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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 A

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

Difficulty: first skills applying them stretch

Skills

  1. Calculate Spearman's rank correlation coefficient for the data. x: 5, 9, 2, 14, 7, 11, 4, 16 y: 18, 25, 12, 20, 22, 31, 15, 27

  2. Calculate Spearman's rank correlation coefficient for the data. x: 2.4, 7.1, 5.3, 9.8, 3.6, 6.2, 8.4 y: 12.1, 9.6, 15.2, 7.9, 14.3, 4.8, 10.4

  3. Calculate Spearman's rank correlation coefficient for the data. x: −8, −3, 5, −6, 2, 9, −1, 0 y: 14, −4, −10, 9, −2, −15, 3, 6

  4. Calculate Spearman's rank correlation coefficient for the data. x: 3.5, 5.2, 3.5, 7.8, 5.2, 6.1, 3.5 y: 9, 4, 7, 4, 2, 6, 7

  5. The scatter graph shows the data. Which pair of values fits the graph? A: r = 1, r_s = 1 B: r = 0.90, r_s = 1 C: r = 0.50, r_s = 0.70

    2468105101520253035xy

In context

  1. A letting agency lists eight flats. The table gives the distance from the city centre (km) and the monthly rent ($). x (km): 1.2, 3.5, 0.8, 5.1, 2.4, 6.3, 3.5, 4.2 y ($): 1450, 1100, 1580, 820, 1250, 760, 1020, 930

    Distance (km)1.23.50.85.12.46.33.54.2Rent ($)14501100158082012507601020930
    1. (a) Write down the ranks of the distance data.
    2. (b) Calculate \(r_s\).
    3. (c) Describe the association in context.

  2. A sample of a radioactive substance is weighed over 14 days. x is the time in days and y the mass remaining in grams. x: 0, 2, 4, 6, 8, 10, 12, 14 y: 80, 52, 35, 24, 15, 11, 7, 5

    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 kiosk records the hours of sunshine each day and the number of ice creams sold. x: 4.5, 6.2, 7, 3.8, 8.1, 5.5, 6.8, 2.9 y: 120, 165, 150, 110, 190, 142, 171, 95

    Sunshine (h)4.56.273.88.15.56.82.9Sold12016515011019014217195
    1. (a) Calculate Pearson's \(r\) and Spearman's \(r_s\).
    2. (b) Describe the correlation between hours of sunshine and ice creams sold.

Stretch

  1. A student records the hours she revised (x) and her score (y) in eight weekly quizzes. The last pair is from a quiz that was unusually easy. Her data: x: 1, 2, 3, 4, 6, 7, 8, 5 y: 12, 15, 19, 22, 28, 33, 36, 70

    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