Correlation Coefficient

What is $r$?

The correlation coefficient $r$ measures the strength and direction of a linear relationship. $r$ ranges from $-1$ to $+1$.

Step-by-step

Data: $(1,2), (3,6), (5,7), (7,8), (9,10)$. $n = 5$.

Step 1 — Compute means: $\bar{x} = 5$, $\bar{y} = 6.6$.

Step 2 — Compute deviations and products:

| $x$ | $y$ | $x - \bar{x}$ | $y - \bar{y}$ | product | $(x-\bar{x})^2$ | $(y-\bar{y})^2$ | |---|---|---|---|---|---|---| | 1 | 2 | -4 | -4.6 | 18.4 | 16 | 21.16 | | 3 | 6 | -2 | -0.6 | 1.2 | 4 | 0.36 | | 5 | 7 | 0 | 0.4 | 0 | 0 | 0.16 | | 7 | 8 | 2 | 1.4 | 2.8 | 4 | 1.96 | | 9 | 10 | 4 | 3.4 | 13.6 | 16 | 11.56 |

Step 3 — Sums: $\sum = 36, \; 40, \; 35.2$

Step 4 — Formula:

$r = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2 \cdot \sum(y_i-\bar{y})^2}} = \frac{36}{\sqrt{40 \times 35.2}} = \frac{36}{\sqrt{1408}} = \frac{36}{37.52} = 0.960$

Interpretation: $r = 0.96$ — strong positive correlation. As $x$ increases, $y$ strongly tends to increase too.

Strength guide

$|r| > 0.7$: strong. $0.3$–$0.7$: moderate. $< 0.3$: weak.

Try it in the visualization

Scatter plot with the computed $r$ value. The tighter the cluster around a line, the higher $|r|$.