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I know that if you calculate variance, you can square root it to get the standard deviation.

What does it mean / what is it called if you square root a scalar value which is the covariance of two variables?

Alan Wolfe
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1 Answers1

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The standard deviation of a random variable has the same units as its mean. For example, the variance of GDP is measured in dollars-squared, not usually a unit that we can understand. That's why we look at ratios such as the mean divided by the standard deviation. It is a scalar without units. In the case of GDP, it would show the percent that a recession is below trend growth as a fraction of the usual standard deviation that characterizes the business cycle.

Now think about the covariance between the USA's GDP and that of France. It would be measured in in dollars-euros, a completely incomprehensible measure. That's why we use the correlation. It has no units, and it has a meaning in every physical or social scientific application of statistics.

You're asking about the case of the square root of the covariance. It is never used, but you could almost call it the co-standard deviation.

Neil G
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Eric Fisher
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    Interesting info. I was thinking the reverse, that perhaps by square rooting the covariance, we'd get something in units of measurement that was somehow more applicable to the data set. I wasn't expecting the reverse POV :P – Alan Wolfe Dec 16 '17 at 05:17
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    In finance, the mean of an asset's return divided by its standard deviation is called the Sharpe ratio. You could calculate the product of two means and divide it by your co-standard deviation. That measure would be a scalar, but I don't know what it would mean. – Eric Fisher Dec 16 '17 at 05:24