Further Maths Spearman’s rank correlation Practice Questions

Free Further Maths Spearman’s rank correlation practice questions with full step-by-step worked solutions. Covers spearman, rank-correlation, formula, rearrangement. Practise exam-style problems and check your method.

spearmanrank-correlationformularearrangementperfect-negativeranking
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
A set of n=8n = 8 bivariate observations is ranked, with no tied ranks, and dd denotes the difference between the two ranks of an observation. Given that ∑d2=30\sum d^{2} = 30, calculate Spearman's rank correlation coefficient rsr_{s}, giving your answer correct to 33 decimal places.
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Worked solution

  1. Quote the formula for Spearman's rank correlation coefficient

    rs=1−6∑d2n(n2−1)r_{s}=1-\dfrac{6\sum d^{2}}{n\left(n^{2}-1\right)}

    This is the standard form, valid when there are no tied ranks.

  2. Substitute nn and ∑d2\sum d^{2}

    rs=1−6×308(82−1)r_{s}=1-\frac{6\times 30}{8\left(8^{2}-1\right)}

    Only the sample size and the total of the squared rank differences are needed.

  3. Evaluate the fraction

    6×30504=514\frac{6\times 30}{504}=\frac{5}{14}

    Keeping the fraction exact means no rounding error can creep in.

  4. State the value of rsr_{s}

    rs=1−514=914=0.643r_{s}=1-\frac{5}{14}=\frac{9}{14}=0.643

    This is Spearman's rank correlation coefficient for the data.

Answer
0.6430.643
Question 2
2 markseasy
A researcher records the finishing position of each of 1212 runners in two different cross-country races. The only information available is the order in which the runners finished. Which of the following statements about the choice of correlation coefficient is correct?
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Worked solution

  1. Identify what each coefficient measures

    r: linear association;rs: monotonic associationr:\ \text{linear association};\qquad r_{s}:\ \text{monotonic association}

    The product moment coefficient measures how close the points lie to a straight line; Spearman's coefficient measures how well the order of one variable matches the order of the other.

  2. Identify the feature of these data that decides the choice

    ordinal data\text{ordinal data}

    This is the feature that makes the product moment coefficient unsuitable here.

  3. Apply the criterion

    ⇒ use rs\Rightarrow\ \text{use }r_{s}

    Because the data are ordinal: only the order is known, and no meaningful arithmetic can be done with finishing positions, so a rank method is the only one available.

  4. State which coefficient should be used

    rs (Spearman’s rank correlation coefficient)r_{s}\ \text{(Spearman's rank correlation coefficient)}

    A rank method is preferred here because the data are ordinal: only the order is known, and no meaningful arithmetic can be done with finishing positions, so a rank method is the only one available.

Answer
rs (Spearman’s rank correlation coefficient)r_{s}\ \text{(Spearman's rank correlation coefficient)}
Question 3
4 marksintermediate
A scatter diagram of 1010 observations shows the points lying very close to a smooth increasing curve which is definitely not a straight line. Which of the following statements about the choice of correlation coefficient is correct?
Show worked solution

Worked solution

  1. Identify what each coefficient measures

    r: linear association;rs: monotonic associationr:\ \text{linear association};\qquad r_{s}:\ \text{monotonic association}

    The product moment coefficient measures how close the points lie to a straight line; Spearman's coefficient measures how well the order of one variable matches the order of the other.

  2. Identify the feature of these data that decides the choice

    a monotonic but non-linear relationship\text{a monotonic but non-linear relationship}

    This is the feature that makes the product moment coefficient unsuitable here.

  3. Apply the criterion

    ⇒ use rs\Rightarrow\ \text{use }r_{s}

    Because the relationship is monotonic but not linear, so the product moment coefficient understates a relationship that the ranks describe perfectly.

  4. Quote Spearman's rank correlation coefficient

    rs=1−6∑d2n(n2−1)r_{s}=1-\frac{6\sum d^{2}}{n\left(n^{2}-1\right)}

    This is the formula in the booklet: dd is the difference between the two ranks of a single item and nn is the number of items.

  5. Note what dd means

    d=rank of x−rank of yd=\text{rank of }x-\text{rank of }y

    The sign of dd never matters, because only d2d^{2} is used.

  6. Check the ranks against a known total

    ∑ranks=n(n+1)2\sum\text{ranks}=\frac{n\left(n+1\right)}{2}

    Both sets of ranks must add up to this, even when tied ranks have been averaged.

  7. State which coefficient should be used

    rs (Spearman’s rank correlation coefficient)r_{s}\ \text{(Spearman's rank correlation coefficient)}

    A rank method is preferred here because the relationship is monotonic but not linear, so the product moment coefficient understates a relationship that the ranks describe perfectly.

Answer
rs (Spearman’s rank correlation coefficient)r_{s}\ \text{(Spearman's rank correlation coefficient)}
Question 4
6 markshard
A scatter diagram of 99 observations shows 88 points in a loose cluster and one point a very long way from all the others. Which of the following statements about the choice of correlation coefficient is correct?
Show worked solution

Worked solution

  1. Identify what each coefficient measures

    r: linear association;rs: monotonic associationr:\ \text{linear association};\qquad r_{s}:\ \text{monotonic association}

    The product moment coefficient measures how close the points lie to a straight line; Spearman's coefficient measures how well the order of one variable matches the order of the other.

  2. Identify the feature of these data that decides the choice

    an outlier\text{an outlier}

    This is the feature that makes the product moment coefficient unsuitable here.

  3. Apply the criterion

    ⇒ use rs\Rightarrow\ \text{use }r_{s}

    Because a single outlier can have an enormous effect on the product moment coefficient, but it changes the ranks by only a small amount.

