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But we are binning! That means that we have a distribution of masses in our bin that we must integrate over. This integral takes the form:  \begin{equation}  \langle \hat{\sigma} \rangle = \frac{\int dM \frac{d \langle n \rangle}{dM} P(\hat{\sigma} | M)}{\int} M)}{\int  dM \frac{d \langle n \rangle}{dM}} \end{equation}  So this is what we want to compare the measured observable with. A scatter convolved, mass-function weighted expectation value.