# What are the differences of Standard Deviation, Variance, Co-variance, Coefficient of Variation when it comes to measures of dispersion or variation?

When we talk about measures of dispersion or variation(not sure if they are 2 different tools), then for what purpose we use what exactly?

Where are we bound to use only Standard Deviation, not Variance and where are we left with only using Coefficient of Variation instead of the former measures?

I need clear explanation with real life examples.

## 1 Answer

Variance is a measure of how uncertain you should be about the outcome of a single random variable. If for example your random variable is the year-end return on your stock portfolio, a high variance portfolio is one whose returns are hard to predict in any given year. Maybe it returns a lot this year, but loses you a lot of money next year. Think junk bonds or something. A low variance portfolio would be one whose yearly return you can be fairly certain of, say a AAA rated bond portfolio. Standard deviation is just the square root of variance.

Covariance is a measure of how much two different random variables move together. Say you have two portfolios. A high covariance between them means if you observe the year end return of one of them, you can be fairly sure what the year end return of the other would be. A low covariance means seeing the year end return of one will not give you much of an idea about how the other performed.

Coefficient of variation gives you the ratio of the standard deviation to the mean of a single random variable. It gives you a sense of how uncertain you should be about your portfolio given your expectation about how much it should return. This is useful because random variables with larger means tend to have larger variances. As an example, say I tell you the standard deviation of your year end returns is 1000 dollars. Ok, so you got some information out of me, but that number isn't so useful by itself. If your mean or "expected return" is 1,000,000 dollars, this standard deviation is by comparison almost nothing. However, if the expected return is 10 dollars, you might have more cause for concern.

Hope this helps.