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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 galaxy-conditioned neighbour count is the integral of the conditional number density $\bar{n}[1+\xi (r)]$, rather than of $\bar{n}$ alone. For a power law $\xi =(r_0/r{)}^{\gamma }$ with $\gamma <3$,With the specified $\gamma =2$, $R=10h^{-1}\mathrm{Mpc}$, $r_0=5h^{-1}\mathrm{Mpc}$ and $\bar{n}=0.01(h^{-1}\mathrm{Mpc}{)}^{-3}$, the factors of $h$ cancel andA uniformly chosen point in space is not conditioned to lie on a galaxy. Its expected value is simply the mean number density times the volume:The difference $10\pi \simeq 31.42$ is the clustering excess. The first count means other galaxies; including the chosen central galaxy adds one. Although the assumed galaxy two-point correlation function diverges at $r=0$, its contribution is integrable here because $r^2\xi (r)$ is constant.

For a statistically homogeneous matter distribution, the cosmological two-point correlation function of the mean-zero density contrast $\delta =(\rho -\bar{\rho })/\bar{\rho }$ isSpatial homogeneity removes dependence on $\mathbf{x}$, and isotropy reduces the separation argument to its magnitude. For discrete galaxies, the galaxy two-point correlation function equivalently specifies the excess pair probability at distinct positions:Consequently the mean density of other galaxies, conditioned on a galaxy at the origin, is $\bar{n}[1+\xi (r)]$. The self-pair at zero separation is excluded from this definition; its Dirac delta contribution is the galaxy power spectrum shot noise of the unsmoothed number-density field.