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162 lines
4.9 KiB
HTML
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<TITLE>Simple average</TITLE>
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HREF="node56.html">Zero-skipping average</A>
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HREF="node54.html">Average vs. weighted average</A>
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HREF="node54.html">Average vs. weighted average</A>
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<H3><A NAME="SECTION00625100000000000000">
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Simple average</A>
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</H3>
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<P>
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Suppose we have <!-- MATH
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$N_{\mathrm{obs}}$
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-->
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<IMG
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WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img142.png"
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ALT="$ N_{\mathrm{obs}}$">
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Poisson-variate experimental evaluations
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<!-- MATH
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$C_j,\quad j=1\ldots N_{\mathrm{obs}}$
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-->
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<IMG
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WIDTH="138" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img143.png"
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ALT="$ C_j,\quad j=1\ldots N_{\mathrm{obs}}$">
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,
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of the same quantity <IMG
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WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img144.png"
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ALT="$ x$">
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.
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There are different ways to obtain from all <!-- MATH
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$N_{\mathrm{obs}}$
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-->
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<IMG
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WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img142.png"
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ALT="$ N_{\mathrm{obs}}$">
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data values a single estimate of the observable which is better than
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any of them. The most straightforward and the best is the simple average
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<P><!-- MATH
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\begin{displaymath}
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x=\langle x\rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ .
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="173" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
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SRC="img145.png"
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ALT="$\displaystyle x=\langle x\rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{...
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...aystyle{ N_{\mathrm{obs}}}}}}}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ .
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$">
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</DIV><P>
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</P>
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As the sum of Poisson variates is a Poisson variate, the standard deviation
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<P><!-- MATH
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\begin{displaymath}
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\sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j^2-{\ensuremath{\left({
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j
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}\right)}}
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}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="394" HEIGHT="87" ALIGN="MIDDLE" BORDER="0"
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SRC="img146.png"
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ALT="$\displaystyle \sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{
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{\en...
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...\mathrm{obs}}}}}}}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j
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}\right)}}
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}
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$">
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</DIV><P>
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</P>
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can be evaluated more comfortably as
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<P><!-- MATH
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\begin{displaymath}
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\sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}\sqrt{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j }
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=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="216" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
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SRC="img147.png"
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ALT="$\displaystyle \sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displayst...
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...style{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}}}}}
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$">
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</DIV><P>
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</P>
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<P>
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<BR><HR>
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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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