Explaining Pearson's Correlation Range

Solution & Explanation

Pearson's correlation coefficient, denoted as $r$, is a measure of the linear relationship between two variables $X$ and $Y$. Its value ranges from -1 to 1. To understand why this is the case, we can delve into the mathematical underpinnings of the formula and the principles that confine $r$ within this range.

Mathematical Definition

The correlation coefficient $r$ is defined as:

$\text{corr}(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$

Where:

• $\text{Cov}(X, Y)$ is the covariance between $X$ and $Y$.
• $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$ respectively.

Covariance and Variance

Covariance measures how much two random variables change together, while variance measures how much a single variable deviates from its mean. The covariance can be expressed as:

$\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$

Where $E$ denotes the expected value, and $\mu_X$ and $\mu_Y$ are the means of $X$ and $Y$ respectively.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states:

$|E[(X - \mu_X)(Y - \mu_Y)]|^2 \leq E[(X - \mu_X)^2]E[(Y - \mu_Y)^2]$

This can be rewritten in terms of variance:

$|\text{Cov}(X, Y)|^2 \leq \text{Var}(X) \cdot \text{Var}(Y)$

Taking the square root of both sides gives:

$|\text{Cov}(X, Y)| \leq \sigma_X \sigma_Y$

Deriving the Range

Substituting this result into the correlation formula:

$-1 \leq \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \leq 1$

Thus, the correlation coefficient $r$ is bounded between -1 and 1.

Geometric Interpretation

Pearson's correlation can also be interpreted geometrically as the cosine of the angle $\theta$ between the centered data vectors $X - \mu_X$ and $Y - \mu_Y$:

$\text{corr}(X, Y) = \cos(\theta)$

Since the range of cosine is $[-1, 1]$, this provides an intuitive geometric reason why $r$ must lie between -1 and 1.

Conclusion

The Pearson correlation coefficient is a standardized measure of the linear relationship between two variables, confined to the range [-1, 1] due to the properties of covariance, variance, and the Cauchy-Schwarz inequality. This range is further reinforced by the geometric interpretation of correlation as the cosine of the angle between two vectors.