Editing Standard deviation (section)

===Geometric interpretation===
To gain some geometric insights and clarification, we will start with a population of three values, {{math|{{var|x}}{{sub|1}}, {{var|x}}{{sub|2}}, {{var|x}}{{sub|3}}}}. This defines a point {{math|1={{var|P}} = ({{var|x}}{{sub|1}}, {{var|x}}{{sub|2}}, {{var|x}}{{sub|3}})}} in {{math|'''R'''{{sup|3}}}}. Consider the line {{math|1={{var|L}} = {{mset|({{var|r}}, {{var|r}}, {{var|r}}) : {{var|r}} ∈ '''R'''}}}}. This is the "main diagonal" going through the origin. If our three given values were all equal, then the standard deviation would be zero and {{mvar|P}} would lie on {{mvar|L}}. So it is not unreasonable to assume that the standard deviation is related to the ''distance'' of {{mvar|P}} to {{mvar|L}}. That is indeed the case. To move orthogonally from {{mvar|L}} to the point {{mvar|P}}, one begins at the point:

<math display="block">M = \left(\bar{x}, \bar{x}, \bar{x}\right)</math>

whose coordinates are the mean of the values we started out with. 
{{Collapse top|title=Derivation of <math>M = \left(\bar{x}, \bar{x}, \bar{x}\right)</math>}}
<math>M</math> is on <math>L</math> therefore <math>M = (\ell,\ell,\ell)</math> for some <math>\ell \in \mathbb{R}</math>.

The line {{mvar|L}} is to be orthogonal to the vector from {{mvar|M}} to {{mvar|P}}. Therefore:

<math display="block">\begin{align} 
                                       L \cdot (P - M) &= 0    \\[4pt]
  (r, r, r) \cdot (x_1 - \ell, x_2 - \ell, x_3 - \ell) &= 0    \\[4pt]
               r(x_1 - \ell + x_2 - \ell + x_3 - \ell) &= 0    \\[4pt]
                      r\left(\sum_i x_i - 3\ell\right) &= 0    \\[4pt]
                                    \sum_i x_i - 3\ell &= 0    \\[4pt]
                                 \frac{1}{3}\sum_i x_i &= \ell \\[4pt]
                                               \bar{x} &= \ell
\end{align}</math>
{{Collapse bottom}}

A little algebra shows that the distance between {{mvar|P}} and {{mvar|M}} (which is the same as the [[orthogonal distance]] between {{mvar|P}} and the line {{mvar|L}}) <math display="inline">\sqrt{\sum_i \left(x_i - \bar{x}\right)^2}</math> is equal to the standard deviation of the vector {{math|({{var|x}}{{sub|1}}, {{var|x}}{{sub|2}}, {{var|x}}{{sub|3}})}}, multiplied by the square root of the number of dimensions of the vector (3 in this case).