Correlation

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.

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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 $Y$

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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.

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Types of Correlation

1. Positive Correlation – height and weight (generally both go up).
2. Negative Correlation – price and demand (usually, as price rises, demand falls).
3. Zero Correlation – shoe size and intelligence (no relationship).

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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).

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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).

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Other Correlation Measures

• Spearman’s rank correlation: Measures monotonic (not necessarily linear) relationships.
• Kendall’s tau: Rank-based correlation, useful for ordinal data.

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In short:
Correlation = “Do two variables move together?”
But it does not mean “Does one cause the other?”