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F07 Spearman's Rank Interpretation

EDEXCEL 1ST0 · Calculator allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

At a village fete, two judges independently rank 6 cakes from 1 (best) to 6 (worst). Their rankings are shown in the table.

Cake	Judge A rank	Judge B rank
Victoria sponge	1	1
Lemon drizzle	2	2
Carrot cake	3	3
Chocolate fudge	4	5
Banana loaf	5	4
Coffee and walnut	6	6

The Spearman's rank correlation coefficient for these two sets of rankings is rs = 0.94.

(a)State whether rs = 0.94 shows a positive or negative association between the two judges' rankings.(1)

(b)Write down whether the value 0.94 is closer to 0 or closer to 1.(1)

(c)Using the value of rs, describe how closely the two judges agree on the order of the cakes.(2)

(Total for Question 1 is 4 marks)

Two teachers, Mr Ahmed and Mrs Patel, each rank the same 5 pupils' handwriting from 1 (neatest) to 5 (least neat), without seeing each other's results.

Pupil	Mr Ahmed rank	Mrs Patel rank
Priya	1	1
Sam	2	2
Owen	3	3
Fatima	4	4
Leo	5	5

The Spearman's rank correlation coefficient for the two teachers' rankings is rs = 1.

(a)Write down what is true about the two teachers' rank orders when rs = 1.(1)

(b)State what rs = 1 tells you about the strength and direction of the agreement between the two teachers.(1)

(c)A third teacher, Mr Owusu, ranks the same 5 pupils and gets rs = 0.9 when compared with Mr Ahmed. Explain why this shows slightly less agreement than Mrs Patel's ranking.(1)

(Total for Question 2 is 3 marks)

Before a school egg-and-spoon race, a PE teacher predicts the finishing order of 6 runners. In the actual race, every runner finishes in exactly the reverse of the predicted order.

Runner	Predicted rank	Actual rank
Freya	1	6
Marcus	2	5
Ines	3	4
Tyler	4	3
Zara	5	2
Callum	6	1

The Spearman's rank correlation coefficient for the predicted and actual rankings is rs = -1.

(a)Write down whether rs = -1 shows a positive or negative association.(1)

(b)State what -1 tells you about the strength of the association.(1)

(c)Explain what rs = -1 means about the relationship between the predicted rank and the actual rank of each runner.(2)

(Total for Question 3 is 4 marks)

In a quiz night, 8 teams are ranked from 1st to 8th on a General Knowledge round and, separately, on a Sports round.

Team	General Knowledge rank	Sports rank
A	1	4
B	2	7
C	3	2
D	4	6
E	5	1
F	6	8
G	7	3
H	8	5

The Spearman's rank correlation coefficient for the two rounds is rs = 0.

(a)Write down whether rs = 0 shows a positive, negative, or no clear association.(1)

(b)State what rs = 0 tells the quiz organisers about how a team's General Knowledge rank compares with its Sports rank.(2)

(c)Chloe says: 'A coefficient of 0 means the teams are all equally good at quizzing.' Give one reason why Chloe is wrong.(1)

(Total for Question 4 is 4 marks)

For each statement about Spearman's rank correlation coefficient, write down whether it is True or False.

(i)A Spearman's rank correlation coefficient can only take values between -1 and 1.(1)

(ii)A coefficient of rs = -0.8 shows a weaker association than a coefficient of rs = 0.3.(1)

(iii)If two rankings are identical, the Spearman's rank correlation coefficient must be 1.(1)

(iv)A coefficient close to 0 always means the two quantities being ranked are completely unrelated in every way.(1)

(Total for Question 5 is 4 marks)

Four students each describe a scatter graph they have drawn comparing two sets of rankings.

(a)Which description best matches a Spearman's rank correlation coefficient of rs = -0.85?(1)

A) Points lie close to a straight line that rises from bottom-left to top-right.
B) Points lie close to a straight line that falls from top-left to bottom-right.
C) Points are scattered with no clear pattern at all.
D) Points lie exactly on a horizontal line.

(b)Give one reason why your chosen description fits a strongly negative coefficient.(1)

(Total for Question 6 is 2 marks)

A food magazine ranks 7 local restaurants by hygiene inspection score, and separately by average customer review score, both from 1 (best) to 7 (worst). The Spearman's rank correlation coefficient between the hygiene ranking and the review ranking is rs = -0.93.

