Ten days of statistics (8) - Correlationship

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Covariance

This is a measure of how two random variables $X$ and $Y$ change together. Formally

cov(X,Y) = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x}) \times (y_i - \bar{y})

Where

$\bar{x}$ is the mean of $x$
$\bar{y}$ is the mean of $y$

Pearson correlation coefficient

The Pearson correlation coefficient $\rho(X, Y)$ is given by:

\rho(X,Y) = \frac{cov(X,Y)}{\sigma_X\sigma_Y}

Where

$\sigma_X$ is the standard deviation of $X$
$\sigma_Y$ is the standard deviation of $Y$

Spearman’s rank correlation coefficient

Given 2 random variables $X$ and $Y$ with the same sample size. Let $rank_X$ and $rank_Y$ denotes the ranks of each data point on $X$ and $Y$ respectively. Let $r_S$ is the Spearman’s rank correlation coefficient of $X$ and $Y$ , which equal to the Pearson correlation coefficient of $rank_X$ and $rank_Y$

r_S = \frac{cov(rank_X, rank_Y)}{\sigma_{rank_X}\sigma_{rank_Y}}

If X and Y contains no duplicates

r_S = \frac{cov(rank_X, rank_Y)}{\sigma^2}

Practice

Hackerrank has some exercises for you to test your knowledge:

Next lesson: Linear regression