Teach Me Maths

Scatter Graphs and Correlation

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How to read and plot points on a scatter graph, describe the correlation, spot an outlier, draw a line of best fit and use it to make estimates, and decide when an estimate can be trusted.

Scatter graphs

A scatter graph shows two sets of data about the same things, such as the number of pages and the weight of each book in a pile. Each book is plotted as one cross, using its two values as coordinates. The crosses are not joined up.

A scatter graph shows whether the two sets of data are linked: for example, whether books with more pages tend to be heavier.

In the exam you are usually asked to write down the type of correlation or describe the relationship, give the coordinates of an outlier, or draw a line of best fit and use it to estimate a value, for 1 or 2 marks each. Harder parts ask you whether someone is right about the graph, why an estimate may not be reliable, what is wrong with someone's line of best fit, or how much one value goes up for each unit of the other.

Reading and plotting points

Before you read or plot anything, work out what one small square is worth on each axis. The two axes often have different scales. If the labels go up in 100s with 5 small squares between them, each small square is worth 100\div5=20

Each cross has two coordinates. Write the value on the horizontal axis first, then the value on the vertical axis. So (340, 420) means 340 across and 420 up.

To plot a point, go along the horizontal axis to the first value, then up to the second value, and mark a small cross. Then read it back from both axes to check it.

Worked example 1: Reading and plotting a point

The scatter graph shows the number of pages and the weight, in grams, of 8 books.

(a) Point P shows one of the books. Write down the number of pages and the weight of this book.

(b) Another book has 160 pages and weighs 240 g. Plot this information on the scatter graph.

  1. On both axes the labels go up in 100s with 5 small squares between them. So each small square is worth 100\div5=20
  2. (a) Point P is 2 small squares to the right of 300, so this book has 300+2\times20=340 pages.

  3. Your turn. Point P is 1 small square above 400. How heavy is this book, in grams?

    400+20=420, so the book weighs 420 g

  4. (b) 160 is 3 small squares past 100, and 240 is 2 small squares above 200. Go along to 160, then up to 240, and mark a cross.
  5. Answer: (a) 340 pages and 420 g (b) the new cross is shown in orange.

Types of correlation

Correlation describes the link between the two sets of data. Look at the general direction of the crosses as you go from left to right.

  • Positive correlation: the crosses go up from left to right. As one value increases, the other tends to increase.
  • Negative correlation: the crosses go down from left to right. As one value increases, the other tends to decrease.
  • No correlation: the crosses are scattered with no pattern. There is no link between the two values.

Always write the word correlation: “positive” on its own is not enough. If you are asked to describe the relationship, say it in words using the context, such as “books with more pages tend to weigh more”.

Outliers

An outlier is a point that does not fit the pattern of the rest. It is well away from the other crosses.

Give an outlier's coordinates in the right order: the value on the horizontal axis first, then the value on the vertical axis.

Ignore an outlier when you describe the correlation or draw a line of best fit. One outlier does not mean there is no correlation: the other crosses still show it.

To explain how an outlier is different, use the context, for example “this book is much heavier than the other books with about the same number of pages”.

Worked example 2: Correlation and an outlier

A walker recorded the air temperature at 10 places on a hill, at different heights above sea level. The scatter graph shows the results.

(a) One of the points is an outlier. Write down its coordinates.

(b) Ignoring the outlier, what type of correlation does the scatter graph show?

(c) Describe the relationship between the height and the temperature.

  1. Across, the labels go up in 200s with 4 small squares between them, so each small square is 200\div4=50 m. Up, each small square is 1^\circC.
  2. (a) Most of the crosses go down from left to right in a narrow band. One cross is far above the others near it.

  3. Your turn. Write down the coordinates of the outlier.

    Write your answer as (x, y)

    The outlier is (600, 13): the height first, then the temperature

  4. (b) Ignoring the outlier, the crosses go down from left to right. This is negative correlation.
  5. (c) Say it in words, using the context: as the height increases, the temperature tends to decrease.
  6. Answer: (a) (600, 13) (b) negative correlation (c) the higher the place, the lower the temperature tends to be.

Drawing a line of best fit

A line of best fit is a straight line drawn through the middle of the crosses, following their direction. It shows the trend. Only draw one when there is correlation.

  • Use a ruler: draw one straight line, not a curve, and don't join the crosses dot to dot.
  • Follow the direction of the crosses, with about the same number of crosses on each side of the line.
  • The line does not have to go through (0, 0), or start at a corner of the grid.
  • Ignore any outlier.

Line A is a good line of best fit. Line B was drawn from (0, 0) to the corner of the grid, so almost all the crosses are on one side of it.

Different people draw slightly different lines, so in the exam an estimate from your own line is accepted within a range. In these notes the line is drawn for you, so you can check your reading.

Using a line of best fit to estimate

To estimate a value, start at the value you know on its axis. Go straight to the line, then straight across, or straight down, to the other axis, and read the value there.

Draw these guide lines on the graph with a ruler, so the examiner can see your method. Then read the value carefully, using what one small square is worth, and give the units.

Use the line, not the nearest cross. A cross is one real item; the line shows the general pattern, so it gives a better estimate.

