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IB Mathematics: Applications and Interpretation SL · 13.3: Statistics · 13.3.8

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Spearman's rank correlation and comparison with Pearson's r Problem set B

Show the ranks, then use the calculator to find the correlation of the ranks. Give coefficients to 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Calculate Spearman's rank correlation coefficient for the data. x: 6, 13, 3, 10, 15, 8, 1, 12 y: 40, 27, 45, 19, 31, 35, 48, 22

  2. Calculate Spearman's rank correlation coefficient for the data. x: 0.8, 3.5, 1.9, 4.2, 2.7, 5.1, 3.1 y: 6.4, 9.9, 7.2, 13.5, 8.8, 12.6, 11.3

  3. Calculate Spearman's rank correlation coefficient for the data. x: −12, 4, −5, 7, −9, 1, −2, 10 y: −20, 7, −8, 3, −15, −6, 12, −1

  4. Calculate Spearman's rank correlation coefficient for the data. x: 11.2, 8.4, 9.7, 8.4, 12.6, 9.7, 8.4 y: 15, 22, 18, 25, 12, 20, 25

  5. The scatter graph shows the data. Which pair of values fits the graph? A: r = −0.81, r_s = −1 B: r = −1, r_s = −1 C: r = −0.50, r_s = −0.70

    246810102030405060xy

In context

  1. Eight students record the hours they spend gaming online each week and their mark (%) in an exam. x (hours): 4, 10, 7, 12, 7, 2, 9, 5 y (%): 58, 41, 64, 35, 48, 77, 45, 52

    Online (h)4107127295Mark (%)5841643548774552
    1. (a) Write down the ranks of the time online data.
    2. (b) Calculate \(r_s\).
    3. (c) Describe the association in context.

  2. The value of a car is recorded each year. x is the age of the car in years and y its value in thousands of dollars. x: 1, 2, 3, 4, 5, 6, 7, 8 y: 24, 19.5, 16, 13, 11.2, 9.4, 8.1, 7

    1. (a) Sketch a scatter graph of y against x.
    2. (b) Use your graph to explain why Spearman's rank correlation is more appropriate than Pearson's to measure the association.
    3. (c) Calculate \(r_s\) and Pearson's \(r\), and comment on them.

  3. A shop records the number of online adverts shown each week and its weekly sales (number of items). x: 3, 8, 5, 12, 7, 10, 2, 6 y: 210, 340, 260, 480, 275, 395, 190, 330

    Adverts3851271026Sales210340260480275395190330
    1. (a) Calculate Pearson's \(r\) and Spearman's \(r_s\).
    2. (b) Describe the correlation between number of adverts and sales.

Stretch

  1. A family records the hours of sunshine (x) and the energy bought from the grid (y kWh) on eight days. On the last pair of values a faulty heater ran all day. Their data: x: 2, 3, 5, 6, 8, 9, 11, 7 y: 41, 37, 33, 30, 24, 22, 17, 95

    1. (a) Calculate Pearson's \(r\) and Spearman's \(r_s\) for all eight pairs.
    2. (b) The last pair is an outlier. Remove it and calculate \(r\) and \(r_s\) again for the remaining seven pairs.
    3. (c) Which coefficient was affected more by the outlier? Explain why.

jmaths.xyz · AI SL · 13.3.8 Spearman's rank correlation and comparison with Pearson's r