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 (Xμ)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

Famous quotes containing the words definition of, definition, population and/or values:

    Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.
    Nadine Gordimer (b. 1923)

    The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.
    Ralph Waldo Emerson (1803–1882)

    We in the West do not refrain from childbirth because we are concerned about the population explosion or because we feel we cannot afford children, but because we do not like children.
    Germaine Greer (b. 1939)

    The return to solid values is always hard.... Distress, panic, and hard times have marked our pathway in returning to solid values.
    James A. Garfield (1831–1881)