All properties of variance in one place
Certainty is the mother of quiet and repose, and uncertainty the cause of variance and contentions. Edward Coke
Preliminaries: study properties of means with proofs.
Definition. Yes, uncertainty leads to variance, and we measure it by . It is useful to use the name deviation from mean for and realize that , so that the mean of the deviation from mean cannot serve as a measure of variation of around .
Property 1. Variance of a linear combination. For any random variables and numbers one has
The term in (1) is called an interaction term. See this post for the definition and properties of covariance.
(using linearity of means)
(grouping by variable)
(using linearity of means and definitions of variance and covariance)
Property 2. Variance of a sum. Letting in (1) we obtain
Property 3. Homogeneity of degree 2. Choose in (1) to get
Exercise. What do you think is larger: or ?
Property 4. If we add a constant to a variable, its variance does not change:
Property 5. Variance of a constant is zero: .
Property 6. Nonnegativity. Since the squared deviation from mean is nonnegative, its expectation is nonnegative: .
Property 7. Only a constant can have variance equal to zero: If , then , see the definition of the expected value. Since all probabilities are positive, we conclude that for all , which means that is identically constant.
Property 8. Shortcut for variance. We have an identity . Indeed, squaring out gives
(expectation of a constant is constant)
All of the above properties apply to any random variables. The next one is an exception in the sense that it applies only to uncorrelated variables.
Property 9. If variables are uncorrelated, that is , then from (1) we have In particular, letting , we get additivity: Recall that the expected value is always additive.
Generalizations. and if all are uncorrelated.
Among my posts, where properties of variance are used, I counted 12 so far.