Karl Pearson's coefficient of correlation is independent from:

Question

Accepted Answer

The correct option is 4:
The change in both the origin and the scale.

Karl Pearson's coefficient of correlation ($r$) is a pure number used to measure the strength and direction of the linear relationship between two variables, $X$ and $Y$. One of the most important mathematical properties of $r$ is that it remains unchanged when the data sets undergo a linear transformation, which includes changes in origin (shifting the data) and changes in scale (rescaling the data). A change in origin means adding or subtracting a constant value to all data points (e.g., $X' = X + a$). A change in scale means multiplying or dividing all data points by a constant (e.g., $X'' = bX$). Since the correlation coefficient is based on standardized scores (z-scores), which are calculated after deviations from the mean are measured, these linear transformations do not affect the relative distance or spread of the data points, thus preserving the calculated correlation coefficient. Therefore, $r$ is independent of changes in both the origin and the scale.

The incorrect option is 1:
Neither the origin nor the scale change. This option incorrectly implies that the coefficient is only independent if neither change occurs, which contradicts the mathematical property of the coefficient. The coefficient is inherently independent of these changes.

The incorrect option is 2:
Only by change in origin. This is partially correct but incomplete. While the coefficient is independent of a change in origin (shifting the data points), it is also independent of a change in scale. Selecting this option would miss the complete property.

The incorrect option is 3:
Only by change in scale. Similar to Option 2, this is also partially correct but incomplete. The coefficient is indeed independent of a change in scale (rescaling the data), but it is also independent of a change in origin. The correlation coefficient is invariant under both types of linear transformation.