Karl Pearson’s Coefficient of Correlation: Methods & Properties

1. What Is Karl Pearson’s Coefficient of Correlation?

Introduced by British mathematician Karl Pearson, the Product Moment Correlation Coefficient (denoted by the symbol r) is the most widely used mathematical method for measuring the degree of linear association between two quantitative variables.

Mathematical Concept:
Pearson’s r is defined as the ratio of the Covariance between two variables to the product of their individual Standard Deviations.

Covariance-Based Definition Formula

Conceptual Formula

r = Cov(X, Y) ÷ (σx × σy)

Cov(X, Y) = [ Σ(X - x̄)(Y - ȳ) ] ÷ N

2. Practical Calculation Formulas

Depending on whether the actual means (x̄ and ȳ) are whole integers or decimals, statisticians choose one of two computational formulas:

A. Deviations from Actual Mean Method (Best when Means are whole numbers):

Let x = X - x̄ and y = Y - ȳ

r = (Σxy) ÷ √[ (Σx²) × (Σy²) ]

B. Shortcut Method (Assumed Mean Method — Best for large/decimal data):

Let dx = X - Ax and dy = Y - Ay (where Ax and Ay are assumed means)

The Shortcut Pearson Formula

Master Shortcut Formula

r = [ N(Σdx·dy) - (Σdx)(Σdy) ] ÷ √[ [N·Σdx² - (Σdx)²] × [N·Σdy² - (Σdy)²] ]

N = number of paired observations | Eliminates all decimal fractions during calculation

3. Solved Numerical Example (Actual Mean Method)

Problem: Calculate Karl Pearson’s Coefficient of Correlation between Advertising Spend (X in ₹ Lakhs) and Sales (Y in ₹ Crores) for 5 quarters:

Quarter	1	2	3	4	5
Ad Spend (X)	4	6	8	10	12
Sales (Y)	10	15	20	25	30

Step-by-Step Solution:

Find Means:
ΣX = 4 + 6 + 8 + 10 + 12 = 40 → x̄ = 40 ÷ 5 = 8
ΣY = 10 + 15 + 20 + 25 + 30 = 100 → ȳ = 100 ÷ 5 = 20
Construct Calculation Table:

X	Y	x = X - 8	y = Y - 20	x²	y²	xy
4	10	-4	-10	16	100	+40
6	15	-2	-5	4	25	+10
8	20	0	0	0	0	0
10	25	+2	+5	4	25	+10
12	30	+4	+10	16	100	+40
Total	—	0	0	Σx² = 40	Σy² = 250	Σxy = 100

Apply Formula:
r = (Σxy) ÷ √[ (Σx²) × (Σy²) ]
r = 100 ÷ √[ 40 × 250 ]
r = 100 ÷ √[ 10,000 ] = 100 ÷ 100 = +1.0

Conclusion: There is a Perfect Positive Linear Correlation (r = +1.0) between advertising expenditure and sales revenue!

4. Fundamental Properties of Correlation Coefficient (r)

University examiners love testing these 5 mathematical properties:

Strict Limits: The numerical value of r strictly lies between -1 and +1:
-1 ≤ r ≤ +1
Independent of Change of Origin & Scale:
If you add/subtract a constant number to X (origin) or multiply/divide by a constant (scale), the value of r remains 100% unchanged!
Pure Dimensionless Number: r has no units of measurement (it is not in ₹, kg, or cm).
Symmetrical Property: The correlation between X and Y is identical to the correlation between Y and X:
r_xy = r_yx
Geometric Mean of Regression Coefficients:
r = ± √(b_yx × b_xy)
(The sign of r is always identical to the sign of the regression coefficients!)

5. Degree of Correlation Guidelines

Range of ‘r’ Value	Qualitative Interpretation
r = +1.0	Perfect Positive Correlation
+0.75 ≤ r < +1.0	High Degree Positive Correlation
+0.25 ≤ r < +0.75	Moderate Degree Positive Correlation
0 < r < +0.25	Low Degree Positive Correlation
r = 0	Zero / No Linear Correlation
-0.75 < r ≤ -1.0	High Degree Negative Correlation
r = -1.0	Perfect Negative Correlation

6. Probable Error (PE) of Correlation Coefficient

To determine whether an observed correlation coefficient r is genuine or merely a chance result of sampling variation, statisticians calculate the Probable Error (PE):

Formula:
PE = 0.6745 × [ (1 - r²) ÷ √N ]
(where Standard Error (SE) = (1 - r²) ÷ √N and PE = 0.6745 × SE)

Decision Rules for Significance:

If r < PE: The correlation is insignificant and unreliable (likely a random fluke).
If r > 6 × PE: The correlation is statistically significant, robust, and highly dependable for management forecasting!

Karl Pearson's Coefficient of Correlation: Methods & Properties | BBA Notes