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\end{equation}  This relationship has a very small lognormal scatter of $S_{\ln(\sigma) | \ln(M)}\sim 4\%$. So:  \begin{equation}  P(\sigma | M) = \frac{1}{\sqrt{2\pi}S_{\ln{\sigma | M}} \frac{1}{\sqrt{2\pi}}  e^{\frac{(\log(\sigma) - \log(\langle \sigma | M \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: