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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Simple average</TITLE>
<META NAME="description" CONTENT="Simple average">
<META NAME="keywords" CONTENT="slsDetectors-FAQ">
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<B> Next:</B> <A NAME="tex2html874"
HREF="node56.html">Zero-skipping average</A>
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HREF="node54.html">Average vs. weighted average</A>
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<H3><A NAME="SECTION00625100000000000000">
Simple average</A>
</H3>
<P>
Suppose we have <!-- MATH
$N_{\mathrm{obs}}$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img142.png"
ALT="$ N_{\mathrm{obs}}$">
Poisson-variate experimental evaluations
<!-- MATH
$C_j,\quad j=1\ldots N_{\mathrm{obs}}$
-->
<IMG
WIDTH="138" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img143.png"
ALT="$ C_j,\quad j=1\ldots N_{\mathrm{obs}}$">
,
of the same quantity <IMG
WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img144.png"
ALT="$ x$">
.
There are different ways to obtain from all <!-- MATH
$N_{\mathrm{obs}}$
-->
<IMG
WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img142.png"
ALT="$ N_{\mathrm{obs}}$">
data values a single estimate of the observable which is better than
any of them. The most straightforward and the best is the simple average
<P><!-- MATH
\begin{displaymath}
x=\langle x\rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ .
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="173" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
SRC="img145.png"
ALT="$\displaystyle x=\langle x\rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{...
...aystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ .
$">
</DIV><P>
</P>
As the sum of Poisson variates is a Poisson variate, the standard deviation
<P><!-- MATH
\begin{displaymath}
\sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j^2-{\ensuremath{\left({
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j
}\right)}}
}
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="394" HEIGHT="87" ALIGN="MIDDLE" BORDER="0"
SRC="img146.png"
ALT="$\displaystyle \sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{
{\en...
...\mathrm{obs}}}}}}}}}
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j
}\right)}}
}
$">
</DIV><P>
</P>
can be evaluated more comfortably as
<P><!-- MATH
\begin{displaymath}
\sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}\sqrt{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j }
=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}}}}}
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="216" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img147.png"
ALT="$\displaystyle \sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displayst...
...style{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}}}}}
$">
</DIV><P>
</P>
<P>
<BR><HR>
<ADDRESS>
Thattil Dhanya
2018-08-23
</ADDRESS>
</BODY>
</HTML>