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
(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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