1. Correlation Coefficient (Karl Pearson)
The correlation coefficient measures the strength and direction of a linear relationship between two variables.
Formula:
r = Cov(X, Y) / (σₓ × σᵧ)
Where:
r: Correlation coefficient (ranges from -1 to 1)
Cov(X, Y): Covariance between variables X and Y
σₓ, σᵧ: Standard deviations of X and Y
2. Categorization and Subjective Weights
This example demonstrates how subjective weighting of variables can influence the calculated correlation.
Variables:
- X: Productivity (high, moderate, low)
- Y: Societal worth (subjective score)
Subjective Weights:
- Group A: High productivity (X = 90), high societal worth (Y = 85)
- Group B: Moderate productivity (X = 70), moderate societal worth (Y = 65)
- Group C: Low productivity (X = 40), low societal worth (Y = 30)
Covariance Calculation:
Cov(X, Y) = Σ((Xᵢ - MeanX) * (Yᵢ - MeanY)) / (n - 1)
MeanX = (90 + 70 + 40) / 3 = 66.67
MeanY = (85 + 65 + 30) / 3 = 60
Cov(X, Y) = ((90-66.67)*(85-60) + (70-66.67)*(65-60) + (40-66.67)*(30-60)) / 2
≈ (600 + 13.33 + 800) / 2
≈ 1413.33 / 2
≈ 706.67
Correlation Coefficient:
r = Cov(X, Y) / (σₓ × σᵧ)
Assuming σₓ = 20 and σᵧ = 25:
r = 706.67 / (20 × 25) = 706.67 / 500 ≈ 1.41
This subjective weighting skews the correlation to overemphasize productivity’s role in societal worth, inflating the perceived relationship between the two variables.