  4. Quote Spearman's rank correlation coefficient

    rs=1−6∑d2n(n2−1)r_{s}=1-\frac{6\sum d^{2}}{n\left(n^{2}-1\right)}

    This is the formula in the booklet: dd is the difference between the two ranks of a single item and nn is the number of items.

  5. Note what dd means

    d=rank of x−rank of yd=\text{rank of }x-\text{rank of }y

    The sign of dd never matters, because only d2d^{2} is used.

  6. Check the ranks against a known total

    ∑ranks=n(n+1)2\sum\text{ranks}=\frac{n\left(n+1\right)}{2}

    Both sets of ranks must add up to this, even when tied ranks have been averaged.

  7. Check that the differences sum to zero

    ∑d=0\sum d=0

    The two sets of ranks have the same total, so the differences must cancel; this also forces ∑d2\sum d^{2} to be even.

  8. Note the range of the coefficient

    −1≤rs≤1-1\le r_{s}\le 1

    A value outside this range can only be an arithmetic error.

  9. Recall what rsr_{s} actually is

    rs=PMCC of the ranksr_{s}=\text{PMCC of the ranks}

    Spearman's coefficient is the product moment correlation coefficient applied to the ranks instead of to the raw data.

  10. State which coefficient should be used

    rs (Spearman’s rank correlation coefficient)r_{s}\ \text{(Spearman's rank correlation coefficient)}

    A rank method is preferred here because a single outlier can have an enormous effect on the product moment coefficient, but it changes the ranks by only a small amount.

Answer
rs (Spearman’s rank correlation coefficient)r_{s}\ \text{(Spearman's rank correlation coefficient)}
Question 5
9 markschallenging
The table shows the mass, in kg, and the price, in pounds, of each of 88 items sold by a shop. One item is a very heavy and very expensive machine. ItemABCDEFGHMass (kg)1112131415161890Price (£)3331302826252295\begin{array}{l|cccccccc} \text{Item} & A & B & C & D & E & F & G & H \\ \hline \text{Mass (kg)} & 11 & 12 & 13 & 14 & 15 & 16 & 18 & 90 \\ \text{Price (\pounds)} & 33 & 31 & 30 & 28 & 26 & 25 & 22 & 95 \end{array} Rank 11 is given to the largest value in each row. Which of the following gives the correct values of Spearman's rank correlation coefficient rsr_{s} and of the product moment correlation coefficient rr of the raw data, together with the correct reason for preferring the rank coefficient here?
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Worked solution

  1. Rank both variables

    ItemABCDEFGHRank of x87654321Rank of y23456781\begin{array}{l|cccccccc} \text{Item} & A & B & C & D & E & F & G & H \\ \hline \text{Rank of x} & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\ \text{Rank of y} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 1 \end{array}

    Rank 11 is given to the largest value of each variable.

  2. Find Spearman's coefficient from the ranks

    rs=−0.333r_{s}=-0.333

    This depends only on the order of the data, not on the values themselves.

  3. Find the product moment correlation coefficient of the raw data

    r=SxySxxSyy=0.974r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=0.974

    This uses the actual values, so it is sensitive to the extreme observation.

  4. Compare the two coefficients

    rs=−0.333,r=0.974r_{s}=-0.333,\qquad r=0.974

    They measure different things: rr measures LINEAR association and rsr_{s} measures MONOTONIC association.

  5. Quote Spearman's rank correlation coefficient

    rs=1−6∑d2n(n2−1)r_{s}=1-\frac{6\sum d^{2}}{n\left(n^{2}-1\right)}

    This is the formula in the booklet: dd is the difference between the two ranks of a single item and nn is the number of items.

  6. Note what dd means

    d=rank of x−rank of yd=\text{rank of }x-\text{rank of }y

    The sign of dd never matters, because only d2d^{2} is used.

  7. Check the ranks against a known total

    ∑ranks=n(n+1)2\sum\text{ranks}=\frac{n\left(n+1\right)}{2}

    Both sets of ranks must add up to this, even when tied ranks have been averaged.

  8. Check that the differences sum to zero

    ∑d=0\sum d=0

    The two sets of ranks have the same total, so the differences must cancel; this also forces ∑d2\sum d^{2} to be even.

  9. Note the range of the coefficient

    −1≤rs≤1-1\le r_{s}\le 1

    A value outside this range can only be an arithmetic error.

  10. Recall what rsr_{s} actually is

    rs=PMCC of the ranksr_{s}=\text{PMCC of the ranks}

    Spearman's coefficient is the product moment correlation coefficient applied to the ranks instead of to the raw data.

  11. Note when the closed formula may be used

    no tied ranks⇒1−6∑d2n(n2−1)=PMCC of the ranks\text{no tied ranks}\Rightarrow 1-\frac{6\sum d^{2}}{n\left(n^{2}-1\right)}=\text{PMCC of the ranks}

    The closed formula is derived on the assumption that the ranks are 1,2,…,n1,2,\ldots,n in some order; once ranks are averaged that is no longer true.

  12. Recall the rule for tied values

    tied values share the mean of the ranks they would occupy\text{tied values share the mean of the ranks they would occupy}

    Two values tied for 33rd and 44th place each receive the rank 3.53.5.

  13. Recall the product moment correlation coefficient

    r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

    Applied to the ranks, this is exactly Spearman's coefficient.

  14. Note what perfect agreement means

    ∑d2=0  ⟺  rs=1\sum d^{2}=0\iff r_{s}=1

    The two rankings are then identical, item by item.

  15. Decide which coefficient is appropriate

    use rs\text{use }r_{s}

    A rank method is preferred here because one extreme observation would otherwise dominate the product moment coefficient, while ranking reduces it to a single ordinary rank.

Answer
use rs\text{use }r_{s}

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