Editing Covariance (section)

{{Short description|Measure of the joint variability}}
{{About|the degree to which random variables vary similarly}}
[[File:covariance_trends.svg|thumb|upright|The sign of the covariance of two random variables ''X'' and ''Y'']]

In [[probability theory]] and [[statistics]], '''covariance''' is a measure of the joint variability of two [[random variable]]s.<ref>{{Cite book | title=Mathematical Statistics and Data Analysis | last=Rice | first=John | publisher=Brooks/Cole Cengage Learning | year=2007 | isbn=9780534399429 | page=138}}</ref>

The sign of the covariance, therefore, shows the tendency in the [[linear relationship]] between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive.<ref>{{MathWorld | urlname=Covariance | title=Covariance}}</ref> In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The [[Pearson product-moment correlation coefficient|correlation coefficient]] normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables.

A distinction must be made between (1) the covariance of two random variables, which is a [[Statistical population|population]] [[Statistical parameter|parameter]] that can be seen as a property of the [[joint probability distribution]], and (2) the [[sample (statistics)|sample]] covariance, which in addition to serving as a descriptor of the sample, also serves as an [[Statistical estimation|estimated]] value of the population parameter.