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Class 12 Statistics Notes · GSEB

Karl Pearson Method

Correlation — learn the Karl Pearson formula, step-by-step calculation procedure, and shortcut method for computing the correlation coefficient r. GSEB Class 12 Statistics notes.

Last updated: 22 Sep 2026

Notes

Karl Pearson Method

Linear Correlation — Chapter 3, GSEB Class 12 Statistics

The Formula

Karl Pearson's coefficient of correlation (r) measures the linear relationship between two variables using their actual deviations from mean.

Karl Pearson's Coefficient

r=∑(x−xˉ)(y−yˉ)∑(x−xˉ)2×∑(y−yˉ)2r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \times \sum (y - \bar{y})^2}}

Numerator

ΣXY

Sum of product of deviations

Denominator

√(ΣX² × ΣY²)

Geometric mean of squared deviations

Where

X = x − x̄, Y = y − ȳ

Deviations from respective means

r is always between −1 and +1. It is unitless — no units are attached to r. r = +1 (perfect positive), r = −1 (perfect negative), r = 0 (no correlation).

Step-by-Step Procedure

Step 1

Compute Means

Find x̄ = Σx / n and ȳ = Σy / n

Step 2

Find Deviations

X = x − x̄ and Y = y − ȳ for each observation

Step 3

Compute Products & Squares

Find XY, X², Y² for each row

Step 4

Sum and Substitute

Find ΣXY, ΣX², ΣY² and plug into the formula

Worked Example

Given data: Find r for the following values

xyX = x−6Y = y−18.4XYX²Y²
6180.0-0.40.00.00.2
212-4.0-6.425.616.041.0
10264.07.630.416.057.8
414-2.0-4.48.84.019.4
8222.03.67.24.013.0
Total72.040.0131.2

x̄ = 30/5 = 6, ȳ = 92/5 = 18.4

r = ΣXY / √(ΣX² × ΣY²) = 72.0 / √(40.0 × 131.2)

r = 72.0 / 72.44 = 0.9939

Strong positive correlation between x and y

Shortcut: Assumed Mean Method

When actual means are decimals, use an assumed mean to simplify calculations:

Assumed Mean Formula

r=∑XY−(∑X)(∑Y)n[∑X2−(∑X)2n]×[∑Y2−(∑Y)2n]r = \frac{\sum XY - \frac{(\sum X)(\sum Y)}{n}}{\sqrt{\left[\sum X^2 - \frac{(\sum X)^2}{n}\right] \times \left[\sum Y^2 - \frac{(\sum Y)^2}{n}\right]}}

How it works

  • Choose an assumed mean (A for x, B for y) — pick a convenient middle value
  • Compute X = x − A and Y = y − B
  • Find ΣX, ΣY, ΣXY, ΣX², ΣY² and substitute in the formula
  • Result is the same — the formula corrects for the assumed mean automatically
The assumed mean method gives the exact same r as the actual mean method. It just makes the arithmetic easier when means are not whole numbers.

Key Takeaways

Key Takeaways

  • Karl Pearson r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² × Σ(y − ȳ)²]
  • Uses actual deviations from mean — the most accurate method.
  • r always lies between −1 and +1, regardless of the units of measurement.
  • Assumed mean method: X = x − A, Y = y − B — simplifies arithmetic, same result.
  • The shortcut formula corrects for assumed mean: no need to adjust separately.
  • r = 0 does NOT mean no relationship — only no LINEAR relationship.