From sample variance to population variance For a sample of size n, the variance is the frequency-weighted mean squared distance from the sample mean: s2=(_i f_i(x_i- x)2)/(_i f_i)= _i(x_i- x)2(f_i)/(n) As n to infinity, (f_i)/(n) to P(X=x_i) and x to mu. The variance of the whole population measures spread about mu. sigma2=Var(X)= _all x(x-mu)2P(X=x)=E((X-mu)2) Expanding the square and using mu=E(X) gives Var(X)=E(X2)-[E(X)]2, E(X2)= _all xx2P(X=x) Var(X) 0. It measures the expected spread or dispersion of the distribution of X. Standard deviation sigma=SD(X)=sqrt(Var(X)) Standard deviation is an alternative measure of spread. Properties of variance and standard deviation Var(f(X))=E([f(X)]2)-[E(f(X))]2 Var(a)=0, Var(aX+- b)=a2Var(X)SD(a)=0, SD(aX+- b)=|a|sqrt(Var(X))

From sample variance to population variance

For a sample of size \(n\), the variance is the frequency-weighted mean squared distance from the sample mean:

\[s^2=\frac{\sum_i f_i(x_i-\bar x)^2}{\sum_i f_i}=\sum_i(x_i-\bar x)^2\frac{f_i}{n}\]

Sample variance

As \(n\to\infty\), \(\frac{f_i}{n}\to P(X=x_i)\) and \(\bar x\to\mu\). The variance of the whole population measures spread about \(\mu\).

\[\sigma^2=\operatorname{Var}(X)=\sum_{\text{all }x}(x-\mu)^2P(X=x)=E((X-\mu)^2)\]

Population variance

Expanding the square and using \(\mu=E(X)\) gives

\[\operatorname{Var}(X)=E(X^2)-[E(X)]^2,\qquad E(X^2)=\sum_{\text{all }x}x^2P(X=x)\]

Computational form

\(\operatorname{Var}(X)\ge0\). It measures the expected spread or dispersion of the distribution of \(X\).

Variance

\[\sigma=\operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}\]

Standard deviation

Standard deviation is an alternative measure of spread.

Properties of variance and standard deviation

\[\operatorname{Var}(f(X))=E([f(X)]^2)-[E(f(X))]^2\]

Variance of a function

\[\operatorname{Var}(a)=0,\qquad \operatorname{Var}(aX\pm b)=a^2\operatorname{Var}(X)\]\[\operatorname{SD}(a)=0,\qquad \operatorname{SD}(aX\pm b)=|a|\sqrt{\operatorname{Var}(X)}\]

Variance and Standard Deviation

Variance and Standard Deviation