CCV masks the impact of changing background concentration. When ${\ensuremath{\sigma_{\Delta}}}= 0$, gradient sensing is limited by ligand-receptor fluctuations, and increases as $\bar{c}$ moves away from $K_D$ ([Fig. ]{}\[fig:ccv\_illus\]d) – accuracy decreases if either few receptors are bound, or if receptors are saturated. As CCV increases, ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ no longer depends strongly on $\bar{c}$ ([Fig. ]{}\[fig:ccv\_illus\]d). Finally, Eq. \[eq:combined\_error\] shows that gradient sensing accuracy depends on the shape parameter $\chi$ and, therefore, on cluster size. For hexagonally packed clusters of cells with unit spacing[^1], a cluster with $Q$ layers has $N = 1 + 3Q + 3Q^2$ cells and $\chi(Q) = (5/8) Q^4 + (5/4) Q^3 + (7/8) Q^2 + (1/4) Q$ (Appendix \[app:Q\]), i.e. $\chi(Q) \sim Q^4 \sim N^2$. Clusters of increasing size then have an error that decreases as $1/N^2$ ([Fig. ]{}\[fig:ccv\_illus\]c); [this scaling is similar to earlier results for single cells (Appendix \[app:Q\]).]{}

Reducing estimation error by time-averaging {#sec:timeaverage}
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If a cluster made $n$ independent measurements, it could reduce ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ by a factor of $n$. In single-cell gradient sensing, independent measurements can be made by averaging over time – improving errors by a factor $\sim T/\tau_{\textrm{corr}}$, where $T$ is the averaging time, and $\tau_{\textrm{corr}}$ the measurement correlation time. 