Negative Correlation
Statistics & ProbabilityNegative correlation means that as one variable increases, the other variable tends to decrease.
Definition
Negative correlation means two things tend to move in opposite directions: when one goes up, the other tends to go down.
Example
As the number of absences increases, test grades tend to decrease. Missing school and good grades have a negative correlation.
Key Insight
On a scatter plot, a negative correlation looks like dots drifting downward from left to right, like a slope going downhill.
Definition
A negative correlation exists when the correlation coefficient r is between -1 and 0. A value near -1 indicates a strong negative linear relationship; near 0 indicates a weak one. The line of best fit for negatively correlated data has a negative slope.
Example
Price and quantity demanded for most goods are negatively correlated ($r \approx -0.6$ to $-0.9$): as price increases, consumers tend to buy less. This is the economic law of demand.
Key Insight
A perfect negative correlation ($r = -1$) means that knowing one variable tells you exactly the other via a linear equation with a negative slope. Few real-world relationships are this precise.
Definition
Negative correlation corresponds to negative covariance: $\text{Cov}(X,Y) < 0$. In portfolio theory, assets with negative correlation reduce overall portfolio variance: $\text{Var}(aX+bY) = a^2\text{Var}(X) + b^2\text{Var}(Y) + 2ab\,\text{Cov}(X,Y)$. When $\text{Cov}(X,Y) < 0$, combining assets reduces total variance below the weighted average of individual variances.
Example
Gold prices and equity returns have historically had a weak negative correlation. Including gold in an equity portfolio reduces variance, improving the risk-return tradeoff (Sharpe ratio). This is the mathematical basis of diversification.
Key Insight
The Cauchy-Schwarz inequality guarantees $-1 \le \rho \le 1$: $|\text{Cov}(X,Y)| \le \sigma_X \sigma_Y$. Equality holds only when $X$ and $Y$ are perfectly linearly related ($Y = aX+b$ with $a < 0$ for $\rho = -1$).