Galaxy two-point correlation function ($d{P}_{12}={\bar{n}}^2[1+\xi (r)]d{V}_{1}d{V}_2$)

For a homogeneous isotropic point process of galaxies with mean number density $\bar{n}$, the galaxy two-point correlation function gives the excess pair probability at distinct positions. Dividing the factorial pair density by the probability of a central galaxy gives conditional neighbouring-galaxy density $\bar{n}[1+\xi (r)]$. Choosing a central location uniformly in volume does not perform this conditioning.

The mean number of other galaxies within distance $R$ of a randomly chosen galaxy is the integral of its conditional number density. A random location instead has mean count $4\pi \bar{n}{R}^3/3$. For a power law $\xi (r)=(r_0/r{)}^{\gamma }$, the extra count is $4\pi \bar{n}{r}_0^{\gamma }{R}^{3-\gamma }/(3-\gamma )$ when $\gamma <3$. The central galaxy is excluded; including it adds one. This distinction matters even when the unconditional mean number density is spatially uniform.