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\section{Introduction}   While investigating the $\sim 7\%$ bias Nick was seeing in his velocity dispersion plot, it became obvious that this bias is a result of binning a sample with scatter in the observables and inherant functions in mass. Originally we thought these relationships would not manifest themselves in the median or mean values of the true observables (table values) that we compare with, but in this walk through we will show otherwise.     There are several key probabilities we need to know in order to accurately predict the expectation value of some observable with mass. In more statistical language, we would like to know $P(\hat{\theta}|M)$. That is, given a cluster of mass $M$, what is the probability the observable is detected as $\hat{\theta}$. For example, the observable $\hat{\theta}$ can be velocity dispersion, some richness, or total luminosity.     Let's take velocity dispersion as an example here. There are two relevant velocity dispersions in our studies. The first is the true 3D/2D velocity dispersion $\sigma$ which we cannot measure in the real universe. \citet{Evrard08} measured $\sigma$ for halos in N-body simulations and showed that they relate to the critical mass $M_{200}$ of the host halo on a very tight relation   \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 $S_{\ln\sigma | \ln M}\sim 4\%$. So:   \begin{equation}   P(\sigma | M) = \frac{1}{\sqrt{2\pi}S_{\ln\sigma | \ln M}} e^{\frac{(\ln\sigma - \ln\langle \sigma | M \rangle)^{2}}{2 S_{\ln\sigma | \ln M}^{2}}}   \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 of $\hat{\sigma}$ for a given $M$. The other term 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 $\sigma$. 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_{\ln\hat{\sigma} | \ln\sigma}} e^{\frac{(\ln\hat{\sigma} - \ln\sigma)^{2}}{2 S_{\ln\hat{\sigma} | \ln\sigma}^2}}   \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}