(a)State the type of association (positive or negative) shown by rs = -0.93.(1)

(b)Describe, using the word 'strong' or 'weak', the strength of this association.(1)

(c)Interpret rs = -0.93 in the context of the two rankings.(2)

(Total for Question 7 is 4 marks)

Two film critics, writing for different magazines, each rank the same 6 films by how 'original' they are and, separately, by how much audience members said they 'enjoyed' the film. The Spearman's rank correlation coefficient between the originality ranking and the enjoyment ranking is rs = -0.15.

(a)State whether rs = -0.15 shows a strong or weak association.(1)

(b)Explain what this value suggests about the link between how original a film is and how much audiences enjoy it.(2)

(Total for Question 8 is 3 marks)

At a talent show, two panels of judges independently rank the acts in three categories. The Spearman's rank correlation coefficient between the two panels is given for each category.

Category	rs
Costume	0.88
Dance	0.45
Singing	-0.91

(a)Write down which category shows the weakest association between the two panels.(1)

(b)Write down which category shows the strongest association between the two panels, regardless of whether it is positive or negative.(2)

(c)Explain what the coefficient for the Singing category tells the organisers about the two panels' rankings.(2)

(Total for Question 9 is 5 marks)

In a photography competition, Assessor A and Assessor B each score 5 entries out of 20. Assessor B gives two entries the same score.

Photo	Assessor A score	Assessor A rank	Assessor B score	Assessor B rank
1	18	1	19	1
2	14	3	15	?
3	12	4	15	?
4	16	2	13	4
5	9	5	10	5

(a)Photo 2 and Photo 3 both scored 15 from Assessor B. Work out the rank that should be given to each of these two photos.(2)

(b)Explain why Photo 2 and Photo 3 are given the same rank instead of one being ranked above the other.(1)

(c)Give one reason why it would not be appropriate to simply give Photo 2 rank 2 and Photo 3 rank 3 (in either order).(1)

(Total for Question 10 is 4 marks)

A teacher wants to find out whether pupils' mock exam results are a good predictor of their final exam results. She carries out a statistical enquiry in four stages: plan, collect, process and interpret.

(a)(Plan) Give one reason why the teacher chooses to rank the pupils' marks, rather than compare the raw marks directly, when checking the relationship between the two exams.(1)

(b)(Collect) The table shows the mock exam rank and final exam rank for 6 pupils.

Pupil	Mock exam rank	Final exam rank
Aisha	1	2
Ben	2	1
Chloe	3	4
Dev	4	3
Ella	5	6
Farah	6	5

Write down the name of the pupil whose mock exam rank and final exam rank are the same.(1)

(c)(Process) Using this data, the Spearman's rank correlation coefficient is calculated to be rs = 0.83. State whether this value is consistent with the ranks in the table.(1)

(d)(Interpret) The teacher concludes: 'Mock exam rank is a reliable predictor of final exam rank for this class.' Use the value of rs to comment on whether this conclusion is reasonable.(2)

(Total for Question 11 is 5 marks)

At a dog show, two independent assessments are made of the same 7 dogs: a professional groomer ranks the dogs by coat condition, and a judge ranks the dogs by overall appearance. The scatter graph shows the groomer's rank plotted against the judge's rank for each dog.

(a)Describe the pattern shown by the scatter graph.(1)

(b)The Spearman's rank correlation coefficient for this data is rs = 0.89. State whether this value is consistent with the pattern in the scatter graph.(1)

(c)Explain why rs = 0.89 rather than rs = 1, even though the scatter graph shows a clearly rising pattern.(2)

(Total for Question 12 is 4 marks)

Two classes each have their coursework marked independently by two different teachers, who then rank the pupils from best to worst. The Spearman's rank correlation coefficient between the two teachers' rankings is rs = 0.91 for Class P and rs = 0.58 for Class Q.

(a)Write down which class shows the stronger agreement between its two teachers.(1)

(b)A moderator says the two teachers marking Class Q should discuss their marking before the grades are finalised, but the teachers marking Class P do not need to. Explain why the moderator's decision makes sense.(2)

(Total for Question 13 is 3 marks)

A student is asked to interpret a Spearman's rank correlation coefficient of rs = 0.4, calculated between two judges' rankings of a bake-off competition. The student writes: 'Since rs = 0.4, the two judges completely disagree with each other.'

(a)Identify what is wrong with the student's statement.(1)

(b)Give a correct interpretation of rs = 0.4.(1)

(c)Write down the value of rs that would represent the two judges completely disagreeing with each other.(1)

(Total for Question 14 is 3 marks)

In a gymnastics competition, Judge A and Judge B each rank 8 gymnasts from 1st to 8th. The Spearman's rank correlation coefficient between their two rankings is rs = 0.35. The organising committee must decide whether to use these two judges' rankings alone to award medals, or to bring in a third judge.

