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

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

• Ranks of x (smallest = 1): 3, 7, 2, 5, 8, 4, 1, 6
• Ranks of y: 6, 3, 7, 1, 4, 5, 8, 2
• From the calculator, correlation of the ranks: \(r_s = −0.762\)

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

• Ranks of x (smallest = 1): 1, 5, 2, 6, 3, 7, 4
• Ranks of y: 1, 4, 2, 7, 3, 6, 5
• From the calculator, correlation of the ranks: \(r_s = 0.929\)

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

• Ranks of x (smallest = 1): 1, 6, 3, 7, 2, 5, 4, 8
• Ranks of y: 1, 7, 3, 6, 2, 4, 8, 5
• From the calculator, correlation of the ranks: \(r_s = 0.667\)

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

• Ranks of x (smallest = 1): 6, 2, 4.5, 2, 7, 4.5, 2
• Ranks of y: 2, 5, 3, 6.5, 1, 4, 6.5
• From the calculator, correlation of the ranks: \(r_s = −0.963\)

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

• The points always fall as x rises, so every increase in x goes with a decrease in y: \(r_s = -1\).
• The points lie on a curve, not a straight line, so \(r\) is not \(\pm 1\).
• From the calculator: \(r = −0.81\). The pair is A.

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

1. (a) Write down the ranks of the time online data.

   • The value 7 appears twice, so the two ranks are averaged.
   • Ranks: 2, 7, 4.5, 8, 4.5, 1, 6, 3

2. (b) Calculate \(r_s\).

   • Ranks of exam mark: 6, 2, 7, 1, 4, 8, 3, 5
   • From the calculator, correlation of the ranks: \(r_s = −0.898\)

3. (c) Describe the association in context.

   • \(r_s = −0.898\): a strong negative association. As time online increases, exam mark tends to decrease. This does not show that one causes the other.

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.

   • Plot the eight points: The points fall from left to right along a curve that flattens out.

2. (b) Use your graph to explain why Spearman's rank correlation is more appropriate than Pearson's to measure the association.

   • The value always falls as age increases but the points lie on a curve, not near a straight line.
   • Pearson's r only measures how close the points are to a straight line.

3. (c) Calculate \(r_s\) and Pearson's \(r\), and comment on them.

   • Ranks of x: 1, 2, 3, 4, 5, 6, 7, 8
   • Ranks of y: 8, 7, 6, 5, 4, 3, 2, 1
   • From the calculator: \(r_s = −1\)
   • From the calculator, raw data: \(r = −0.971\)
   • \(r_s\) shows a perfect monotonic relationship; \(r\) is weaker because the points are curved.

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

1. (a) Calculate Pearson's \(r\) and Spearman's \(r_s\).

   • From the calculator, raw data: \(r = 0.968\)
   • Ranks of x: 2, 6, 3, 8, 5, 7, 1, 4
   • Ranks of y: 2, 6, 3, 8, 4, 7, 1, 5
   • From the calculator, ranks: \(r_s = 0.976\)

2. (b) Describe the correlation between number of adverts and sales.

   • \(r_s = 0.976\): a strong positive association. As number of adverts increases, sales tends to increase. This does not show that one causes the other.
   • \(r\) and \(r_s\) are close, so the relationship is close to linear.

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.

   • Ranks of x: 1, 2, 3, 4, 6, 7, 8, 5
   • Ranks of y: 7, 6, 5, 4, 3, 2, 1, 8
   • From the calculator: \(r = −0.242\), \(r_s = −0.762\)

2. (b) The last pair is an outlier. Remove it and calculate \(r\) and \(r_s\) again for the remaining seven pairs.

   • Ranks of x: 1, 2, 3, 4, 5, 6, 7
   • Ranks of y: 7, 6, 5, 4, 3, 2, 1
   • From the calculator: \(r = −0.998\), \(r_s = −1\)

3. (c) Which coefficient was affected more by the outlier? Explain why.

   • The change in r is 0.756; the change in \(r_s\) is 0.238.
   • Pearson's r uses the actual values, so one extreme value has a large effect. Spearman's uses only ranks.