Sample Standard Deviation - Definition of Population Values

Definition of Population Values

Let X be a random variable with mean value μ:

Here the operator E denotes the average or expected value of X. Then the standard deviation of X is the quantity

\begin{align} \sigma & = \sqrt{\operatorname E}\\ & =\sqrt{\operatorname E + \operatorname E + \operatorname E} =\sqrt{\operatorname E -2 \mu \operatorname E + \mu^2}\\ &=\sqrt{\operatorname E -2 \mu^2 + \mu^2} =\sqrt{\operatorname E - \mu^2}\\ & =\sqrt{\operatorname E-(\operatorname E)^2}. \end{align}

(derived using the properties of expected value)

In other words the standard deviation σ (sigma) is the square root of the variance of X; i.e., it is the square root of the average value of (X − μ)2.

The standard deviation of a (univariate) probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation, since these expected values need not exist. For example, the standard deviation of a random variable that follows a Cauchy distribution is undefined because its expected value μ is undefined.

Read more about this topic:  Sample Standard Deviation

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