State the formula for rs.

rs=1-\frac{6\sum d^2}{n(n^2-1)}

What is d in the formula?

The difference between the two ranks for each item.

What does rs=+1 mean?

The two rankings are identical.

What does rs=-1 mean?

One ranking is the exact reverse of the other.

How are tied values ranked?

Each gets the mean of the ranks they would have occupied, e.g. values in 4th and 5th place both get 4.5.

What happens to rs if one ranking is reversed?

The sign changes and the size stays the same.

When is Spearman's coefficient better than the product moment correlation coefficient?

When the relationship is monotonic but not linear, when the data are ranks, or when there are outliers or the data are not bivariate Normal.

Give one limitation of Spearman's coefficient.

It uses only the order, so it ignores the size of the differences between values.

Does a high rs show causation?

No. Another factor could cause both variables to change.

How should you interpret rs in an exam?

State the strength and direction of the correlation between the ranks, in the context of the question.

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

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What does Spearman's rank correlation coefficient measure?

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What does Spearman's rank correlation coefficient measure?: The agreement between two sets of ranks, from $-1$ (reverse order) to $+1$ (same order).
State the formula for $r_s$ .: $r_s=1-\frac{6\sum d^2}{n(n^2-1)}$
What is $d$ in the formula?: The difference between the two ranks for each item.
What does $r_s=+1$ mean?: The two rankings are identical.
What does $r_s=-1$ mean?: One ranking is the exact reverse of the other.
How are tied values ranked?: Each gets the mean of the ranks they would have occupied, e.g. values in 4th and 5th place both get 4.5.
What happens to $r_s$ if one ranking is reversed?: The sign changes and the size stays the same.
When is Spearman's coefficient better than the product moment correlation coefficient?: When the relationship is monotonic but not linear, when the data are ranks, or when there are outliers or the data are not bivariate Normal.
Give one limitation of Spearman's coefficient.: It uses only the order, so it ignores the size of the differences between values.
Does a high $r_s$ show causation?: No. Another factor could cause both variables to change.
What is $\sum d$ for any two sets of ranks?: $0$
Six items, $\sum d^2=8$ . Find $r_s$ .: $1-\frac{48}{210}=0.771$
How should you interpret $r_s$ in an exam?: State the strength and direction of the correlation between the ranks, in the context of the question.

Exam questions on Spearman's rank correlation coefficient

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.
Interpret the value of the coefficient for the two judges.2 marks
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 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
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.
Rank the blends by price and by rating, and calculate Spearman's rank correlation coefficient.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).