corr-samp.md

layout	mathjax	author	affiliation	e_mail	date	title	chapter	section	topic	definition	sources	def_id	shortcut	username
definition	true	Joram Soch	BCCN Berlin	joram.soch@bccn-berlin.de	2021-12-13 23:23:00 -0800	Sample correlation coefficient	General Theorems	Probability theory	Correlation	Sample correlation coefficient	authors year title in pages url Wikipedia 2021 Pearson correlation coefficient Wikipedia, the free encyclopedia retrieved on 2021-12-14 https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#For_a_sample	D168	corr-samp	JoramSoch

Definition: Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ and $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ be samples from random variables $X$ and $Y$. Then, the sample correlation coefficient of $x$ and $y$ is given by

$$ \label{eq:corr-samp} r_{xy} = \frac{\sum_{i=1}^n (x_i-\bar{x}) (y_i-\bar{y})}{\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2} \sqrt{\sum_{i=1}^n (y_i-\bar{y})^2}} $$

where $\bar{x}$ and $\bar{y}$ are the sample means.