Worked example 3: Estimating with a line of best fit

The scatter graph shows the number of hours some people spent on a typing course and their typing speed, in words per minute, at the end of the course. A line of best fit has been drawn.

(a) Estimate the typing speed of a person who spent 13 hours on the course.

(b) Another person types at 30 words per minute. Estimate the number of hours this person spent on the course.

  1. Across, each small square is 1 hour. Up, the labels go up in 10s with 5 small squares between them, so each small square is 10\div5=2 words per minute.
  2. (a) Go up from 13 hours to the line, then across to the vertical axis. It meets the axis 4 small squares above 30, so the speed is 30+4\times2=38

  3. Your turn. (b) Go across from 30 words per minute to the line, then down to the horizontal axis. How many hours does it show?

    The line is at 30 words per minute above 9 hours

  4. Both readings are shown by the guide lines.
  5. Answer: (a) about 38 words per minute (b) about 9 hours

When an estimate can be trusted

An estimate inside the range of the data is usually reliable, because there are crosses nearby showing the pattern. This is called interpolation.

An estimate outside the range of the data is not reliable, because you don't know that the pattern carries on. This is called extrapolation. A line for children aged 5 to 12 might show them growing about 6 cm a year. Carried on to age 40, it would give a height of over 3 m.

Worked example 4: Is the estimate reliable?

The scatter graph shows the distance driven, in km, and the fuel used, in litres, by 9 delivery vans on one day. A line of best fit has been drawn.

(a) Estimate the fuel used by a van that is driven 150 km.

(b) Another van will be driven 400 km. Explain why the line of best fit may not give a reliable estimate of the fuel it will use.

  1. Across, each small square is 50\div5=10 km. Up, each small square is 1 litre.
  2. (a) The data go from 40 km to 220 km, so 150 km is inside the range and the estimate should be reliable. Go up from 150 to the line, then across.

  3. Your turn. How many litres does the line show at 150 km?

    The line is at 14 litres above 150 km

  4. The guide lines show the reading.
  5. (b) The data only go up to 220 km. 400 km is far outside the range of the data, so you don't know that the pattern carries on that far.
  6. Answer: (a) about 14 litres (b) 400 km is outside the range of the data, so the pattern may not carry on.

Worked example 5: Negative correlation and a claim

The scatter graph shows the distance from the town centre, in km, and the monthly rent, in £, of 10 flats. A line of best fit has been drawn.

(a) What type of correlation does the scatter graph show?

(b) Another flat is 3.5 km from the town centre. Estimate its monthly rent.

Priya says, “Flats closer to the town centre tend to have lower rents.”

(c) Does the scatter graph support what Priya says? Give a reason for your answer.

  1. Across, the labels go up in 2s with 4 small squares between them, so each small square is 2\div4=0.5 km. Up, the axis starts at £500, not at 0. The labels go up in 100s with 5 small squares between them, so each small square is £20.
  2. (a) The crosses go down from left to right, so this is negative correlation. The further a flat is from the town centre, the lower its rent tends to be.

  3. Your turn. (b) Go up from 3.5 km to the line, then across to the vertical axis. What monthly rent, in pounds, does the line show?

    The line is 1 small square above £800, so the rent is about £820.

  4. The guide lines show the reading. Use the line, not the crosses at 3 km and 4 km.
  5. (c) Start your answer with yes or no. The crosses go down from left to right, so flats closer to the town centre tend to have higher rents, not lower. The flat 1 km from the centre costs £940 a month; the flat 9 km out costs £580.
  6. Answer: (a) negative correlation (b) about £820 (c) No. The graph shows negative correlation, so flats closer to the town centre tend to have higher rents.

Worked example 6: The change for each extra degree

The scatter graph shows the midday temperature and the number of swimmers at an outdoor pool on 10 days. A line of best fit has been drawn.

Use the line of best fit to work out how many more swimmers there are, on average, for each extra 1^\circC.

  1. The horizontal axis starts at 10^\circC, and each small square across is 1^\circC. Up, each small square is 100\div5=20 swimmers.
  2. Choose two points on the line that are far apart and easy to read. At 14^\circC the line is at 200 swimmers.

  3. Your turn. How many swimmers does the line show at 26^\circC?

    At 26^\circC the line is at 500 swimmers

  4. From 14^\circC to 26^\circC is 26-14=12 degrees, and the line goes up by 500-200=300 swimmers.
  5. For each extra degree, that is 300\div12=25 swimmers.
  6. Answer: about 25 more swimmers for each extra 1^\circC

Common mistakes

  • Writing an outlier's coordinates the wrong way round. Give the value on the horizontal axis first, then the value on the vertical axis.
  • Writing just “positive”, or saying the values are “in proportion”. Write “positive correlation” or “negative correlation”, or describe the relationship in words using the context.
  • Drawing the line of best fit from (0, 0), or joining the crosses dot to dot. Draw one straight line through the middle of the crosses, following their direction.
  • Guessing an estimate without using a line. Draw a line of best fit if there isn't one, and draw guide lines to it and on to the axis.
  • Taking each small square to be worth 1. Work out what one small square is worth on each axis before you read or plot a value.
  • Not saying yes or no when asked if someone is right. Start with “Yes” or “No”, then give a reason from the graph.

Practice questions

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