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Class 12 Statistics Notes · GSEB

Properties of r

Correlation — learn the 6 key properties of the correlation coefficient including bounded range, independence from origin and scale, and relationship with regression coefficients. GSEB Class 12 Statistics notes.

Last updated: 22 Sep 2026

Notes

Properties of r

Linear Correlation — Chapter 4, GSEB Class 12 Statistics

Properties of the Correlation Coefficient

The correlation coefficient r has six important properties that govern how it behaves:

1

Bounded Between −1 and +1

−1 ≤ r ≤ +1 always. No value of r can exceed this range.

If you get r = 1.2 or r = −1.5, there is a calculation error. The coefficient is mathematically bounded.

2

Independent of Change of Origin

Adding or subtracting a constant from all values of x or y does not change r.

If you add 10 to every x value (shifting the origin), r remains the same. The correlation depends on the pattern, not the position.

3

Independent of Change of Scale

Multiplying or dividing all values of x or y by a constant does not change r.

If you convert height from cm to inches (multiply by 0.3937), r stays the same. The relationship pattern is unchanged.

4

r = +1 or −1 Means Perfect Correlation

r = +1: perfect positive linear relationship. r = −1: perfect negative linear relationship.

All dots lie exactly on a straight line. In practice, perfect correlation is extremely rare with real data.

5

Geometric Mean of Regression Coefficients

r = √(by × bxy) — the geometric mean of the two regression coefficients.

by = regression coefficient of y on x; bxy = regression coefficient of x on y. Their product r² ≤ 1 always.

6

r is Unitless

The correlation coefficient has no units. It is a pure number.

Whether x is measured in kg, metres, or rupees, r is just a number between −1 and +1.

Number Line Visualization

-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Perfect NegativeNo CorrelationPerfect Positive
Common misconception: r = 0 does NOT mean there is no relationship. It means there is no linear relationship. A strong curved (non-linear) relationship can exist with r = 0.

Key Takeaways

Key Takeaways

  • r is always between −1 and +1 (bounded). If you get a value outside this range, check your calculation.
  • r is independent of change of origin — adding a constant to all values does not change r.
  • r is independent of change of scale — multiplying all values by a constant does not change r.
  • r = +1 or −1 means perfect linear correlation; r = 0 means no linear correlation.
  • r = √(by × bxy) — geometric mean of the two regression coefficients.
  • r is unitless — it has no units attached, regardless of what x and y measure.