point.

Worked solution

The graph compares the age and value of used bicycles. What type of correlation is shown?

  1. Positive correlation
  2. Negative correlation
  3. No correlation
  1. As bicycle age increases, value tends to decrease. The graph shows negative correlation.

Question 31 mark

A class records shoe sizes and spelling marks. What type of correlation does the graph show?

Hint

Check whether the points follow a clear upward or downward trend.

Worked solution

A class records shoe sizes and spelling marks. What type of correlation does the graph show?

  1. Positive correlation
  2. Negative correlation
  3. No correlation
  1. Higher shoe sizes are not consistently paired with higher or lower marks. There is no clear correlation.

Question 41 mark

Point A represents one day. Write down its coordinates in the order (temperature, ice creams sold).

Write your answer as (x, y)

Hint

Read across the horizontal axis first, then up the vertical axis.

Worked solution

Point A represents one day. Write down its coordinates in the order (temperature, ice creams sold).

  1. Point A has temperature 22 °C and 60 ice creams sold. Its coordinates are (22,60).

Question 53 marks

Box P has width 24 cm and length 32 cm. Enter the coordinates needed to plot P. The site will then plot the point.

(a)

Coordinates of P:

1 mark

Write your answer as (x, y)

(b)

How many centimetres wider is P than the plotted box with length 24 cm?

2 marks

Hint

Match each measurement to its axis.

Worked solution

Box P has width 24 cm and length 32 cm. Enter the coordinates needed to plot P. The site will then plot the point.

Part (a)

Coordinates of P:

  1. Length is on the horizontal axis, so it is the first coordinate. Width is on the vertical axis, so P is (32,24).

Part (b)

How many centimetres wider is P than the plotted box with length 24 cm?

  1. The box with length 24 cm has width 18 cm. P has width 24 cm.
  2. The difference in width is 24-18=6 cm.

Question 61 mark

One point does not follow the general pattern. Write down the coordinates of this outlier.

Write your answer as (x, y)

Hint

Find the point well away from the main group, then read its coordinates.

Worked solution

One point does not follow the general pattern. Write down the coordinates of this outlier.

  1. Most values fall as age increases. The point at age 3 and value 3 is far below the other nearby points, so the outlier is (3,3).

Question 71 mark

Which statement describes the relationship shown?

Hint

Describe what tends to happen to one variable as the other increases.

Worked solution

Which statement describes the relationship shown?

  1. As the outside temperature increases, heating use tends to decrease.
  2. As the outside temperature increases, heating use tends to increase.
  3. Every home uses exactly the same amount of heating.
  4. Doubling the outside temperature always halves heating use.
  1. The cloud of points slopes downwards. Warmer outside temperatures are associated with lower heating use; the scattered points do not show an exact rule for every home.

Question 81 mark

Use the line of best fit shown to estimate the plant height after 5 weeks, in cm.

Hint

Go from 5 on the horizontal axis to the line, then read across.

Worked solution

Use the line of best fit shown to estimate the plant height after 5 weeks, in cm.

  1. The vertical through 5 weeks meets the line at a height of 30 cm. The estimated height is 30 cm; an individual plant may differ.

Question 91 mark

Use the line shown to estimate the length of a leaf with width 32 mm. Give the length in cm.

Hint

Start at 32 on the vertical axis and read across to the line.

Worked solution

Use the line shown to estimate the length of a leaf with width 32 mm. Give the length in cm.

  1. The horizontal through width 32 mm meets the line above length 6 cm. The estimated length is 6 cm.

Question 101 mark

Which of the three lines is the most suitable line of best fit?

Hint

Look for the line through the middle of the cloud, with points on both sides.

Worked solution

Which of the three lines is the most suitable line of best fit?

  1. Line A
  2. Line B
  3. Line C
  1. Line A follows the upward trend through the middle of the points. Line B sits above the cloud, and line C slopes the wrong way. A line of best fit does not need to pass through every point.

Question 112 marks

The axes do not start at zero. Use the line shown to estimate the width of a picture frame with length 60 cm. Give your answer in cm.

