Editing Standard deviation (section)

===Weighted calculation===
<!--N.B. the apparently superfluous trailing \0 in the <math> equations prevents conversion of simple formulas to HTML, resulting in more consistent formatting.-->
When the values <math>x_k</math> are weighted with unequal weights <math>w_k</math>, the power sums {{math|{{var|s}}{{sub|0}}, {{mvar|s}}{{sub|1}}, {{var|s}}{{sub|2}}}} are each computed as:

<math display="block">s_j = \sum_{k=1}^N w_k x_k^j.\,</math>

And the standard deviation equations remain unchanged. {{math|{{var|s}}{{sub|0}}}} is now the sum of the weights and not the number of samples {{mvar|N}}.

The incremental method with reduced rounding errors can also be applied, with some additional complexity.

A running sum of weights must be computed for each {{mvar|k}} from 1 to {{mvar|n}}:
<math display="block">\begin{align}
  W_0 &= 0 \\
  W_k &= W_{k-1} + w_k
\end{align}</math>

and places where {{math|1/{{var|k}}}} is used above must be replaced by <math>w_k/W_k</math>:
<math display="block">\begin{align}
  A_0 &= 0 \\
  A_k &= A_{k-1} + \frac{w_k}{W_k}\left(x_k - A_{k-1}\right) \\
  Q_0 &= 0 \\
  Q_k &= Q_{k-1} + \frac{w_k W_{k-1}}{W_k}\left(x_k  -A_{k-1}\right)^2 = Q_{k-1} + w_k\left(x_k-A_{k-1}\right)\left(x_k - A_k\right)
\end{align}</math>

In the final division,
<math display="block">\sigma^2_n = \frac{Q_n}{W_n}\,</math>

and
<math display="block">s^2_n = \frac{Q_n}{W_n - 1},</math>

or 
<math display="block">s^2_n = \frac{n'}{n' - 1} \sigma^2_n,</math>

where {{mvar|n}} is the total number of elements, and {{mvar|{{prime|n}}}} is the number of elements with non-zero weights.

The above formulas become equal to the simpler formulas given above if weights are taken as equal to one.