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HTML
174 lines
5.5 KiB
HTML
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<TITLE>Comparison</TITLE>
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<B> Next:</B> <A NAME="tex2html934"
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HREF="node61.html">Analytical comparison of averages</A>
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<B> Up:</B> <A NAME="tex2html930"
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HREF="node54.html">Average vs. weighted average</A>
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<B> Previous:</B> <A NAME="tex2html924"
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HREF="node59.html">Mighell-Poisson weighted average</A>
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<B> <A NAME="tex2html932"
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<H3><A NAME="SECTION00625600000000000000">
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Comparison</A>
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</H3>
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<P>
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We have seen four different ways to take an average -
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two simple averages (the second skipping zero values)
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and two weighted averages (using straight Poisson and Poisson-Mighell [6] <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$">
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formulations).
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We know that the simple average (not skipping zeros) is the best possible result. However,
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there are inconveniences with it. If for instance we need to scale our data before averaging, then the
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simple average is no more usable (it will give the correct average but a bad estimate of the s.d.) .
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In any case, the passage to normal statistics (using Mighell's correction) needs to be done before or later.
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Therefore a comparison is due in order to ascertain
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how wrong can it be using the different methods.
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<P>
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We have to give a measure of what is negligible first.
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The relative error is a measure of the smallest relative variation of an estimate <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img144.png"
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ALT="$ x$">
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that is <I>not</I> negligible:
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<P><!-- MATH
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\begin{displaymath}
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\epsilon_x = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{x}}}}}}}
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\end{displaymath}
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-->
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</P>
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<IMG
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WIDTH="61" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
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ALT="$\displaystyle \epsilon_x = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{x}}}}}}}
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$">
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</DIV><P>
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</P>
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We shall then consider negligible
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(w.r.t. <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img144.png"
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ALT="$ x$">
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) terms whose relative magnitude is <!-- MATH
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$O(\epsilon_x^2)$
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-->
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<IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img172.png"
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ALT="$ O(\epsilon_x^2)$">
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.
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As the s.d. of <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img144.png"
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ALT="$ x$">
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is <!-- MATH
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$\propto\epsilon_x$
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-->
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<IMG
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WIDTH="36" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img173.png"
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ALT="$ \propto\epsilon_x$">
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, we may not discard terms <!-- MATH
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$O(\epsilon_x^2)$
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-->
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<IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img172.png"
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ALT="$ O(\epsilon_x^2)$">
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on the s.d.;
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there instead we may neglect terms <!-- MATH
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$O(\epsilon_x^3)$
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-->
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<IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img174.png"
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ALT="$ O(\epsilon_x^3)$">
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.
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<B> Next:</B> <A NAME="tex2html934"
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HREF="node61.html">Analytical comparison of averages</A>
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<B> Up:</B> <A NAME="tex2html930"
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HREF="node54.html">Average vs. weighted average</A>
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<B> Previous:</B> <A NAME="tex2html924"
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HREF="node59.html">Mighell-Poisson weighted average</A>
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<B> <A NAME="tex2html932"
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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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2018-09-28
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</ADDRESS>
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