Standard deviation s

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The standard deviation is the positive square root of the variance, i.e. s = √V. Thus, if the variance has units of cows², the standard deviation has units of cows, the same as the variable x. The standard deviation is therefore more popularly used to express a measure of spread.

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The variance V of the Uniform(1,3) distribution is calculated as follows: 

                        

LaTeX Math Block alignment left E\big(x^2 \big)=\int \frac{1}{2}x^2 dx= {\bigg[\frac{x^3}{6}\bigg] }_1^3 = \frac{27-1}{6}= \frac{13}{3}

LaTeX Math Inline body \bar{x}=2

from here

           

and therefore 

LaTeX Math Block
alignment left
V=\frac{26}{6}-2^2=\frac{1}{3}

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and the standard deviation s is therefore:

LaTeX Math Block
alignment left
\sigma=\sqrt V= \frac{1}{\sqrt 3}

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Variance and standard deviation have the following properties, where a is some constant and X, Xi are random variables: 

LaTeX Math Inline body V\big(X\big)\geq 0

and

LaTeX Math Inline body \sigma\big(X\big)\geq 0

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LaTeX Math Block alignment left V\bigg( \displaystyle\sum_{i=1}^{n} X_i \bigg)=\displaystyle\sum_{i=1}^{n}V(X_i)

providing the Xis are uncorrelated.

 

 

 

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