Hint

Read the labelled values on both axes before using the line.

Worked solution

The axes do not start at zero. Use the line shown to estimate the width of a picture frame with length 60 cm. Give your answer in cm.

  1. At length 60 cm, the line is at width 32 cm. The first vertical-axis label is 20, so the answer is 32 cm, not 12 cm.

Question 121 mark

A survey finds positive correlation between children’s shoe sizes and reading scores. A pupil says, “Giving children larger shoes will improve their reading.” Which response is justified?

Hint

An association does not show what causes a change.

Worked solution

A survey finds positive correlation between children’s shoe sizes and reading scores. A pupil says, “Giving children larger shoes will improve their reading.” Which response is justified?

  1. The claim is proved because the correlation is positive.
  2. The claim is not proved; age could affect both shoe size and reading score.
  3. Positive correlation means every child has the same reading score.
  4. The claim is proved because shoe size is measured first.
  1. The survey shows an association, not that shoe size causes reading ability. Older children may tend to have both larger feet and higher reading scores. Changing shoes alone is not shown to improve reading.

Question 133 marks

Use the line shown to estimate how much taller a plant at 8 weeks is than a plant at 3 weeks. Give the difference in cm.

Hint

Read both estimated heights from the line before comparing them.

Worked solution

Use the line shown to estimate how much taller a plant at 8 weeks is than a plant at 3 weeks. Give the difference in cm.

  1. The line gives 42 cm at 8 weeks and 22 cm at 3 weeks.
  2. Estimated difference =42-22=20 cm. These are predictions from the line, not guaranteed heights for individual plants.

Question 142 marks

A café records temperature and cold drinks sold. The line shown extends beyond the observed data.

(a)

Use the line to estimate the number of drinks sold at 28 °C.

1 mark

(b)

Why is this prediction less reliable than a prediction at 18 °C?

1 mark

Hint

Use the line for the estimate, and the points to judge its range.

Worked solution

A café records temperature and cold drinks sold. The line shown extends beyond the observed data.

Part (a)

Use the line to estimate the number of drinks sold at 28 °C.

  1. The line at 28 °C gives an estimate of 80 drinks.

Part (b)

Why is this prediction less reliable than a prediction at 18 °C?

  1. 28 °C is outside the observed temperature range.
  2. 28 °C is inside the observed temperature range.
  3. Positive correlation makes every prediction exact.
  1. The observed temperatures run from 10 °C to 24 °C. Predicting at 28 °C extends the trend beyond those data: this is extrapolation. The trend may not continue.

Question 153 marks

Point A is a pupil who revised for 4 hours. How many marks higher is this pupil’s actual score than the score estimated by the line?

Hint

Read the point and the line at the same horizontal position.

Worked solution

Point A is a pupil who revised for 4 hours. How many marks higher is this pupil’s actual score than the score estimated by the line?

  1. Point A gives an actual score of 64. At 4 hours the line gives an estimated score of 52.
  2. The actual score is 64-52=12 marks higher.

Question 16Challenge3 marks

The graph shows ages and resale values of some devices. A line of best fit for the main group is shown.

(a)

Write down the coordinates of the outlier.

1 mark

Write your answer as (x, y)

(b)

Use the line to estimate the age of a device with value £280. Give the age in years.

2 marks

Hint

Keep the point coordinates and the axis units separate.

Worked solution

The graph shows ages and resale values of some devices. A line of best fit for the main group is shown.

Part (a)

Write down the coordinates of the outlier.

  1. The point at age 6 and value 58 is well above the main group. The outlier is (6,58).

Part (b)

Use the line to estimate the age of a device with value £280. Give the age in years.

  1. The vertical axis uses £10s, so £280 is 28 on that axis.
  2. Reading from 28 to the line and down gives an estimated age of 8 years.

Question 17Challenge4 marks

The graph shows two fitted lines for used bicycles: model A and model B.

(a)

For bicycles aged 2 years, how much greater is the estimated value of model A than model B? Give the difference in pounds.

2 marks

(b)

At what age do the two lines give the same estimated value? Give the age in years.

