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\begin{equation}  \langle \sigma | M \rangle = 1093 \left(\frac{h(z) M}{1e15 M_{\odot}}\right)^{0.34}  \end{equation}  This relationship has a very small lognormal  scatter of $S_{\log(\sigma) | \log(M)}\sim 4\%$. So: \begin{equation}  P(\sigma | M) = \frac{1}{\sqrt{2\pi}S_{\sigma | M}} e^{\frac{(\log(\sigma) - \langle \log(\langle  \sigma | M \rangle)^{2}}{2 \rangle))^{2}}{2  S_{\sigma | M}}} \end{equation}  The second velocity dispersion is the observed velocity dispersion $\hat{\sigma}$. In \citet{Gifford13a}, we define this as the l.o.s velocity dispersion. This has all kinds of ugly things in it including cluster shape effects, cluster environment contamination, substructure, redshift-space interlopers, and non-gaussianity. Not to mention the low number statistics that exist at low mass. Even though this is a messy observable, most are, and this is what we need to predict for a given mass $M$. So here is the generative model for observable:  \begin{equation}  P(\hat{\sigma} | M) = \sum_{\sigma} P(\hat{\sigma} | \sigma) P(\sigma | M)  \end{equation}  Really there are completeness and purity terms in there as well, but lets ignore those for a second. So that is our expected distribution for a given $M$, but $M$. The other term present is equally important $P(\hat{\sigma} | \sigma)$. This represents the probability of observing a velocity dispersion $\hat{\sigma}$ given $\sigma$. Why is this important? When we observe clusters in the real universe, we don't measure the "Evrard" velocity dispersion. We are randomly drawing from a distribution where the $\sigma$ is the expectation value. This is what \citet{Gifford13a} means by l.o.s effects. So what is that distribution? It's approximately lognormal with $S_{\log(\hat{\sigma}) | \log(\sigma)}\sim 25\%$. So:   \begin{equation}   P(\hat{\sigma} | \sigma) = \frac{1}{\sqrt{2\pi}S_{\hat{\sigma} | \sigma}} e^{\frac{(\log(\hat{\sigma}) - \log(\sigma))^{2}}{2 S_{\hat{\sigma} | \sigma}}}   \end{equation}     But  we are binning! That means that we have a distribution of masses in our bin that we must integrate over. What does this integral look like? \begin{equation}  \langle \hat{\sigma} \rangle = \int_{min(bin)}^{max_bin} dM \frac{d \langle n \rangle}{dM} P(\hat{\sigma} | M)  \end{equation}