Spearman's rank correlation coefficientEdexcel International A Level Further Maths: Subtopic test

1. 1
   Two judges, X and Y, each rank six dancers A to F, with rank 1 for the best. Judge X ranks them A, B, C, D, E, F in that order, so A is 1st and F is 6th. Judge Y gives A rank 3, B rank 1, C rank 2, D rank 5, E rank 4 and F rank 6.

   (a)

   Find $\sum d^2$, where $d$ is the difference between the two ranks given to a dancer.

   [1 mark]

   • A$6$
   • B$8$
   • C$0$
   • D$48$

   (b)

   Calculate Spearman's rank correlation coefficient for the two judges.

   [1 mark]

   • A$0.229$
   • B$0.784$
   • C$0.771$
   • D$0.829$

   (c)

   Interpret the value of the coefficient for the two judges.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A teacher ranks eight students A to H from the highest mark (rank 1) to the lowest in maths, and again in physics. The maths ranks of A to H are 1 to 8 respectively. The physics ranks of A to H are 3, 1, 2, 5, 4, 8, 6 and 7 respectively.

   (a)

   Find $\sum d^2$ for the two sets of ranks.

   [1 mark]

   • A$10$
   • B$0$
   • C$84$
   • D$14$

   (b)

   Calculate Spearman's rank correlation coefficient.

   [1 mark]

   • A$0.833$
   • B$0.167$
   • C$0.838$
   • D$-0.833$

   (c)

   A second teacher gives the physics ranks with rank 1 for the lowest mark instead of the highest, and leaves the maths ranks unchanged. State the new value of the coefficient, with a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A café owner compares seven coffee blends A to G. The prices per kilogram, in pounds, are A 12, B 18, C 15, D 22, E 9, F 25 and G 20. The customer rating scores out of 10 are A 6.1, B 7.0, C 5.8, D 8.2, E 4.9, F 7.9 and G 7.4.

   (a)

   Rank the blends by price and by rating, and calculate Spearman's rank correlation coefficient.

   [3 marks]

   (b)

   The owner says: 'My results show that charging more for a blend makes customers rate it higher.' Evaluate this conclusion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A coach records the 100 m time in seconds and the long jump distance in metres for eight athletes A to H. The times are A 11.2, B 10.9, C 11.5, D 11.0, E 12.1, F 11.8, G 11.3 and H 10.8. The jumps are A 6.9, B 7.4, C 6.6, D 7.1, E 6.0, F 6.8, G 6.7 and H 7.3.

   (a)

   Rank the athletes by time (rank 1 for the fastest) and by jump (rank 1 for the longest). Calculate Spearman's rank correlation coefficient and interpret it in context.

   [6 marks]

   (b)

   A ninth athlete, I, runs the 100 m in 11.2 seconds, the same time as A.
   (i) State the ranks given to A, to I and to G for the times of all nine athletes, explaining how they are found.
   (ii) Give one reason why Spearman's coefficient might be preferred to the product moment correlation coefficient for these data, one limitation of it, and explain why a value close to 1 does not show that fast sprinting causes long jumps.

   [6 marks]

   Total for question 4: 12 marks

End of questions