Karl Pearson’s Coefficient of Correlation (r)

Karl Pearson’s method is the most widely used numerical measure of correlation. It tells us both the direction and strength of the linear relationship between two variables.

It is also called the Product Moment Correlation Coefficient.

Meaning

Karl Pearson’s r measures the degree of linear relationship between two variables (X and Y). It indicates:

• Positive correlation → r > 0
• Negative correlation → r < 0
• Zero correlation → r = 0

The value of r always lies between –1 and +1.

Formula (Ungrouped Data)

          Σ(X − X̄)(Y − Ȳ)
r =  --------------------------------
      √[ Σ(X − X̄)² × Σ(Y − Ȳ)² ]

Where:

• X̄ = Mean of X
• Ȳ = Mean of Y

Shortcut (Coding) Formula

          nΣXY − (ΣX)(ΣY)
r = ---------------------------------------------
      √[ nΣX² − (ΣX)² ] × √[ nΣY² − (ΣY)² ]

This method is preferred because it saves time in exams.

Properties of Pearson’s r

• r lies between –1 and +1.
• r = +1 → Perfect positive linear correlation.
• r = –1 → Perfect negative linear correlation.
• r = 0 → No linear correlation.
• r is unit-free (no measurement units).
• Sensitive to extreme values (outliers).

Steps to Calculate Pearson’s r (Shortcut Method)

1. Prepare a table with X, Y, X², Y², and XY.
2. Compute ΣX, ΣY, ΣX², ΣY², and ΣXY.
3. Substitute values into the shortcut formula.
4. Compute r and interpret.

Solved Example

Calculate Pearson’s r using the following data:

Step 1: Prepare Table

Step 2: Totals

• ΣX = 150
• ΣY = 175
• ΣX² = 5500
• ΣY² = 7125
• ΣXY = 6250
• n = 5

Step 3: Apply Formula

          nΣXY − (ΣX)(ΣY)
r = ---------------------------------------------
      √[ nΣX² − (ΣX)² ] × √[ nΣY² − (ΣY)² ]

Substitute:

r = [5(6250) − 150×175] ÷ √[(5×5500 − 150²)(5×7125 − 175²)]

r = [31250 − 26250] ÷ √[(27500 − 22500)(35625 − 30625)]

r = 5000 ÷ √[(5000)(5000)]

r = 5000 ÷ 5000

r = +1

Interpretation

Perfect positive correlation.

Interpretation Guidelines

Advantages of Pearson’s Method

• Simple and widely used.
• Measures both direction and strength.
• Based on all observations.
• Mathematically sound.

Limitations

• Only for linear relationships.
• Affected by outliers.
• Requires quantitative data.
• Cannot establish cause-effect.

Summary

• Pearson’s r measures linear correlation between X and Y.
• Value lies between –1 and +1.
• Shortcut formula is used in exams.
• Provides direction (positive/negative) and strength.
• Does not imply causation.

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