2 marks

Hint

Compare the two line readings, not the heights of individual data points.

Worked solution

The graph shows two fitted lines for used bicycles: model A and model B.

Part (a)

For bicycles aged 2 years, how much greater is the estimated value of model A than model B? Give the difference in pounds.

  1. At 2 years, line A gives £1200 and line B gives £1000.
  2. Difference =1200-1000=£200.

Part (b)

At what age do the two lines give the same estimated value? Give the age in years.

  1. The lines meet at age 6 years and value £800. Their predicted values are equal at 6 years.

Question 18Challenge3 marks

Two pupils draw the lines A and B shown. Both are reasonable lines of best fit for these data.

(a)

At x=8, how much greater is the estimate from A than the estimate from B?

2 marks

(b)

Which statement is correct?

1 mark

Hint

Distinguish the value read from a chosen line from an exact rule for the data.

Worked solution

Two pupils draw the lines A and B shown. Both are reasonable lines of best fit for these data.

Part (a)

At x=8, how much greater is the estimate from A than the estimate from B?

  1. Line A gives 26 and line B gives 24 at x=8.
  2. The difference is 26-24=2.

Part (b)

Which statement is correct?

  1. Different reasonable lines can give different estimates.
  2. Only a line through every point can be correct.
  3. One line must be wrong because the estimates differ.
  1. A line of best fit is an estimate of the trend. More than one nearby line can reasonably represent the same scattered data, so their predictions need not be identical.

Question 19Challenge2 marks

The data were collected from adults. A line of best fit is shown.

(a)

Use the line to estimate the height of an adult with arm span 180 cm. Give the height in cm.

1 mark

(b)

Someone uses the same line for a young child with arm span 60 cm. Which is the best criticism?

1 mark

Hint

Check who was measured as well as the range of measurements.

Worked solution

The data were collected from adults. A line of best fit is shown.

Part (a)

Use the line to estimate the height of an adult with arm span 180 cm. Give the height in cm.

  1. The line gives an estimated height of 175 cm at arm span 180 cm.

Part (b)

Someone uses the same line for a young child with arm span 60 cm. Which is the best criticism?

  1. 60 cm is outside the adult data range, and the child is from a different population.
  2. The prediction must be correct because the line is straight.
  3. It is interpolation because the child has an arm span.
  4. The adult graph proves arm span causes height.
  1. The plotted adult arm spans run from 145 to 195 cm. A value of 60 cm lies far outside this range.
  2. The data were also collected from adults, so they may not describe young children. Both limitations make that prediction unreliable.

Question 20Challenge5 marks

A shop records ages and prices of used cameras. The line of best fit shown follows the main group.

(a)

Write down the coordinates of the outlier.

1 mark

Write your answer as (x, y)

(b)

A buyer budgets £350 for three cameras, each aged 6 years. Using the line, estimate how many pounds of the budget will remain.

3 marks

(c)

Why should the line not be relied on to predict the price of a camera aged 20 years?

1 mark

Hint

Read the fitted trend for a prediction and keep its limits in mind.

Worked solution

A shop records ages and prices of used cameras. The line of best fit shown follows the main group.

Part (a)

Write down the coordinates of the outlier.

  1. The camera aged 4 years priced at £210 is well above the main group. Its coordinates are (4,210).

Part (b)

A buyer budgets £350 for three cameras, each aged 6 years. Using the line, estimate how many pounds of the budget will remain.

  1. At age 6 the line gives an estimated price of £110.
  2. Three cameras would cost about 3\times110=£330.
  3. Estimated money remaining =350-330=£20. Individual prices may differ from the fitted line.

Part (c)

Why should the line not be relied on to predict the price of a camera aged 20 years?

  1. 20 years is outside the ages observed, so the trend may not continue.
  2. 20 years is between the smallest and largest observed ages.
  3. An outlier means a line can never be used for any prediction.
  1. The observed ages are between 1 and 9 years. Predicting at 20 years would extrapolate far beyond those data, where the trend may change.

More on this topic: Scatter Graphs and Correlation worksheet with full solutions

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