Correlation Under Linear Transformations
The correlation between random variables $X$ and $Y$ is $0.56$. What is the correlation between $8X$ and $Y + 7$?
Hints
- Correlation is invariant to linear transformations of the variables -- adding constants or multiplying by positive constants does not change it.
- Write out the definition $\rho = \text{Cov}(U,V) / (\text{SD}(U) \cdot \text{SD}(V))$ and apply the rules $\text{Cov}(aX, Y+c) = a \cdot \text{Cov}(X,Y)$ and $\text{SD}(aX) = |a| \cdot \text{SD}(X)$.
- The factor of 8 appears in both the numerator (from the covariance) and the denominator (from $\text{SD}(8X)$), so it cancels exactly.
Worked Solution
How to Think About It: Correlation measures the linear relationship between two variables after stripping out their individual scales. It is defined as covariance divided by the product of standard deviations -- and that ratio is invariant to rescaling or shifting either variable. Multiplying $X$ by 8 scales both the covariance and the standard deviation of $X$ by 8, so they cancel. Adding 7 to $Y$ shifts the mean but does not change how $Y$ moves around its mean, so covariance is unaffected and the standard deviation of $Y$ is unchanged.
Quick Estimate: The answer is obviously still 0.56. The transformation $Z = 8X$ is just a change of units (like converting meters to centimeters). The transformation $W = Y + 7$ is just a level shift. Neither changes how tightly $Z$ and $W$ move together.
Formal Solution:
Recall the definition of correlation:
$$\rho(U, V) = \frac{\text{Cov}(U, V)}{\text{SD}(U) \cdot \text{SD}(V)}$$
Compute each piece for $U = 8X$ and $V = Y + 7$:
- $\text{Cov}(8X, Y + 7) = 8 \cdot \text{Cov}(X, Y + 7) = 8 \cdot \text{Cov}(X, Y)$ (covariance is linear in each argument; additive constants drop out)
- $\text{SD}(8X) = 8 \cdot \text{SD}(X)$ (scaling by a positive constant scales the standard deviation by the same factor)
- $\text{SD}(Y + 7) = \text{SD}(Y)$ (adding a constant does not change the spread)
Therefore:
$$\rho(8X, Y + 7) = \frac{8 \cdot \text{Cov}(X, Y)}{8 \cdot \text{SD}(X) \cdot \text{SD}(Y)} = \frac{\text{Cov}(X, Y)}{\text{SD}(X) \cdot \text{SD}(Y)} = \rho(X, Y)$$
Answer: $\rho(8X, Y + 7) = 0.56$.
Intuition
Correlation is a dimensionless measure of linear co-movement -- it has been engineered to be unit-free by construction. This is precisely why it is so useful: you can correlate daily returns in dollars with daily returns in yen and the answer does not depend on the exchange rate. The covariance alone would change if you switched units; correlation stays the same.
The subtlety worth memorizing: this invariance holds for positive scaling constants. If the scaling constant is negative (e.g., $-8X$ instead of $8X$), the sign of the correlation flips. And the invariance is only to linear (affine) transformations -- if you replace $X$ with $X^2$, you will generally change the correlation. These edge cases come up on quant interviews more often than the basic version, so know them cold.