Editing Standard deviation (section)

==Definition of population values==
Let {{mvar|μ}} be the [[expected value]] (the average) of [[random variable]] {{mvar|X}} with density {{math|{{var|f}}({{mvar|x}})}}:
<math display="block"> \mu \equiv \operatorname{\mathbb E}[X] = \int_{-\infty}^{+\infty} x\ f(x)\ {\mathrm d} x </math>
The standard deviation {{mvar|σ}} of {{mvar|X}} is  defined as 
<math display="block"> \sigma \equiv \sqrt{\operatorname{\mathbb E}\left[ \left(X - \mu\right)^2 \right]}  = \sqrt{ \int_{-\infty}^{+\infty} \left( x - \mu \right)^2 f(x) \ {\mathrm d} x \;}\ , </math>
which can be shown to equal <math display="inline"> \sqrt{\ \operatorname{\mathbb E}\left[\ X^2\ \right] - \left(\ \operatorname{\mathbb E}\left[ X \right]\ \right)^2 \;} ~.</math>
 
Using words, the standard deviation is the square root of the [[variance]] of {{mvar|X}}.

The standard deviation of a probability distribution is the same as that of a random variable having that distribution. 

Not all random variables have a standard deviation. If the distribution has [[fat tails]] going out to infinity, the standard deviation might not exist, because the integral might not converge. The [[normal distribution]] has tails going out to infinity, but its mean and standard deviation do exist, because the tails diminish quickly enough. The [[Pareto distribution]] with parameter <math> \alpha \in (1,2] </math> has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The [[Cauchy distribution]] has neither a mean nor a standard deviation.

===Discrete random variable===
In the case where {{mvar|X}} takes random values from a finite data set {{math|{{var|x}}{{sub|1}}, {{var|x}}{{sub|2}}, ..., {{var|x}}{{sub|{{var|N}}}}}}, with each value having the same probability, the standard deviation is

<math display="block">\sigma = \sqrt{\frac{1}{N}\ \left[ \left( x_1 - \mu \right)^2 + \left( x_2 - \mu \right)^2 + \cdots + \left( x_N - \mu \right)^2 \right] \;}\ , ~~\text{ where }~~ \mu \equiv \frac{1}{N} \left(x_1 + \cdots + x_N \right)\ ,</math>
Note: The above expression has a built-in bias. See the discussion on [[Bessel's correction]] further down below.
 
or, by using [[summation]] notation,

<math display="block">\sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N \left(x_i - \mu \right)^2 \;}\ , ~~\text{ where }~~ \mu \equiv \frac{1}{N} \sum_{i=1}^N x_i ~.</math><!-- In the previous, not N - 1 but N.  This is the whole population. -->

If, instead of having equal probabilities, the values have different probabilities, let {{math|{{var|x}}{{sub|1}}}} have probability {{math|{{var|p}}{{sub|1}}}}, {{math|{{var|x}}{{sub|2}}}} have probability {{math|{{var|p}}{{sub|2}}, ..., {{var|x}}{{sub|{{var|N}}}}}} have probability {{nobr|{{math|{{var|p}}{{sub|{{var|N}}}}}} .}} In this case, the standard deviation will be
<math display="block">\sigma = \sqrt{ \sum_{i=1}^N p_i(x_i - \mu)^2 \;}\ , ~~\text{ where }~~ \mu \equiv \sum_{i=1}^N p_i\ x_i ~.</math>

===Continuous random variable===
The standard deviation of a [[continuous distribution|continuous real-valued random variable]] {{mvar|X}} with [[probability density function]] {{math|{{var|p}}({{var|x}})}} is
<math display="block">\sigma = \sqrt{ \int_\mathbf{X} \left( x - \mu \right)^2\ p(x)\ {\mathrm d} x \;}\ , ~~\text{ where }~~ \mu \equiv \int_\mathbf{X} x\ p(x)\ {\mathrm d} x\ ,</math>

and where the integrals are [[definite integral]]s taken for {{mvar|x}} ranging over '''{{math|X}}''', which represents the set of possible values of the random variable&nbsp;{{mvar|X}}.

In the case of a [[parametric model|parametric family of distributions]], the standard deviation can often be expressed in terms of the parameters for the underlying distribution. For example, in the case of the [[log-normal distribution]] with parameters {{mvar|μ}} and {{math|{{var|σ}}{{sup|2}}}} for the underlying normal distribution, the standard deviation of the log-normal variable is given by the expression
<math display="block"> \sqrt{ \left(e^{\sigma^2} - 1\right)\ e^{2\mu + \sigma^2} \;} ~.</math>