Example: how to calculate Spearman Correlation?

Sample question

The scores for nine students in history and algebra are as follows:

History: 35, 23, 47, 17, 10, 43, 9, 6, 28
Algebra: 30, 33, 45, 23, 8, 49, 12, 4, 31

Compute the Spearman rank correlation.

Answer

Step 1: rank each student

Student #	History Score	History Rank	Algebra Score	Algebra Rank
1	35	3	30	5
2	23	5	33	3
3	47	1	45	2
4	17	6	23	6
5	10	7	8	8
6	43	2	49	1
7	9	8	12	7
8	6	9	4	9
9	28	4	31	4

step 2: calculate difference between the ranks (d) and square d

Student #	History Score	History Rank	Algebra Score	Algebra Rank	d	d²
1	35	3	30	5	2	4
2	23	5	33	3	2	4
3	47	1	45	2	1	1
4	17	6	23	6	0	0
5	10	7	8	8	1	1
6	43	2	49	1	1	1
7	9	8	12	7	1	1
8	6	9	4	9	0	0
9	28	4	31	4	0	0

step 3: sum (add up) all the d² scores

Σ d² = 4 + 4 + 1 + 0 + 1 + 1 + 1 + 0 + 0 = 12

step 4: insert the values in the formula. These ranks are not tied, so use the first formula:

$\rho = 1 - \frac{6\sum{d^2_i}}{n(n^2-1)}$

= 1 – (6*12)/(9(81-1))
= 1 – 72/720
= 1-0.1
= 0.9

> The Spearman Rank Correlation for this set of data is 0.9.

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