Which of following is not correct about properties of correlation coefficient?
(A) Depends on the origin.
(B) Depends on the scale.
(C) Depends on both origin and scale.
(D) Is independent with respect to origin.
(E) Is independent with respect to unit of scale.
Choose the correct answer from the options given below :

Question

Which of following is not correct about properties of correlation coefficient?

(A) Depends on the origin.

(B) Depends on the scale.

(C) Depends on both origin and scale.

(D) Is independent with respect to origin.

(E) Is independent with respect to unit of scale.

Choose the correct answer from the options given below :

Prepp Team · Accepted Answer

Understanding Correlation Coefficient Properties

The correlation coefficient, often denoted by \(r\), is a statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables. Its value ranges from -1 to +1. A value of +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.

One of the important properties of the correlation coefficient is its invariance under linear transformations. This means that changing the origin or scale of the variables does not affect the value of the correlation coefficient.

Analyzing Statements on Correlation Coefficient Properties

Let's examine each statement given in the question:

Statement (A): Depends on the origin.
This statement claims that the correlation coefficient changes if you shift the data points by adding a constant to one or both variables (changing the origin). This is incorrect. The correlation coefficient is independent of the change of origin.
Statement (B): Depends on the scale.
This statement claims that the correlation coefficient changes if you multiply the variables by a constant (changing the scale). This is also incorrect, provided the scaling constant is positive. If a variable is scaled by a negative number, the sign of the correlation coefficient will flip, but its magnitude (the strength of the relationship) remains the same. However, the common understanding of 'independent of scale' includes invariance in magnitude. Thus, saying it 'depends on the scale' is generally considered incorrect in the context of its magnitude.
Statement (C): Depends on both origin and scale.
Since the correlation coefficient is independent of both origin and scale individually (with positive scaling), it does not depend on both. This statement is incorrect.
Statement (D): Is independent with respect to origin.
This statement says that changing the origin (adding a constant) does not affect the correlation coefficient. This is a correct property of the correlation coefficient.
Statement (E): Is independent with respect to unit of scale.
This statement says that changing the unit of measurement (scaling the variable) does not affect the correlation coefficient. This is also a correct property of the correlation coefficient (for positive scaling).

Identifying Incorrect Statements

Based on our analysis of the properties of the correlation coefficient, the statements that are *not correct* are:

Statement (A): Depends on the origin.
Statement (B): Depends on the scale.
Statement (C): Depends on both origin and scale.

These are the statements that incorrectly describe the behavior of the correlation coefficient under changes in origin and scale.

Conclusion on Incorrect Properties

The question asks which of the given statements is *not correct* about the properties of the correlation coefficient. We have identified statements (A), (B), and (C) as incorrect statements regarding its properties.

Statement	Property Claimed	Correctness	Is it NOT correct?
(A)	Depends on the origin	Incorrect	Yes
(B)	Depends on the scale	Incorrect	Yes
(C)	Depends on both origin and scale	Incorrect	Yes
(D)	Independent with respect to origin	Correct	No
(E)	Independent with respect to unit of scale	Correct	No

Therefore, the statements that are not correct are (A), (B), and (C).

Revision Table: Key Properties of Correlation Coefficient

Property	Description
Range	Always between -1 and +1 (\(-1 \le r \le +1\)).
Independence of Origin	Unaffected by adding a constant to the variables. If \(u = x + a\) and \(v = y + b\), then \(r_{uv} = r_{xy}\).
Independence of Scale	Unaffected by multiplying variables by a positive constant. If \(u = cx\) and \(v = dy\) with \(c > 0, d > 0\), then \(r_{uv} = r_{xy}\). If one constant is negative (e.g., \(c < 0, d > 0\)), \(r_{uv} = -r_{xy}\). The magnitude \(\lvert r \rvert\) is always invariant.
Symmetry	The correlation between X and Y is the same as the correlation between Y and X (\(r_{xy} = r_{yx}\)).
Linear Relationship Measure	Only measures the strength and direction of a linear relationship.

Additional Information: Correlation Coefficient Concepts

The correlation coefficient is a dimensionless quantity, meaning it does not have any units. This is a direct consequence of its independence from the unit of scale. It is calculated using the covariance of the two variables divided by the product of their standard deviations.

Formula for Pearson Correlation Coefficient \(r\):

\(r_{xy} = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2}\sum{(y_i - \bar{y})^2}}}\)

Alternative form:

\(r_{xy} = \frac{Cov(x, y)}{s_x s_y}\)

Where:

\(Cov(x, y)\) is the covariance between x and y.
\(s_x\) is the standard deviation of x.
\(s_y\) is the standard deviation of y.

Standard deviation and covariance are both affected by changes in origin and scale, but the way they combine in the correlation formula cancels out these effects (or just changes the sign in case of negative scaling for standard deviation and covariance). This cancellation makes the correlation coefficient robust to these linear transformations.