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Cambridge IGCSE0607

Scatter diagrams

International Mathematics 0607 Chapter Notes

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Scatter diagrams
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In this chapter

  1. 1Drawing a scatter diagram
  2. 2Positive, negative and zero correlation
  3. 3Drawing a line of best fit
  4. 4Using the line of best fit
  5. 5Linear regression on a calculator

1. Drawing a scatter diagram

A scatter diagram plots pairs of values as points. One value goes along the horizontal axis. The other goes up the vertical axis.

Mark each point as a small cross (×). Do not join the points up.

Before you plot, work out what one small square is worth on each axis. Then read the scale carefully.

Examiner insight

“Although many candidates gained both marks, a number of candidates could not cope with the vertical scale and points were plotted incorrectly.” “Generally, points were plotted correctly but some candidates misread the scale and made errors.”

Worked example 12 marks

Do not use a calculator.

On the vertical axis of a scatter diagram, each large square is 10 units. Each large square is divided into 5 small squares. A point is 3 small squares above the line for 40. Find the value of the point.

  1. 1

    one small square = 10 ÷ 5 = 2 [1]

  2. 2

    value = 40 + 3 × 2 = 46 [1]

2. Positive, negative and zero correlation

Correlation describes how two sets of values are linked. There are three types.

Positive correlation: as one value increases, the other increases. Negative correlation: as one value increases, the other decreases. Zero correlation: there is no trend linking the two values.

When a question asks for the type of correlation, the answer is one word: positive, negative or none. Do not add the strength, and describe the trend of all the points, not a single point.

Examiner insight

“To describe correlation all that is needed is positive, negative or none. Candidates do not need to qualify the type with the strength.” “For the description, candidates must describe the trend and not pick on a single point as comments such as, ‘When the temperature is hot a lot of ice creams are sold.’ were seen and did not receive the mark.”

Worked example 12 marks

As the temperature rises, the number of ice creams sold rises. State the type of correlation between temperature and ice creams sold, and describe the trend in words.

  1. 1

    positive correlation [1]

  2. 2

    as the temperature increases, the number of ice creams sold increases [1]

3. Drawing a line of best fit

A line of best fit is a single ruled line. Use a ruler. Never join the points up with line segments or draw a curve.

The line must pass through the mean point. Find the mean of the x values and the mean of the y values, then plot that point.

The line should extend across the full data set. It does not have to touch any point, but there should be a roughly even distribution of points either side of it.

mean = (total of the values)/(number of values)

Examiner insight

“A line of best fit needed to pass through the mean point and be drawn with a straight edge.” “The line of best fit should not extend much beyond the area of the data, so the large number of candidates who forced their line to go through the origin, often from the top point, were not correct.”

Worked example 14 marks

The scatter diagram shows points connecting values of t and values of m.

(a) Write down the type of correlation shown in the scatter diagram.

(b) The mean value of t is 8.5 and the mean value of m is 6.5. Plot the mean point and draw a line of best fit.

(c) Use your line of best fit to estimate the value of t when m = 5.

  1. 1

    (a) the points fall from top left to bottom right: negative [1]

  2. 2

    (b) plot the mean point (8.5, 6.5) and draw a ruled line through the mean point [1]

  3. 3

    the line slopes down, follows the trend and is within tolerance [1]

  4. 4

    (c) at m = 5, read across from the m axis to the line, then down to the t axis: t ≈ 11 for the line drawn here [1]

4. Using the line of best fit

To estimate a value, find the given value on the correct axis. Go across or up to the line, then read the other axis.

Check which axis the question gives you. If it gives m, start on the m axis.

The answer is an estimate, so it will not match any single point. Give it in the form the context needs. If the answer counts whole things, give a whole number.

Examiner insight

“Many understood that they needed to find 5 on the vertical axis to read over to their line and then down rather than go up from 5 on the horizontal axis which is a more commonly asked question.” “Candidates who gave as an answer the decimal value read off from their line of best fit at age 11, or who rounded this value up, could only gain part marks since they were not giving the number of complete lengths the student could swim.”

Worked example 12 marks

Do not use a calculator.

A line of best fit for test score s against hours of revision h passes through (2, 40) and (6, 72). Use the line to estimate the score for 5 hours of revision.

  1. 1

    gradient = (72 - 40)/(6 - 2) = 8 [1]

  2. 2

    s = 40 + 8 × (5 - 2) = 64 [1]

5. Linear regression on a calculator

The equation of linear regression comes from a graphic display calculator. Enter the x values in one list and the y values in another. Choose linear regression and read off the two constants.

