Square of Sample Correlation Coefficient is no greater than 1

Definition

Let $S$ be a set of paired observations of a statistic from a population.

The sample correlation coefficient $r$ of $S$ satisfies the inequality:

$r^2 \le 1$

Proof

Let $\ds \bar x = \frac 1 n \sum_{i \mathop = 1}^n x_i$ and $\ds \bar y = \frac 1 n \sum_{i \mathop = 1}^n y_i$.

\(\ds r^2\)

\(=\)

\(\ds \frac {\paren {s_{x y} }^2} {s_{xx} s_{yy} }\)

Definition of Sample Correlation Coefficient

\(\ds \)

\(=\)

\(\ds \frac {\ds \paren {\sum_{i \mathop = 1}^n \paren {x_i - \bar x} \paren {y_i - \bar y} }^2} {\ds \paren {\sum_{i \mathop = 1}^n \paren {x_i - \bar x}^2} \paren {\sum_{i \mathop = 1}^n \paren {y_i - \bar y}^2} }\)

\(\ds \)

\(\le\)

\(\ds \frac {\ds \paren {\sum_{i \mathop = 1}^n \paren {x_i - \bar x} \paren {y_i - \bar y} }^2} {\ds \paren {\sum_{i \mathop = 1}^n \paren {x_i - \bar x} \paren {y_i - \bar y} }^2}\)

Cauchy's Inequality

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy-Schwarz inequality: $(2)$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy-Schwarz inequality: $(2)$