Independence of Origin and Scale in Karl Pearson’s Correlation Coefficient
Definition of Correlation Coefficient
The correlation coefficient \( r(X, Y) \) is defined as:
\[
r(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}.
\]
-
- Covariance:
\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n (x_i – \bar{X})(y_i – \bar{Y})
\]
-
- Variance of \( X \):
\[
\text{Var}(X) = \frac{1}{n} \sum_{i=1}^n (x_i – \bar{X})^2
\]
-
- Variance of \( Y \):
\[
\text{Var}(Y) = \frac{1}{n} \sum_{i=1}^n (y_i – \bar{Y})^2
\]
Transforming Variables
We apply the transformations:
\[
U = \frac{X – a}{h}, \quad V = \frac{Y – b}{k}, \quad h > 0, \quad k > 0.
\]
Substitute into the transformed variables:
\[
x_i = a + h u_i, \quad y_i = b + k v_i.
\]
Transformed means:
\[
\bar{X} = a + h \bar{U}, \quad \bar{Y} = b + k \bar{V}.
\]
Deviations from the Mean
The deviations from the mean are:
-
- For \( x_i \):
\[
x_i – \bar{X} = h (u_i – \bar{U}),
\]
-
- For \( y_i \):
\[
y_i – \bar{Y} = k (v_i – \bar{V}).
\]
Substitute these into the covariance formula:
\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n (x_i – \bar{X})(y_i – \bar{Y}),
\]
Simplifying further:
\[
\text{Cov}(X, Y) = hk \cdot \frac{1}{n} \sum_{i=1}^n (u_i – \bar{U})(v_i – \bar{V}).
\]
Simplified Covariance
-
- Covariance of \( X \) and \( Y \):
\[
\text{Cov}(X, Y) = hk \cdot \text{Cov}(U, V).
\]
-
- Variance of \( X \):
\[
\text{Var}(X) = h^2 \cdot \text{Var}(U).
\]
-
- Variance of \( Y \):
\[
\text{Var}(Y) = k^2 \cdot \text{Var}(V).
\]
Substituting into Correlation Coefficient
Substitute into \( r(X, Y) \):
\[
r(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}.
\]
Substituting the values:
\[
r(X, Y) = \frac{hk \cdot \text{Cov}(U, V)}{\sqrt{h^2 \cdot \text{Var}(U) \cdot k^2 \cdot \text{Var}(V)}}.
\]
Simplifying:
\[
r(X, Y) = r(U, V).
\]
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Conclusion
The correlation coefficient \( r(X, Y) \) remains unchanged under linear transformations:
\[
U = \frac{X – a}{h}, \quad V = \frac{Y – b}{k}.
\]
This demonstrates independence from:
- Changes in origin (\( a, b \)),
- Changes in scale (\( h, k \)).
This makes the correlation coefficient a robust measure of linear association.
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