Pearson correlation coefficient (${\rho }_{X,Y}$)

The Pearson correlation coefficient is the covariance divided by the product of the standard deviations. For real random variables with finite, strictly positive variances,It is the expected value of the product of the two centered variables after dividing each by its standard deviation. The Cauchy-Schwarz inequality gives $-1\le {\rho }_{X,Y}\le 1$. This removes measurement units, while retaining the direction and strength of the linear relationship. Constant random variables have zero standard deviation, so their Pearson correlation coefficient is undefined.

For the covariance model $Y=2X+Z$, with $X,Z$ independently uniform on $\{-1,0,1\}$, the calculation givesThe positive correlation is less than one because the independent $Z$ term introduces extra variation. For $Y=-2X+Z$ the correlation is $-2/\sqrt{5}$. Both examples appear in the covariance diagram.

For nonzero real $a,c$, the correlation of $aX+b$ and $cY+d$ is $\mathrm{sgn}(ac){\rho }_{X,Y}$. Positive rescaling and added offsets therefore leave the correlation unchanged; reversing one scale reverses its sign. Equality $|{\rho }_{X,Y}|=1$ means that one random variable is an affine function of the other almost surely, with a nonzero slope of the corresponding sign. This is the equality condition in the Cauchy-Schwarz inequality, and includes perfect positive correlation.

Zero correlation says that the centered variables are orthogonal in their square-integrable inner product. It permits nonlinear statistical dependence, as the example $Y=X^2$ in uncorrelated random variables demonstrates. More generally, a correlation is a statement about a joint probability distribution; identifying effects of interventions is the separate task of causal inference. This distinction keeps an observed association from being mistaken for a demonstrated mechanism.

An early source for standardized association is Francis Galton's 1888 paper “Co-relations and their Measurement, chiefly from Anthropometric Data”. He compared stature and forearm measurements and changed each measurement to units of its own variability so the two regression directions could be compared. His scale used probable error, so it should not be identified word for word with the modern standard deviation formula above. ^[ref].

The correlation coefficient computed under a conditional distribution, using its conditional covariance and positive conditional variances. It can depend on the observed conditioning value. For a multivariate normal distribution, conditioning on other coordinates gives the corresponding partial correlation.

The partial correlation is the correlation of residuals after linearly projecting each of two variables on the conditioning variables and an intercept. With nonsingular covariance matrix, subtracting the two least-squares predictors gives residual covariance ${\sigma }_{12}-{\sigma }_{13}{\sigma }_{23}/{\sigma }_{33}$ and residual variances ${\sigma }_{ii}-{\sigma }_{i3}^2/{\sigma }_{33}$. Dividing gives the displayed formula. For a multivariate normal distribution, it is also the conditional correlation, independent of the conditioning value; this equality need not hold for general distributions.

For square-integrable variables of positive variance, correlation one is equivalent to $X-\mathbb{E}X=c(Y-\mathbb{E}Y)$ with $c>0$. The squared norm of the difference between the two standardized centered variables is $2(1-\rho )$, proving the equivalence. For uncentered variables the relationship can have a nonzero intercept; it need not be proportionality.

For two strictly positive zero-rate stock martingales with bounded volatilities and the same initial value, proportional terminal prices must be equal because their expectations coincide. Conditional expectation then makes the whole paths equal. Comparing their stochastic integrals by the Itô isometry gives equality of the volatility coefficients almost everywhere in time and probability. Perfect positive correlation alone does not provide the required proportionality.

The stocks $S_t=e^{B_t-t/2}$ and $S_t^{\prime }=(1+S_t)/2$ start at one, are true square-integrable zero-rate martingales, and have perfect positive correlation at each positive time. Their bounded multiplicative volatilities are $1$ and $S_t/(1+S_t)$, which differ throughout. The additive intercept is what allows the same correlation but different stochastic-exponential coefficients.

Responses observed on the same process or individual at different times may have nonzero correlation coefficients. Such serial correlation can arise from a shared random effect or from history-dependent conditional means. A scalar dispersion parameter adjustment alone does not represent this temporal dependence.

The intraclass correlation coefficient measures the correlation between observations in the same group. In a random-intercept model with group variance ${\sigma }_G^2$ and individual variance ${\sigma }_I^2$, it is ${\sigma }_G^2/({\sigma }_G^2+{\sigma }_I^2)$.