Goal: formalize the concept of the strength of a linear relationship.

The correlation (or correlation coefficient) \(R\) between two variables describes the strength of their linear relationship.

\[R = \frac{1}{n-1}\sum_{i=1}^n\left(\frac{x_i - \bar{x}}{s_x}\times\frac{y_i - \bar{y}}{s_y}\right)\]

• \(s_x\) and \(s_y\) are the respective standard deviations for \(x\) and \(y\)
• The sample size \(n\) is the total number of \((x,y)\) pairs.
• \(R\) always takes values between \(-1\) and \(1\).

This is a pretty involved formula! We’ll let a computer handle this one.

Correlations

• close to \(-1\) suggest strong, negative linear relationships.
• close to \(+1\) suggest strong, positive linear relationships.
• close to \(0\) have little-to-no linear relationship.

Note: the sign of the correlation will match the sign of the slope!

• If \(R < 0\), there is a downward trend and \(b_1 < 0\).
• If \(R > 0\), there is an upward trend and \(b_1 > 0\).
• If \(R \approx 0\), there is no relationship and \(b_1 \approx 0\).

When two variables are highly correlated (\(R\) close to \(-1\) or \(1\))

• we know there is a strong relationship between them
• BUT we do not know what causes that relationship

That is, correlation does not imply causation.

A final note:

• Obviously there’s a strong relationship between \(x\) and \(y\).
  • In fact, \(y = x^2\).
• But the correlation between \(x\) and \(y\) is 0!