What Spearman rank correlation means
Big idea: Spearman rank correlation measures how strongly two rankings move together.
It is used for ordinal data or non-linear monotonic patterns.
If higher rank in one variable usually matches higher rank in the other, correlation is positive.
How to calculate Spearman coefficient
Meaning of d: d is the difference between the two ranks for the same observation.
Worked example
For 5 students, rank differences d are: 1, -1, 0, 2, -2.
Find rs.
Step by step
- Square each d: 1,1,0,4,4
- Sum d2 = 10
- Use formula: rs = 1 - 6(10)/(5(25-1))
- rs = 1 - 60/120 = 0.5
Final answer
rs = 0.5, showing moderate positive rank correlation.
IB-style question — calculate and interpret rₛ [6 marks]
A manager records, for 6 sales representatives, the number of training hours they completed last month and their sales for the month (in thousands of pounds).
Rep: A B C D E F
Hours: 20 14 28 11 25 17
Sales (£000): 52 40 60 33 44 55
(a) Calculate Spearman's rank correlation coefficient rₛ between training hours and sales.
(b) Interpret your value of rₛ in context.
Step by step
- Rank each variable from 1 (highest) to 6 (lowest). Hours rank to A→3, B→5, C→1, D→6, E→2, F→4; sales rank to A→3, B→5, C→1, D→6, E→4, F→2.
- Find d (difference of the two ranks) for each rep, then square it. Only E and F differ.
- As a check the rank differences sum to zero, then add the squares.
- Write the formula before substituting (n = 6 pairs).
- Substitute n = 6 and Σd² = 8.
- (b) rₛ is close to +1, so reps who do more training tend to rank higher in sales — a strong positive monotonic association between training hours and sales.
Final answer
(a) rₛ = 0.771 (3 s.f.). (b) A strong positive monotonic association — more training hours tends to go with higher sales.
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Tied ranks and interpretation
Ties: If two values tie, assign each the average of the tied rank positions.
Worked example
Values: 10, 12, 12, 15.
What ranks are assigned?
Step by step
- 10 gets rank 1
- 12 and 12 would be ranks 2 and 3, so each gets (2+3)/2 = 2.5
- 15 gets rank 4
Final answer
Ranks are 1, 2.5, 2.5, 4.
IB-style question — rₛ with tied ranks [6 marks]
Seven students record their average daily screen time (hours) and their average nightly sleep (hours).
Student: 1 2 3 4 5 6 7
Screen (h): 5 3 6 4 5 2 7
Sleep (h): 7 8 5 7 6 9 4
(a) Rank each variable from 1 (highest) to 7 (lowest), giving tied values the average rank.
(b) Calculate Spearman's rank correlation coefficient rₛ.
(c) Interpret your answer in context.
Step by step
- (a) Screen time has two values of 5, which would take positions 3 and 4, so each gets the average rank 3.5.
- Sleep has two values of 7, which would take positions 3 and 4, so each gets 3.5.
- (b) Find each d (screen rank − sleep rank) and square it.
- Check Σd = 0, then sum the squares.
- Apply the formula with n = 7.
- (c) rₛ is close to −1, so students with more screen time tend to rank lower for sleep — a strong negative monotonic association between screen time and sleep.
Final answer
(a) Tied values each take the average rank (3.5). (b) rₛ = −0.938 (3 s.f.). (c) A strong negative monotonic association — more screen time tends to go with less sleep.
Exam Tips:
- Always show tied-rank calculation.
- Interpret sign and strength of rs in words.
- Do not claim causation.
Past-paper style responses
Weak response
- Only gives calculator output
- No context interpretation
- No mention of ties
Strong response
- Shows formula and key steps
- Interprets sign + strength in context
- Handles ties correctly
Full-credit habit: Write one final sentence in context, for example: There is a moderate positive association between revision ranking and test ranking.
IB-style question — Spearman hypothesis test
For n = 8 paired ranks, Spearman's rₛ = 0.83. The critical value at the 5% level is 0.738.
Test, at the 5% level, whether there is a monotonic relationship.
Step by step
- State the hypotheses (Spearman tests a MONOTONIC relationship, not 'linear').
- Compare |rₛ| with the critical value.
- Since it exceeds the critical value, reject H₀.
Final answer
0.83 > 0.738 ⇒ reject H₀: there is evidence of a monotonic relationship. (rₛ is unchanged by any value change that keeps the ranks the same.)