Master Spearman's Rank Correlation for CBSE Class 11 Applied Mathematics. Learn formulas, step-by-step calculations, and practice exam-style problems.
Spearman's Rank Correlation Coefficient, denoted by ρ (rho), is a non-parametric measure of rank correlation that assesses how well the relationship between two variables can be described using a monotonic function. It is used when data are ordinal or when the assumptions of Pearson's correlation are not met. The coefficient ranges from -1 to +1, where +1 indicates a perfect positive correlation of ranks.
Step 1: Given ranks (x, y) = (1, 3), (2, 2), (3, 1).
Step 2: Calculate dᵢ = xᵢ - yᵢ: d₁ = -2, d₂ = 0, d₃ = 2.
Step 3: Calculate dᵢ²: 4, 0, 4. Σ dᵢ² = 8.
Step 4: n = 3. ρ = 1 - (6 × 8) / (3(9 - 1)) = 1 - 48 / 24 = 1 - 2 = -1.
Answer: ρ = -1
The value of ρ is independent of the scale of measurement, as it relies solely on the relative order of the data points. If the ranks are identical for both variables, ρ = 1. If the ranks are in exact reverse order, ρ = -1. This method is particularly useful for qualitative data like beauty contests or performance rankings.
Step 1: Let ranks be (1, 2, 3, 4) and (1, 2, 3, 4).
Step 2: dᵢ = 0 for all i, so Σ dᵢ² = 0.
Step 3: ρ = 1 - (6 × 0) / (4³ - 4) = 1 - 0 = 1.
Answer: ρ = 1
When all values in the dataset are distinct, we assign ranks from 1 to n. The formula remains consistent as there are no tied ranks to adjust. This simplifies the calculation significantly as we do not need to apply correction factors.
Step 1: Σ dᵢ² = 10, n = 5.
Step 2: n³ - n = 125 - 5 = 120.
Step 3: ρ = 1 - (6 × 10) / 120 = 1 - 60 / 120 = 1 - 0.5 = 0.5.
Answer: ρ = 0.5