Evaluate the situation and give a recommendation to the committee, using the value of rs to justify your answer.

(Total for Question 15 is 4 marks)

Three pairs of assessors each rank the same set of entries in a competition. The Spearman's rank correlation coefficient is given for each pair.

Pair	rs
X	0.97
Y	-0.63
Z	0.08

The competition organiser wants to know which pair of assessors can be trusted to judge consistently together, and which pair actively disagree.

(a)Write down which pair shows the strongest agreement.(1)

(b)Explain why pair Y should be described as disagreeing with each other, rather than simply showing a weak association.(2)

(c)Explain why a coefficient of rs = 0.08 for pair Z does not necessarily mean either assessor is bad at judging.(2)

(Total for Question 16 is 5 marks)

Mark scheme · F07 Spearman's Rank Interpretation

Question 1

(a) B1 Positive cao
(a) Answer: Positive
(b) B1 Closer to 1 cao
(b) Answer: Closer to 1
(c) B1 identifies the agreement as strong/very strong
(c) B1 links this to rs being close to +1, e.g. the two judges ranked the cakes in almost the same order
(c) Answer: The judges agree strongly; rs is close to +1 so they ranked the cakes in almost the same order.

Question 2

(a) B1 the two rank orders are identical/exactly the same oe
(a) Answer: The two rank orders are exactly the same.
(b) B1 perfect positive agreement cao (oe: complete/total agreement)
(b) Answer: Perfect positive agreement.
(c) B1 0.9 is not the maximum possible value, so the rankings are not identical, only very similar oe
(c) Answer: 0.9 is slightly less than 1, showing the rankings are very similar but not identical.

Question 3

(a) B1 Negative cao
(a) Answer: Negative
(b) B1 the strongest/maximum possible association (perfect) oe
(b) Answer: The strongest possible association (a perfect correlation).
(c) B1 identifies that the rank order is completely reversed
(c) B1 links this to the context, e.g. the runner predicted to finish 1st actually finished last (6th), and this pattern holds for every runner
(c) Answer: The actual finishing order is the exact reverse of the predicted order for every runner.

Question 4

(a) B1 no (clear) association / neither positive nor negative oe
(a) Answer: No clear association
(b) B1 there is little or no agreement between the two rankings
(b) B1 context: e.g. doing well in the General Knowledge round gives no indication of how a team will do in the Sports round
(b) Answer: There is no real link; doing well on one round tells you nothing about how a team will do on the other.
(c) B1 rs measures the agreement/link between two rankings, not how good the teams are overall oe
(c) Answer: rs = 0 only says there is no link between the two rankings; it says nothing about how good the teams actually are.

Question 5

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 True cao
(iii) Answer: True
(iv) B1 False cao
(iv) Answer: False

Question 6

(a) B1 B cao
(a) Answer: B) Points lie close to a straight line that falls from top-left to bottom-right.
(b) B1 a falling pattern shows that as one ranking increases the other tends to decrease, and rs is close to -1 so the points lie close to a line oe
(b) Answer: As one rank increases the other tends to decrease, and because -0.85 is close to -1 the points lie close to a straight line.

Question 7

(a) B1 Negative cao
(a) Answer: Negative
(b) B1 Strong cao
(b) Answer: Strong
(c) B1 identifies that restaurants ranked well (highly) for hygiene tend to be ranked poorly for reviews, or vice versa
(c) B1 correct overall conclusion, e.g. there is a strong tendency for the two rankings to disagree/run in opposite directions
(c) Answer: Restaurants with a good hygiene ranking tend to have a poor customer review ranking, and vice versa - the two rankings strongly disagree.

Question 8

(a) B1 Weak cao
(a) Answer: Weak
(b) B1 there is little to no agreement/link between the two rankings
(b) B1 context: e.g. a film's originality ranking gives very little indication of its audience-enjoyment ranking (they are almost independent)
(b) Answer: There is very little link; how original a film is ranked gives almost no indication of how much audiences enjoy it.

Question 9

(a) B1 Dance cao
(a) Answer: Dance
(b) M1 compares the size of each coefficient ignoring sign (0.88, 0.45, 0.91)
(b) A1 Singing cao
(b) Answer: Singing
(c) B1 identifies that the two panels strongly disagree (a negative association)
(c) B1 draws a sensible conclusion for the organisers, e.g. the panels are judging the singing acts very differently, so they may want to check the judging criteria for that category
(c) Answer: The two panels strongly disagree on the singing rankings; the organisers may want to review how that category is being judged.

