Recipe

Compute correlation from a joint distribution

Use when: you have a full joint table (or covariance plus both standard deviations) and need \rho_{X,Y}

Goal: a single unitless number between -1 and 1 measuring the strength and direction of the linear relationship between X and Y

Taught in Correlation · extends Compute covariance from a joint distribution

1. 1

   Get E(X), E(Y), and E(XY) from the joint table exactly as in the covariance recipe, then apply \operatorname{Cov}(X,Y)=E(XY)-E(X)E(Y).

   Example

   Using the same table as the covariance recipe, p(0,0)=0.2, p(0,1)=0.1, p(1,0)=0.1, p(1,1)=0.6: E(X)=E(Y)=0.7, E(XY)=0.6, so \operatorname{Cov}(X,Y)=0.6-(0.7)(0.7)=0.11.

   Why: correlation is built directly on top of covariance, so the first real step is producing that number.

2. 2

   Compute each variable's own variance from its marginal distribution (the ordinary variance formula from earlier in the course), then take the square root of each.

   Example

   Both marginals here are \operatorname{Bernoulli}(0.7), so \operatorname{Var}(X)=\operatorname{Var}(Y)=(0.7)(0.3)=0.21, giving \sigma_X=\sigma_Y=\sqrt{0.21}\approx0.458.

   Why: correlation needs both spreads to cancel the units out of covariance's raw, unit-dependent number.

3. 3

   Multiply the two standard deviations together, then divide the covariance from Step 1 by that product.

   Example

   \rho_{X,Y} = \frac{0.11}{(0.458)(0.458)} = \frac{0.11}{0.21} \approx 0.524

   ⚡ Shortcut

   Since \sigma_X and \sigma_Y are always positive (they're square roots), the SIGN of \rho_{X,Y} always matches the sign of \operatorname{Cov}(X,Y). If only the direction of the relationship is needed, not its exact strength, the sign of the covariance alone already answers it, no division required.

   Why: dividing by both standard deviations is exactly what strips the units out, leaving a pure number the next step can check against a fixed bound.

4. 4

   Confirm the result sits between -1 and 1. If it doesn't, a covariance or standard-deviation value earlier in the computation was miscomputed.

   Example

   0.524 sits safely inside [-1,1], confirming no arithmetic slip in the steps above.

   Why: correlation's own defining bound makes this the cheapest possible check on the whole computation, the same reasoning the second worked example below relies on.