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Standard deviation s

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.


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


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}


 from here


and therefore


and the standard deviation s is therefore:

\sigma=\sqrt V= \frac{1}{\sqrt 3}

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

V\big(X\big)\geq 0


\sigma\big(X\big)\geq 0

V\big(aX \big)=a^2V\big(X\big)



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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