Correlation¶

Definition¶

For random variables $X$ and $Y$ with $0<\operatorname{Var}(X)<\infty$ and $0<\operatorname{Var}(Y)<\infty$ $\infty$ , the correlation of $X$ and $Y$ is

\operatorname{Corr}(X,Y)=\frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}} \in[-1,1].

Correlation is undefined if either variance is zero.

Interpretation¶

Correlation is the covariance of the two variables after each has been standardized to mean $0$ and variance $1$ . It is therefore unit-free, lies between $-1$ and $1$ , and measures how closely the pair follows a straight line. Zero correlation rules out linear association but can coexist with dependence.

Three scatter plots: points on an exact decreasing line, a cloud with a positive but imperfect linear trend, and points on a parabola y = x squared that are uncorrelated although y depends on x.

The left panel shows the exact decreasing relation $Y=5-2X$ $2 X$ , with correlation $-1$ . The middle panel shows a positive but imperfect linear association, with correlation about $0.74$ . In the right panel, $X\sim\operatorname{Unif}(-1,1)$ and $Y=X^2$ are uncorrelated but dependent, illustrating that correlation measures linear association only.

Change the slope and the noise: the correlation measures how tightly the points follow a line. With $Y=X^2$ it is $0$ , although $Y$ depends on $X$ .

Dividing by the standard deviations removes the units: measuring $X$ in cents instead of euros multiplies the covariance by $100$ but leaves the correlation unchanged. In general $\operatorname{Corr}(a+bX,c+dY)=\operatorname{Corr}(X,Y)$ when $bd>0$ $0$ , and the sign flips when $bd<0$ $0$ . The values $\pm1$ occur only for an exact linear relation.

There is a useful picture from linear algebra: covariance behaves like an inner product of the centred variables, variance like a squared length, and correlation like the cosine of the angle between them. The bound $|\operatorname{Corr}(X,Y)|\le1$ is then the Cauchy–Schwarz inequality, which returns for random variables later in this course (Cauchy–Schwarz for random variables).

Correlation is not the slope of the line: $Y=5-2X$ and $Y=5-0.1X$ both have correlation $-1$ .

Key Results¶

$-1\le\operatorname{Corr}(X,Y)\le1$ $1$ .
Absolute correlation is $1$ exactly when $Y=a+bX$ almost surely for constants $a,b$ with $b\ne0$ $0$ ; its sign is the sign of $b$ .

Examples¶

If $Y=5-2X$ and $X$ has positive finite variance, the correlation is $-1$ .
For $X\sim\operatorname{Unif}(-1,1)$ and $Y=X^2$ $X^{2}$ , symmetry gives $\mathbb{E}X=\mathbb{E}[X^3]=0$ $0$ , so correlation is $0$ although $Y$ is determined by $X$ .

Prerequisites¶

Covariance

Sources¶

Nicky van Foreest · Probability Distributions · Week 2

Aliases: correlation coefficient, Pearson correlation, Pearson's correlation coefficient, product-moment correlation coefficient, linear correlation coefficient Symbols: element of, infinity, square root

Last updated: 22 October 2026 19:04