Definition: Sample covariance

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Sample covariance

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 covariance of $x$ and $y$ is given by

$$\label{eq:cov-samp} \hat{\sigma}_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y})$$

and the unbiased sample covariance of $x$ and $y$ is given by

$$\label{eq:cov-samp-unb} s_{xy} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y})$$

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

 

Sources:

• Wikipedia (2021): "Covariance"; in: Wikipedia, the free encyclopedia, retrieved on 2020-05-20; URL: https://en.wikipedia.org/wiki/Covariance#Calculating_the_sample_covariance.

Metadata: ID: D144 | shortcut: cov-samp | author: ciaranmci | date: 2021-04-21, 06:53.