Definition
Correlation measures the strength and direction of the linear relationship between two variables.
- If two variables move together (increase or decrease together), they have a positive correlation.
- If one increases while the other decreases, they have a negative correlation.
- If they don’t move in a systematic way, correlation is close to zero.
Formula (Pearson Correlation Coefficient)
For variables $X$ and $Y$ with $n$ data points:
$r = \frac{\sum_{i=1}^{n} (x_i – \bar{x})(y_i – \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i – \bar{x})^2} \sqrt{\sum_{i=1}^{n} (y_i – \bar{y})^2}}$
- $r$ = correlation coefficient (ranges from –1 to +1)
- $\bar{x}, \bar{y}$ = means of $X$ and $Y$
Interpretation of $r$
- $r = +1$: Perfect positive correlation (as $X$ increases, $Y$ increases).
- $r = -1$: Perfect negative correlation (as $X$ increases, $Y$ decreases).
- $r = 0$: No linear correlation.
Important: Correlation only measures linear association, not causation.
Types of Correlation
- Positive Correlation – height and weight (generally both go up).
- Negative Correlation – price and demand (usually, as price rises, demand falls).
- Zero Correlation – shoe size and intelligence (no relationship).
Example
Suppose:
- Hours studied = [2, 4, 6, 8]
- Exam scores = [50, 65, 80, 95]
As hours studied increase, scores increase. rrr will be close to +1 (strong positive correlation).
Correlation vs Causation
- Correlation: Just a statistical relationship — variables move together.
- Causation: One variable actually produces a change in the other.
Example of misleading correlation:
- Ice cream sales and drowning accidents are positively correlated.
- But ice cream doesn’t cause drowning — both are caused by hot weather (a confounder).
Other Correlation Measures
- Spearman’s rank correlation: Measures monotonic (not necessarily linear) relationships.
- Kendall’s tau: Rank-based correlation, useful for ordinal data.
In short:
Correlation = “Do two variables move together?”
But it does not mean “Does one cause the other?”