Positive Correlation
Statistics & ProbabilityPositive correlation means that as one variable increases, the other variable tends to increase as well.
Definition
Positive correlation means two things tend to move in the same direction: when one goes up, the other tends to go up too.
Example
The more hours you study, the higher your test score tends to be. These two variables (study hours and test score) have a positive correlation.
Key Insight
On a scatter plot, a positive correlation looks like dots drifting upward from left to right, like a hill.
Definition
A positive correlation exists when the correlation coefficient r is between 0 and +1. A value near +1 indicates a strong positive linear relationship; near 0 indicates a weak one. On a scatter plot, positively correlated data forms a cloud of points sloping upward from left to right.
Example
Height and weight of adult males have a strong positive correlation ($r \approx 0.7$). A line of best fit with a positive slope summarizes this trend: taller men tend to weigh more.
Key Insight
Strength matters as much as direction. A correlation of $r = 0.3$ is positive but weak; $r = 0.9$ is positive and very strong. Always report the value, not just the direction.
Definition
Positive correlation corresponds to positive covariance: $\text{Cov}(X,Y) = E[XY] - E[X]E[Y] > 0$. For jointly normal $(X,Y)$, positive correlation means knowing $X > \mu_X$ increases the conditional expectation of $Y$ above $\mu_Y$. Partial correlation measures positive association between two variables controlling for the effects of one or more additional variables.
Example
In a multiple regression, two predictors can be positively correlated with each other ($r = 0.8$) and both positively correlated with the response. The partial correlation between each predictor and the response (controlling for the other) reveals their unique contribution, which may differ from the marginal correlation.
Key Insight
Multicollinearity in regression occurs when predictors are strongly positively (or negatively) correlated, inflating standard errors and making individual coefficient estimates unstable. Variance inflation factors (VIF) quantify this problem.