Changeset 744 for trunk/doc/Statistics.tex
 Timestamp:
 Feb 10, 2007, 9:16:11 PM (16 years ago)
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trunk/doc/Statistics.tex
r675 r744 22 22 % 021111307, USA. 23 23 24 \usepackage{html}25 24 26 25 … … 56 55 {\bf Weighted Statistics} 57 56 \normalsize 58 \begin{htmlonly}59 This document is also available in60 \htmladdnormallink{PDF}{Statistics.pdf}.61 \end{htmlonly}62 57 63 58 \tableofcontents … … 96 91 have not chosen one situation for our implementations, so see specific 97 92 function documentation for what assumtions are made. Though, common 98 for implementation are the following:93 for implementations are the following: 99 94 \begin{itemize} 100 95 \item Setting all weights to unity yields the same result as the … … 141 136 \sigma^2=<X^2><X>^2= 142 137 \\\frac{\sum w_ix_i^2}{\sum w_i}\frac{(\sum w_ix_i)^2}{(\sum w_i)^2}= 143 \\\frac{\sum w_i(x_i^2m^2)}{\sum w_i} 144 \\\frac{\sum w_i(x_i^22mx_i+m^2)}{\sum w_i} 138 \\\frac{\sum w_i(x_i^2m^2)}{\sum w_i}= 139 \\\frac{\sum w_i(x_i^22mx_i+m^2)}{\sum w_i}= 145 140 \\\frac{\sum w_i(x_im)^2}{\sum w_i} 146 141 \end{eqnarray} 147 142 This estimator fulfills that it is invariant under a rescaling and 148 143 having a weight equal to zero is equivalent to removing the data 149 point. Having all weight equal to unity we get $\sigma=\frac{\sum144 point. Having all weights equal to unity we get $\sigma=\frac{\sum 150 145 (x_im)^2}{N}$, which is the same as returned from Averager. Hence, 151 146 this estimator is slightly biased, but still very efficient.
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