AI SL · 13.3
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Spearman's rank correlation and comparison with Pearson's r
13.3.8
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Drag a point on the top diagram, or change the numbers in the table. The ranks are worked out for you and drawn below, and Pearson's r and Spearman's rₛ are found side by side. Then add one extreme point.
| x | y | Rank of x | Rank of y |
|---|
Change a number in the table.
Eight members of a running club give their weekly training distance (km) and their time for a 10 km race (minutes). Calculate Spearman's rank correlation coefficient \(r_s\) between distance and time.
Have a go first. Then open the working one line at a time.
Distance ranks: 1, 5, 3, 7, 4, 2, 8, 6
Smallest value gets rank 1. List the ranks in the original order of the students.
Time ranks: 8, 3, 6, 2, 5, 7, 1, 4
Rank the second list the same way, smallest = 1.
From the calculator: \(r_s = −0.976\)
Put the two lists of ranks into the calculator and find the correlation coefficient r.
\(r_s = −0.976\): a strong negative association. As weekly distance increases, race time tends to decrease. This does not show that one causes the other.
Name the direction and the strength, and use the words of the question.
Eight adults record their mean daily steps (thousands) and their resting heart rate (beats per minute). Calculate \(r_s\) and comment on the relationship.
Have a go first. Then open the working one line at a time.
Steps in order: 4, 4, 6, 6, 8, 9, 10, 12. The two 4s take ranks 1 and 2, so each gets 1.5. The two 6s take ranks 3 and 4, so each gets 3.5. Steps ranks: 1.5, 6, 3.5, 8, 3.5, 5, 7, 1.5
Tied values receive the mean of the ranks they occupy.
The two 66s take ranks 3 and 4, so each gets 3.5. Heart rate ranks: 7, 3.5, 5, 1, 6, 3.5, 2, 8
Check that the ranks add to 36, the sum of 1 to 8.
From the calculator: \(r_s = −0.982\)
Correlation coefficient of the two lists of ranks.
\(r_s = −0.982\): a strong negative association. As daily steps increases, resting heart rate tends to decrease. This does not show that one causes the other.
Direction, strength, context.
A house is heated for eight winter days. The table gives the mean outside temperature (°C) and the energy used (kWh). Calculate \(r_s\).
Have a go first. Then open the working one line at a time.
Temperature ranks: 1, 5, 3, 7, 4, 2, 6, 8
A negative value is smaller than a positive one: −4.5 is the smallest, so it has rank 1.
Energy ranks: 8, 4, 6, 2, 5, 7, 3, 1
Decimals rank in the same way. Compare place by place.
From the calculator: \(r_s = −1\)
The warmer the day, the less energy used: a negative association.
A mug of tea cools. Its temperature T (°C) is measured every 5 minutes. t (min): 0, 5, 10, 15, 20, 25, 30, 35 T (°C): 92, 71, 58, 48, 41, 36, 33, 31 (a) Describe the scatter graph. (b) Calculate Pearson's r and Spearman's \(r_s\). (c) Which is the better measure of the association? Explain.
Have a go first. Then open the working one line at a time.
The points fall from left to right along a curve that flattens out.
The temperature always goes down as the time goes up, but not in a straight line.
From the calculator, raw data: \(r = −0.943\)
Pearson measures how close the points are to a straight line.
Time ranks: 1, 2, 3, 4, 5, 6, 7, 8 Temperature ranks: 8, 7, 6, 5, 4, 3, 2, 1 From the calculator: \(r_s = −1\)
Every increase in t comes with a decrease in T, so the ranks are exactly opposite.
\(r_s\) is the better measure here.
The relationship is decreasing but not linear. \(r_s = -1\) shows it always decreases; \(r\) is further from \(-1\) only because the points curve.