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# Sample Covariance

## What is Sample Covariance?

The sample covariance estimator uses the sample data for the expectation operator. The sample correlation is generated from the sample covariance, which assesses the intensity and direction of the association between the elements of two samples.

## Example of Sample Covariance:

$\sigma _{x,y}=\frac{1}{n}\sum_{i=1}^{n}\left ( Xi-\mu _{X} \right )\left ( Yi-\mu _{Y} \right )$

Where

μX is the sample mean of X, and μY is the sample mean of Y. Like the sample variance estimator, the sample covariance estimator is biased toward zero. Dividing by n — 1 rather than n produces an unbiased covariance estimate.

Computing the Sample Covariance

 Year Excelsior Corp Annual Annual Return (percent) Adirondack Corp Annual Annual Return (percent) (Xi – $\overline{X}$) (Yi – $\overline{Y}$)) (Xi – $\overline{X}$)(Yi – $\overline{Y}$)) 2008 1 3 1-1= 0 3-3= 0 (0)(0) = 0 2009 -2 2 -2-1= -3 2-3= -1 (-3)(-1) = 3 2010 3 4 3-1 = 2 4-3 = 1 (2)(1) = 2 2011 0 6 0-1= -1 6-3 = 3 (-1)(3) = -3 2012 3 0 3-1 = 2 0-3 = -3 (2)(-3) = -6 Mean 1 3 Sum -4

## Why is Sample Covariance important?

The sample covariance helps judge the reliability of the sample means as estimators and is also helpful in estimating the population covariance matrix.

Owais Siddiqui
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