Spearman's Rank Correlation
Spearman's rank correlation coefficient measures the strength and direction of a relationship between two ranked variables, giving a value between -1 (perfect negative) and +1 (perfect positive). In GCSE Statistics it is calculated using rs = 1 - (6 x sum of d^2) / (n(n^2 - 1)), where d is the difference between each pair of ranks and n is the number of pairs.
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Method
- Rank each set of data separately, giving equal (averaged) ranks to any tied values.
- Find the difference, d, between the two ranks for each pair of data.
- Square each difference to get d^2, then find the sum of all the d^2 values.
- Substitute the sum of d^2 and the number of pairs, n, into the formula rs = 1 - (6 x sum of d^2) / (n(n^2 - 1)).
- Calculate rs, giving the answer as a decimal, usually to 2 or 3 decimal places, between -1 and 1.
- Interpret the value: close to +1 means strong positive correlation between the rankings, close to -1 means strong negative correlation, and close to 0 means little or no correlation.
Worked example
Five students' scores in a maths test and a science test are ranked below. Maths rank: 1, 2, 3, 4, 5. Science rank: 2, 1, 3, 5, 4. Calculate Spearman's rank correlation coefficient.
- Find d for each pair: 1-2=-1, 2-1=1, 3-3=0, 4-5=-1, 5-4=1.
- Square each difference: 1, 1, 0, 1, 1.
- Sum the squared differences: sum d^2 = 1+1+0+1+1 = 4.
- Substitute into the formula with n = 5: rs = 1 - (6 x 4) / (5(25-1)) = 1 - 24/120.
- Simplify: rs = 1 - 0.2 = 0.8.
- Final answer: rs = 0.8, showing a strong positive correlation between the two sets of ranks.
Practice questions
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Q1Two judges rank 4 dogs in a competition. Judge A's ranks: 1, 2, 3, 4. Judge B's ranks: 1, 2, 3, 4. Find the sum of d^2.Show answer
Answer: 0 (every difference is 0).
Q2Using the data in the previous question, find Spearman's rank correlation coefficient.Show answer
Answer: rs = 1 (perfect agreement, since sum of d^2 = 0).
Q3Two rankings of 5 items give differences d: 1, -1, 2, -2, 0. Find the sum of d^2.Show answer
Answer: 10 (1+1+4+4+0).
Q4Using n = 5 and sum of d^2 = 10, calculate Spearman's rank correlation coefficient.Show answer
Answer: rs = 0.5 (1 - (6x10)/(5x24) = 1 - 60/120).
Q5A calculation gives Spearman's rank correlation coefficient of -0.9. Describe what this shows about the relationship between the two rankings.Show answer
Answer: A strong negative correlation: as one ranking increases, the other tends to decrease.
Q6Six pairs of ranks give sum of d^2 = 14. Calculate Spearman's rank correlation coefficient, giving your answer to 2 decimal places.Show answer
Answer: 0.60 (1 - (6x14)/(6x35) = 1 - 84/210 = 1 - 0.4).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
Two examiners rank 5 essays. Their rankings differ by d = 0, 1, -1, 2, -2 for the five essays. Calculate Spearman's rank correlation coefficient.
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The table shows the ranks given by two panels to 6 films. Panel 1 ranks: 1, 2, 3, 4, 5, 6. Panel 2 ranks: 2, 1, 4, 3, 6, 5. (a) Find the value of sum of d^2. (b) Calculate Spearman's rank correlation coefficient, giving your answer to 2 decimal places.
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Eight athletes are ranked by two different coaches based on performance. The sum of the squared rank differences is sum d^2 = 20. (a) Calculate Spearman's rank correlation coefficient, giving your answer to 2 decimal places. (b) Interpret your value in the context of the two coaches' rankings.
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Free printable worksheet
Want more practice on paper? Download the spearman's rank correlation worksheet pack - 19 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 14 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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