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127 lines
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127 lines
4.5 KiB
HTML
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<TITLE>Zero-skipping average</TITLE>
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<B> Next:</B> <A NAME="tex2html886"
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HREF="node57.html">Weighted average: definition and</A>
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HREF="node54.html">Average vs. weighted average</A>
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HREF="node55.html">Simple average</A>
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<H3><A NAME="SECTION00625200000000000000">
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Zero-skipping average</A>
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</H3>
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<P>
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In some cases, in order to avoid possible singularities,
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values <IMG
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WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img148.png"
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ALT="$ C_j=0$">
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are skipped. Then if <!-- MATH
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$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="img149.png"
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ALT="$ N_{\mathrm{obs}}^*$">
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is the number of non-zero data points,
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we can evaluate the 'zero-skipping' average as
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<P><!-- MATH
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\begin{displaymath}
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x=\langle x\rangle^*={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*}}}}}}}
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\mathop{\sum}_ {\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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C_j={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*}}}}}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\langle x\rangle
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="382" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
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SRC="img150.png"
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ALT="$\displaystyle x=\langle x\rangle^*={\ensuremath{\displaystyle{\frac{{\ensuremat...
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...obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\langle x\rangle
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$">
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</DIV><P>
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</P>
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The standard deviation is then
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{x^*}= {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\sigma_x = \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}
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\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="300" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
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SRC="img151.png"
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ALT="$\displaystyle \sigma_{x^*}= {\ensuremath{\displaystyle{\frac{{\ensuremath{\disp...
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...e{\langle x\rangle^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}
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$">
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</DIV><P>
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</P>
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Note that the s.d. is evaluated exactly as if the non-zero <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img152.png"
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ALT="$ C_j$">
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were the only observations,
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whilst the average is overestimated by the fraction of zero-counting events.
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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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