An example of how to work out Spearman's rank correlation coefficient.

Let's say we have two sets of data:

Set A: 10, 15, 20, 25, 30
Set B: 5, 10, 15, 20, 25

Step 1: Rank the data in each set from smallest to largest.

Set A: 10(2), 15(3), 20(4), 25(5), 30(1)
Set B: 5(2), 10(3), 15(4), 20(5), 25(1)

Step 2: Calculate the difference in ranks for each pair of data points.

10-2 = 8
15-3 = 12
20-4 = 16
25-5 = 20
30-1 = 29

Step 3: Square the differences.

8^2 = 64
12^2 = 144
16^2 = 256
20^2 = 400
29^2 = 841

Step 4: Calculate the sum of the squared differences.

64 + 144 + 256 + 400 + 841 = 1705

Step 5: Use the formula for Spearman's rank correlation coefficient:

r = 1 - (6 * Σd^2) / (n * (n^2 - 1))

where Σd^2 is the sum of the squared differences, and n is the number of data points.

In this case, n = 5, so:

r = 1 - (6 * 1705) / (5 * (5^2 - 1))
r = 1 - (10230) / (5 * 24)
r = 1 - 10230 / 120
r = 1 - 85.25
r = -84.25

Therefore, the Spearman's rank correlation coefficient for the two sets of data is -0.84.

Data:

| Rank X | Rank Y |
|---|---|
| 1 | 4 |
| 3 | 3 |
| 2 | 1 |
| 5 | 2 |

Steps:

1. Calculate the difference in ranks:

| Rank X | Rank Y | Difference (D) |
|---|---|---|
| 1 | 4 | -3 |
| 3 | 3 | 0 |
| 2 | 1 | 1 |
| 5 | 2 | 3 |

2. Square the differences:

| Rank X | Rank Y | Difference (D) | D^2 |
|---|---|---|---|
| 1 | 4 | -3 | 9 |
| 3 | 3 | 0 | 0 |
| 2 | 1 | 1 | 1 |
| 5 | 2 | 3 | 9 |

3. Sum the squared differences:

```
ΣD^2 = 9 + 0 + 1 + 9 = 19
```

4. Calculate the number of pairs:

```
n = 4
```

5. Calculate Spearman's rank correlation coefficient (rs):

```
rs = 1 - (6ΣD^2) / (n^3 - n)
```

```
rs = 1 - (6 - 19) / (4^3 - 4)
```

```
rs = 1 - 114 / 60
```

```
rs = 0.05
```

Interpretation:

The Spearman's rank correlation coefficient (rs) is 0.05, indicating a very weak positive correlation between Rank X and Rank Y. This means that the ranks of the two variables tend to increase or decrease together to a very small degree.