Variance and Standard Deviation of a Random Variable

Statements

Definition. (Ost20, Definition 6.2.1) Let $X$ be a random variable with expectation $E(X)$. The variance $X$ is defined as

$$\text{Var}(X)=E((X-E(X){)}^2)$$

assuming that this expectation exists. The standard deviation of a random variable is defined as the square root of the variance of a random variable,

$$\text{Std}(X)=\sqrt{\text{Var}(X)}.$$

Remark. (Ost20, Section 6.3) The theoretical constructs of expectations, variances, and standard deviations should not be confused with the concepts of sample means, sample variance, and sample standard deviations. The former entities are of theoretical nature and can be evaluated once the distributions of random variables have been specified. The latter entities are of practical nature, can be evaluated numerically based on observed data, and serve as estimators of the former theoretical quantities

Definition. (Ost20, Definition 6.3.1) Let $(X_1,\dots ,X_n)$ be random variables and let $\mu$ be their sample mean. Then the sample variance of $(X_1,\dots ,X_n)$ is defined as

$${\sigma }^2(X)=\frac{1}{n-1}\sum _{i=1}^n(\mu -X_i{)}^2.$$

The sample standard deviation is defined as

$$\sigma (X)=\sqrt{{\sigma }^2(X)}.$$

References

D. Ostwald - The general linear model 20_21