Question 10

(a) M1 recognises the two tied scores would occupy positions 2 and 3, oe
(a) A1 2.5 for both cao
(a) Answer: 2.5 (for both Photo 2 and Photo 3)
(b) B1 they scored exactly the same, so there is no fair way to place one above the other oe
(b) Answer: They scored exactly the same, so neither can fairly be placed above the other.
(c) B1 this would treat the tie unfairly/arbitrarily, making one photo look better than the other when the scores show they are equal oe
(c) Answer: It would arbitrarily favour one photo over the other, even though their scores show they are equally good.

Question 11

(a) B1 ranking removes the effect of the two exams having different total marks/difficulty, so the two sets of results can be compared fairly oe
(a) Answer: Ranking lets the two exams be compared fairly even if they are marked out of different totals or have different levels of difficulty.
(b) B1 no pupil / none cao
(b) Answer: None - every pupil's rank changes between the two exams.
(c) B1 Yes, consistent - the rank orders are similar but not identical, matching a high but not perfect coefficient oe
(c) Answer: Yes - the two rank orders are close but not identical, which fits a high value close to but below 1.
(d) B1 rs = 0.83 is fairly close to 1, showing a strong positive association, so the conclusion has some support
(d) B1 balanced comment, e.g. the coefficient is not exactly 1, so mock rank does not predict final rank perfectly for every pupil
(d) Answer: The conclusion is broadly reasonable - rs = 0.83 shows a strong positive link, but since it is not 1, the mock exam does not predict every pupil's final rank exactly.

Question 12

(a) B1 an upward/rising trend, with points fairly close to a straight line oe
(a) Answer: The points show a rising trend, lying fairly close to a straight line.
(b) B1 Yes - a strong positive coefficient matches a rising pattern of points close to a straight line oe
(b) Answer: Yes, it is consistent.
(c) B1 identifies that the points do not lie exactly on a straight line / some pairs of ranks are swapped from the perfect order
(c) B1 links this to rs being less than 1: only a perfect match of every rank (all points exactly on a rising line) would give rs = 1
(c) Answer: Not every point lies exactly on the line - a few dogs' ranks are swapped slightly, so the coefficient is high but below the maximum of 1.

Question 13

(a) B1 Class P cao
(a) Answer: Class P
(b) B1 rs = 0.58 is noticeably lower/further from 1, showing the two teachers marking Class Q agree less closely, so their marking is more likely to need reconciling
(b) B1 rs = 0.91 is close to 1, showing the two teachers marking Class P already agree closely, so discussion is less necessary
(b) Answer: Class Q's teachers agree less closely (0.58 is well below 1), so their marking is more likely to need reconciling, while Class P's teachers already agree closely (0.91 is close to 1).

Question 14

(a) B1 rs = 0.4 is positive, so it cannot show disagreement/a negative association oe
(a) Answer: The statement is wrong because rs = 0.4 is positive, so it shows some agreement, not disagreement.
(b) B1 a weak/fairly weak positive association between the two judges' rankings oe
(b) Answer: There is a weak positive association - the judges show only a small tendency to agree.
(c) B1 -1 cao
(c) Answer: -1

Question 15

B1 identifies rs = 0.35 as a positive but weak/fairly weak association between the two judges' rankings
B1 explains that this means the judges do not agree closely on the order of the gymnasts
B1 reaches a sensible recommendation, e.g. a third judge should be brought in because the two current judges do not agree closely enough to be confident in the medal order
B1 justifies the recommendation by reference to the size of rs, e.g. because 0.35 is much closer to 0 than to 1, relying on just these two rankings risks an unfair medal decision
Answer: Recommendation: bring in a third judge, because rs = 0.35 shows only a weak positive agreement between Judge A and Judge B, so their combined ranking is not reliable enough on its own to decide medal positions fairly.

Question 16

(a) B1 X cao
(a) Answer: X
(b) B1 rs for Y is negative, showing the two assessors' rankings run in opposite directions oe
(b) B1 -0.63 is a fairly large negative value (not close to 0), so this disagreement is fairly strong, not weak oe
(b) Answer: Y's coefficient is negative and fairly large in size, so the assessors' rankings tend to run in opposite directions - a real disagreement, not just a weak link.
(c) B1 rs measures agreement between the two rankings, not the quality/ability of either assessor oe
(c) B1 a low rs simply means the two assessors are ranking the entries in largely different orders from each other, which is not the same as either ranking being wrong oe
(c) Answer: rs only measures how closely the two rankings match each other, not whether either assessor is good at judging; a low value just means their orders differ, not that one (or both) is incorrect.

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