<dl>
<dd><span class="texhtml"><i>A</i><sub>1</sub> = <i>x</i><sub>1</sub></span></dd>
<dd><img class="tex" alt="
A_i=A_{i-1}+\frac{x_i-A_{i-1}}{i}
" src="http://upload.wikimedia.org/math/8/3/2/8329dff4bacb09ce4a81a069d040c086.png" /></dd>
</dl>
<p>where A is the mean value.</p>
<dl>
<dd><span class="texhtml"><i>Q</i><sub>1</sub> = 0</span></dd>
<dd><img class="tex" alt="
Q_i=Q_{i-1}+\frac{(i-1)(x_i-A_{i-1})^2}{i}
" src="http://upload.wikimedia.org/math/f/0/1/f019c960a8a82d4d5ec955d3c11ae540.png" /></dd>
</dl>
<p>sample variance:</p>
<dl>
<dd><img class="tex" alt="
s^2_n=\frac{Q_n}{n-1}
" src="http://upload.wikimedia.org/math/9/b/e/9be1db86f3cc7e91dd114f2af434eabb.png" /></dd>
</dl>
<p>standard variance</p>
<dl>
<dd><img class="tex" alt="
\sigma^2_n=\frac{Q_n}{n}
" src="http://upload.wikimedia.org/math/2/c/c/2cc8b2dc542fa217eb32117a0a73d0c8.png" /></dd>
</dl>
<p>For weighted distribution it is somewhat more complicated: The mean is given by:</p>
<dl>
<dd><span class="texhtml"><i>A</i><sub>1</sub> = <i>x</i><sub>1</sub></span></dd>
<dd><img class="tex" alt="
A_i=A_{i-1}+(x_i-A_{i-1})\frac{w_i}{\sum^{i}_{j=1} w_j}
" src="http://upload.wikimedia.org/math/4/7/5/475f126ffd2894635351239e5af2177e.png" /></dd>
</dl>
<p>where <span class="texhtml"><i>w</i><sub><i>j</i></sub></span> are the weights.</p>
<dl>
<dd><span class="texhtml"><i>Q</i><sub>1</sub> = 0</span></dd>
<dd><img class="tex" alt="
Q_i=Q_{i-1}+w_i\frac{\sum^{i-1}_{j=1} w_j}{\sum^{i}_{j=1} w_j} ( x_i-A_{i-1})^2
" src="http://upload.wikimedia.org/math/2/3/d/23d7d5775a633b27acd95cb818b80269.png" /></dd>
<dd><img class="tex" alt="
s^2_n=Q_n \frac{n'}{(n'-1)\sum^{n}_{j=1} w_j}
" src="http://upload.wikimedia.org/math/9/4/f/94f2b60a9eaa756d60501d7a1bb8ab7b.png" /></dd>
<dd><img class="tex" alt="
\sigma^2_n=Q_n \frac{1}{\sum^{n}_{j=1} w_j}
" src="http://upload.wikimedia.org/math/f/d/d/fddbf9c877c7b9d4d1d3f2ea2f555322.png" /></dd>
</dl>
<p>where n is the total number of elements, and n' is the number of elements with non-zero weights. The above formulas become equal to the more simple formulas given above if we take all weights equal to 1.</p>
<p><a name="See_also" id="See_also"></a></p>
<h2><span class="editsection">[<a href="/w/index.php?title=Standard_deviation&action=edit&sec tion=19" title="Edit section: See also">edit</a>]</span> <span class="mw-headline">See also</span></h2>
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