mirror of
https://github.com/slsdetectorgroup/slsDetectorPackage.git
synced 2025-04-28 17:10:03 +02:00
265 lines
7.6 KiB
HTML
265 lines
7.6 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2012 (1.2)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* 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>Weighted average: definition and relationship with </TITLE>
|
|
<META NAME="description" CONTENT="Weighted average: definition and relationship with ">
|
|
<META NAME="keywords" CONTENT="slsDetectors-FAQ">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2012">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="slsDetectors-FAQ.css">
|
|
|
|
<LINK REL="next" HREF="node58.html">
|
|
<LINK REL="previous" HREF="node56.html">
|
|
<LINK REL="up" HREF="node54.html">
|
|
<LINK REL="next" HREF="node58.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html897"
|
|
HREF="node58.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html893"
|
|
HREF="node54.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html887"
|
|
HREF="node56.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/share/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html895"
|
|
HREF="node1.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html898"
|
|
HREF="node58.html">Straight Poisson (zero-skipping) weighted</A>
|
|
<B> Up:</B> <A NAME="tex2html894"
|
|
HREF="node54.html">Average vs. weighted average</A>
|
|
<B> Previous:</B> <A NAME="tex2html888"
|
|
HREF="node56.html">Zero-skipping average</A>
|
|
<B> <A NAME="tex2html896"
|
|
HREF="node1.html">Contents</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H3><A NAME="SECTION00625300000000000000">
|
|
Weighted average: definition and relationship with <IMG
|
|
WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1.png"
|
|
ALT="$ \chi ^2$">
|
|
</A>
|
|
</H3>
|
|
|
|
<P>
|
|
A weighted average is the result of the special case of a data fitting to a model function which is a constant.
|
|
It is easy to see that minimizing w.r.t <IMG
|
|
WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img144.png"
|
|
ALT="$ x$">
|
|
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({x-O_j}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="143" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img153.png"
|
|
ALT="$\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\dis...
|
|
...h{\left({x-O_j}\right)}}^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
yields
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
x= \langle x \rangle_{\!\mathrm{w}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
|
|
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="141" HEIGHT="126" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img154.png"
|
|
ALT="$\displaystyle x= \langle x \rangle_{\!\mathrm{w}}={\ensuremath{\displaystyle{\f...
|
|
...th{\displaystyle{1
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
The good-faith s.d. (square-root of twice the inverse of the second derivative of <IMG
|
|
WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1.png"
|
|
ALT="$ \chi ^2$">
|
|
at the minimum)
|
|
is then
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\sigma_{\langle x \rangle_{\!\mathrm{w}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
|
|
1
|
|
}}}}{{\ensuremath{\displaystyle{\sqrt{
|
|
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
}}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="130" HEIGHT="122" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img155.png"
|
|
ALT="$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w}}} = {\ensuremath{\display...
|
|
...h{\displaystyle{1
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
}}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
I use the term 'good-faith' to indicate the case when it is really appropriate to use a constant as a model functions,
|
|
i.e. when the observations are truly different observations of the same observable.
|
|
When this is not the case but we do not know what to do better we can at least increase the s.d.
|
|
In fact, there is a correction factor for the s.d., given - in this case - by
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\mathsf{GoF}=
|
|
\sqrt{
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
|
|
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j^2
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
|
|
{\ensuremath{\left[{
|
|
\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}}
|
|
}\right]}}^2
|
|
}}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
\sigma_j^2
|
|
}}}}}}} }}}}}}}
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
N_{\mathrm{obs}}-1
|
|
}}}}}}}
|
|
}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="230" HEIGHT="267" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img156.png"
|
|
ALT="$\displaystyle \mathsf{GoF}=
|
|
\sqrt{
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremat...
|
|
...
|
|
}}}}}}} }}}}}}}
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
N_{\mathrm{obs}}-1
|
|
}}}}}}}
|
|
}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
so that
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
{\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img157.png"
|
|
ALT="$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
|
|
<P>
|
|
Specializing now to the two cases above,
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html897"
|
|
HREF="node58.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/share/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html893"
|
|
HREF="node54.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/share/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html887"
|
|
HREF="node56.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/share/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html895"
|
|
HREF="node1.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html898"
|
|
HREF="node58.html">Straight Poisson (zero-skipping) weighted</A>
|
|
<B> Up:</B> <A NAME="tex2html894"
|
|
HREF="node54.html">Average vs. weighted average</A>
|
|
<B> Previous:</B> <A NAME="tex2html888"
|
|
HREF="node56.html">Zero-skipping average</A>
|
|
<B> <A NAME="tex2html896"
|
|
HREF="node1.html">Contents</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Thattil Dhanya
|
|
2018-09-28
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|