Give both constants to 3 significant figures. Do not draw a line by eye and find its equation.

To use the equation, substitute the x value and work out y. Round the answer as the context needs.

y = mx + c

Examiner insight

“A large number of candidates omitted to give both parts of the regression equation to the required 3 significant figure accuracy.” “A number of candidates tried to find an equation for the line of best fit.”

Worked example 14 marks

The table shows the marks of 10 students in a French exam and a Spanish exam.

(a) Find the mean French mark.

(b)(i) Find the equation of the regression line for y in terms of x.

(ii) Use your answer to 10(b)(i) to estimate the Spanish mark when the French mark is 25.

French mark (x)18223136424855636976
Spanish mark (y)34424355485859687279
  1. 1

    (a) total = 460, so mean = 460 ÷ 10 = 46 [1]

  2. 2

    (b)(i) enter the French marks in one list and the Spanish marks in another. Choose linear regression: y = 0.708x + 23.2, both to 3 s.f. [2]

  3. 3

    (ii) substitute x = 25: y = 0.708 × 25 + 23.2 = 40.9…, so about 41 [1]

Recap

  • A scatter diagram plots pairs of values; mark each point as a small cross.
  • Read the scale first: work out what one small square is worth.
  • Correlation is positive, negative or none. Give one word, with no strength.
  • A line of best fit is a single ruled line through the mean point.
  • The line follows the trend with points roughly evenly either side, and does not go through the origin unless the data do.
  • Linear regression comes from the calculator, to 3 significant figures. Substitute into it to estimate.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. On the vertical axis of a scatter diagram, each large square is 20 units. Each large square is divided into 4 small squares. A point is 3 small squares above the line for 60. Find the value of the point.1 mark
  2. State whether the correlation is positive, negative or none. (a) The height of adults and the number of letters in their surname. (b) The distance a car has travelled since it was filled up and the fuel left in its tank. (c) The hours of sunshine and the sales of sun cream. (d) The time spent revising and the number of mistakes in a test.4 marks
  3. Do not use a calculator. Four points on a scatter diagram are (1, 9), (3, 7), (5, 4) and (7, 4). (a) Find the mean point. (b) State the type of correlation.3 marks
  4. Do not use a calculator. A line of best fit passes through (10, 30) and (30, 50). Use it to estimate y when x = 25.2 marks
  5. Do not use a calculator. A class records hours of revision and test score for each student. The mean number of hours is 3.5 and the mean score is 62. Mia draws her line of best fit from (0, 0) to the highest point. (a) State the point that her line should pass through. (b) Give one other thing wrong with her line.2 marks
  6. Do not use a calculator. A line of best fit for the number of lengths n a swimmer completes against age a passes through (6, 4) and (10, 13). Estimate the number of complete lengths for a swimmer aged 9.3 marks
  7. Do not use a calculator. A gardener records the height h cm of a sunflower at the end of weeks 1 to 5. The heights are 20, 34, 50, 66 and 80. (a) Find the mean week and the mean height. A line of best fit passes through the mean point and through (5, 82). (b) Use it to estimate the height in week 4. (c) Show that this line does not pass through the origin.5 marks
  8. Use a graphic display calculator. The table gives x and y. (a) Find the equation of linear regression for y in terms of x. (b) Use it to estimate y when x = 7. | x | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | | y | 5 | 9 | 10 | 16 | 17 | 24 |3 marks
  9. Use a graphic display calculator. A scientist records the depth d metres and the water temperature T °C at six places in a lake. (a) Find the equation of linear regression for T in terms of d. (b) Find the mean depth and the mean temperature. Show that the regression line passes through the mean point. (c) Use your equation to estimate the depth at which the temperature is 14 °C. | d (m) | 2 | 5 | 9 | 14 | 20 | 28 | | --- | --- | --- | --- | --- | --- | --- | | T (°C) | 21.5 | 20.1 | 17.8 | 15.9 | 12.4 | 9.5 |7 marks
  10. Use a graphic display calculator. An engineer tests a delivery drone. She records the load m kg and the flight time f minutes for eight flights. (a) Find the equation of linear regression for f in terms of m. (b) Each parcel has a mass of 0.3 kg. A flight must last at least 20 minutes. Use your equation to find the greatest number of parcels the drone can carry. (c) Use your equation to estimate the flight time for a load of 6 kg. Give a reason why this estimate may not be reliable. | m (kg) | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | | f (min) | 31 | 29 | 26 | 25 | 22 | 19 | 18 | 15 |7 marks

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