Editing Standard deviation (section)

===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>