Compute Spearman’s Rho for the following data : Data 1 Data 2 10 7 9 10 7 6 9 4 8 5 3 9 4 8 11 12 12 11 5 2.

1. Introduction to Spearman's Rank Correlation

Spearman's rank correlation coefficient, denoted by ρ (rho), is a non-parametric measure of the strength and direction of the relationship between two variables. It assesses how well the relationship between two variables can be described using a monotonic function. Spearman's rho is suitable for ordinal or ranked data and does not assume that the variables follow a normal distribution.

2. Calculation of Spearman's Rho

Step 1: Rank the Data

Rank each set of data separately, assigning ranks from 1 to n (the number of data points), with 1 being assigned to the smallest value and n to the largest value. Ties are assigned the average of the ranks they would occupy if un-tied.

For Data 1:
10 → 1
9 → 2.5
8 → 4
7 → 5.5
6 → 7
5 → 8.5
4 → 10.5
3 → 13
2 → 14.5
1 → 16

For Data 2:
12 → 1
11 → 2
10 → 3.5
9 → 3.5
8 → 5.5
7 → 6.5
6 → 8
5 → 9
4 → 10.5
3 → 12
2 → 13.5
1 → 15

Step 2: Calculate the Differences in Ranks

Compute the difference between the ranks of corresponding pairs of data points.

For each pair of data points, subtract the rank of the corresponding data point in Data 2 from the rank of the corresponding data point in Data 1.

Step 3: Square the Differences

Square each of the differences calculated in Step 2.

Step 4: Calculate Spearman's Rho

Spearman's Rho (ρ) is given by the formula:

ρ = 1 – (6∑d^2 / (n^3 – n))

Where:

• ∑d^2 is the sum of the squared differences in ranks.
• n is the number of pairs of data points.

3. Calculation of Spearman's Rho

Using the computed ranks and differences, we can calculate Spearman's Rho:

∑d^2 = (1-1)^2 + (2.5-2)^2 + (3.5-3)^2 + (3.5-4)^2 + (5.5-5)^2 + (6-6)^2 + (7-8)^2 + (8.5-9)^2 + (10.5-10)^2 + (12-12)^2 + (13-11)^2 + (14.5-15)^2

∑d^2 = 0^2 + 0.25^2 + 0.5^2 + 0.5^2 + 0.25^2 + 0^2 + 1^2 + 0.25^2 + 0.5^2 + 0^2 + 4^2 + 0.5^2
∑d^2 = 0 + 0.0625 + 0.25 + 0.25 + 0.0625 + 0 + 1 + 0.0625 + 0.25 + 0 + 16 + 0.25
∑d^2 = 18.375

Now, plug this value into the formula for Spearman's Rho:

ρ = 1 – (6 * 18.375 / (20^3 – 20))

ρ = 1 – (6 * 18.375 / (8000 – 20))

ρ = 1 – (110.25 / 7980)

ρ ≈ 1 – 0.0138

ρ ≈ 0.9862

Spearman's Rho (ρ) ≈ 0.9862

4. Interpretation of Spearman's Rho

Spearman's Rho ranges from -1 to 1. A value of 1 indicates a perfect positive monotonic relationship, a value of -1 indicates a perfect negative monotonic relationship, and a value of 0 indicates no monotonic relationship.

In this case, Spearman's Rho is approximately 0.9862, indicating a strong positive monotonic relationship between the two sets of data. This suggests that as the values in one set increase, the values in the other set also tend to increase, and vice versa, albeit not necessarily at a constant rate.