Correlation and Regression Analysis

Unit VIII: Business Statistics and Operations Management

For a given set of pair data, the correlation and regression coefficients have been calculated as being equal to r, byx and bxy respectively. Now, each of values in the x series is divided by 4. What effect does it have on each of their coefficients?

✅ Correct Answer: There is no change in r but byx changes to 4byx and bxy changes to bxy/4.

📘 Explanation:

We have a dataset where:

$r$ = Correlation coefficient
$b_{y x}$ = Regression coefficient of Y on X
$b_{x y}$ = Regression coefficient of X on Y

Now, each value in the x-series is divided by 4. Let’s analyze the effect.

1. Effect on Correlation Coefficient ()

The correlation coefficient () is a unit-free measure of the relationship between X and Y.
Scaling X by dividing by 4 does not affect .

🔹 Conclusion:

2. Effect on Regression Coefficient $b_{y x}$ (Regression of Y on X)

The formula for $b_{y x}$ is:

$b_{y x} = [r \times σ ^{y}^{] / σ x}$

(standard deviation of X) is divided by 4, so the denominator decreases by a factor of 4.
Since division in the denominator leads to an increase in value, $b_{y x}$ is multiplied by 4.

🔹 Conclusion: $b_{y x'} = 4 \times b_{y x}$

2. Effect on Regression Coefficient $b_{x y}$ (Regression of X on Y)

The formula for $b_{x y}$ is:

Since $σ_{x}$ is divided by 4, the numerator decreases by a factor of 4.
This means $b_{x y}$ is divided by 4.

🔹 Conclusion: $b_{x y'} = b ^{x y}$

✅ Final Answer:

Coefficient	Effect
(Correlation Coefficient)	No Change
$b_{y x}$ (Regression of Y on X)	Multiplied by 4
$b_{x y}$ (Regression of X on Y)	Divided by 4

1. Effect on Correlation Coefficient (r)

2. Effect on Regression Coefficient byx​ (Regression of Y on X)

2. Effect on Regression Coefficient bxy​ (Regression of X on Y)

✅ Final Answer:

1. Effect on Correlation Coefficient ()

2. Effect on Regression Coefficient $b_{y x}$ (Regression of Y on X)

2. Effect on Regression Coefficient $b_{x y}$ (Regression of X on Y)