Correlation and Association

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Correlation does not imply causation. Two variables can move together perfectly without one causing the other.

Correlation Coefficient

The Pearson correlation coefficient ( $r$ ) measures the linear relationship between two variables, ranging from -1 to 1.

Rank Correlation

Spearman's Rank Correlation evaluates how well the relationship between two variables can be described using a monotonic function, without requiring it to be perfectly linear.

Homoscedasticity vs. Heteroscedasticity

Homoscedasticity: The variance of errors is constant across all levels of the independent variable.
Heteroscedasticity: The variance of errors changes (often forming a cone shape in a scatter plot), which violates many linear regression assumptions.

Interactive Correlation Explorer

Correlation scatter plot

Adjust r and see how the cloud tightens/loosens. Toggle heteroscedasticity for cone-shaped variance.

Correlation (r)

0.80

-1 is perfect negative, +1 is perfect positive.

Heteroscedasticity

Off

Variance increases with |x| (cone shape).

Enable heteroscedasticity

Test Your Knowledge

Example: Covariance and Pearson r

Two variables $X$ and $Y$ have a Covariance of $Cov(X,Y) = 15$ . The standard deviation of $X$ is $\sigma_x = 4$ and the standard deviation of $Y$ is $\sigma_y = 5$ . Calculate the Pearson correlation coefficient $r$ .

View Step-by-Step Solution

The formula for the Pearson correlation coefficient is: $r = \frac{Cov(X,Y)}{\sigma_x \sigma_y}$

$r = \frac{15}{4 \times 5} = \frac{15}{20} = 0.75$

The correlation is 0.75, indicating a strong positive linear relationship between $X$ and $Y$ .

Exponential Distribution Chi-Square Distribution