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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<!--Converted with LaTeX2HTML 2008 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<HEAD>
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<TITLE>Weighted average: definition and relationship with </TITLE>
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<BR>
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<B> Next:</B> <A NAME="tex2html898"
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HREF="node58.html">Straight Poisson (zero-skipping) weighted</A>
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<B> Up:</B> <A NAME="tex2html894"
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HREF="node54.html">Average vs. weighted average</A>
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<B> Previous:</B> <A NAME="tex2html888"
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HREF="node56.html">Zero-skipping average</A>
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<B> <A NAME="tex2html896"
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HREF="node1.html">Contents</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<H3><A NAME="SECTION00625300000000000000">
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Weighted average: definition and relationship with <IMG
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WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.png"
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ALT="$ \chi ^2$">
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</A>
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</H3>
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<P>
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A weighted average is the result of the special case of a data fitting to a model function which is a constant.
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It is easy to see that minimizing w.r.t <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img144.png"
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ALT="$ x$">
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<P><!-- MATH
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\begin{displaymath}
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\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({x-O_j}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="143" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img153.png"
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ALT="$\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\dis...
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...h{\left({x-O_j}\right)}}^2
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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$">
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</DIV><P>
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</P>
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yields
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<P><!-- MATH
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\begin{displaymath}
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x= \langle x \rangle_{\!\mathrm{w}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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}}}}{{\ensuremath{\displaystyle{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="141" HEIGHT="126" ALIGN="MIDDLE" BORDER="0"
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SRC="img154.png"
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ALT="$\displaystyle x= \langle x \rangle_{\!\mathrm{w}}={\ensuremath{\displaystyle{\f...
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...th{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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}}}}}}}
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$">
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</DIV><P>
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</P>
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The good-faith s.d. (square-root of twice the inverse of the second derivative of <IMG
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WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.png"
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ALT="$ \chi ^2$">
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at the minimum)
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is then
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{\langle x \rangle_{\!\mathrm{w}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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1
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}}}}{{\ensuremath{\displaystyle{\sqrt{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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}}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="130" HEIGHT="122" ALIGN="MIDDLE" BORDER="0"
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SRC="img155.png"
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ALT="$\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w}}} = {\ensuremath{\display...
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...h{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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}}}}}}}}
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$">
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</DIV><P>
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</P>
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I use the term 'good-faith' to indicate the case when it is really appropriate to use a constant as a model functions,
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i.e. when the observations are truly different observations of the same observable.
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When this is not the case but we do not know what to do better we can at least increase the s.d.
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In fact, there is a correction factor for the s.d., given - in this case - by
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<P><!-- MATH
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\begin{displaymath}
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\mathsf{GoF}=
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\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j^2
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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{\ensuremath{\left[{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}}
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}\right]}}^2
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}}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1
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}}}}{{\ensuremath{\displaystyle{
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\sigma_j^2
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}}}}}}} }}}}}}}
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}}}}{{\ensuremath{\displaystyle{
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N_{\mathrm{obs}}-1
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}}}}}}}
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}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="230" HEIGHT="267" ALIGN="MIDDLE" BORDER="0"
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SRC="img156.png"
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ALT="$\displaystyle \mathsf{GoF}=
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\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ensuremat...
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...
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}}}}}}} }}}}}}}
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}}}}{{\ensuremath{\displaystyle{
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N_{\mathrm{obs}}-1
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}}}}}}}
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}
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$">
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</DIV><P>
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</P>
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so that
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<P><!-- MATH
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\begin{displaymath}
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{\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img157.png"
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ALT="$\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}}
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$">
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</DIV><P>
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</P>
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<P>
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Specializing now to the two cases above,
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<P>
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<B> Next:</B> <A NAME="tex2html898"
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HREF="node58.html">Straight Poisson (zero-skipping) weighted</A>
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<B> Up:</B> <A NAME="tex2html894"
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HREF="node54.html">Average vs. weighted average</A>
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<B> Previous:</B> <A NAME="tex2html888"
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HREF="node56.html">Zero-skipping average</A>
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<B> <A NAME="tex2html896"
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HREF="node1.html">Contents</A></B>
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<ADDRESS>
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Thattil Dhanya
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2018-03-12
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</ADDRESS>
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