### 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

(1)

The term in (1) is called an **interaction term**. See this post for the definition and properties of covariance.

**Proof**.

(using linearity of means)

(grouping by variable)

(squaring out)

(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

(distributing expectation)

(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.

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