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<TITLE>Hint for Physicists</TITLE>
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<P><font size="+2" color="green">Hint for physicists</font></P>
<P>
Very often, the data to be fitted is a histogram of physical events. In that
case, since each bin would follow a multinomial distribution, the error is
equal to <code>√<i>f</i></code>, where <i>f</i> is the expression you are trying to fit.
Of course, since you don't know the parameter values yet, you don't actually
know <i>f</i>, so you approximate by using the <i>y</i> data values. In the
limit, these results are the same. In the case of a large number of bins, the
variance can be approximated by <code>√y</code>. Hence, the correct
weighting factor that will give properly normalized errors is <i>w = 1/y</i>,
and the corresponding one standard deviation error,
<font size="+1"><code>σ = E2/sqrt(χ<sup>2</sup>/n)</code></font>, where
<code>E2</code> is the standard error and <code>n</code> is the number of degrees of freedom, usually
equal to the number of data points minus the number of parameters,
<code>(N-M)</code>.</P>
<P>
<a href="chi2andweights.htm"><img src="../shadow_left.gif">
<font size="+1" color="olive">Chi-square and weights</font></a><br />
<a href="degfree.htm"><img src="../shadow_right.gif">
<font size="+1" color="olive">Degrees of freedom</font></a>
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