with equation 17. The direct application of Eq. (17) leads to a sub- stantial over prediction of the variance, since it ignores the effect of binning in pair separation which reduces the covariance in the measurement (Cohn 2006; Smith et al. 2008). An estimate of the correlation function in the i th pair separa- tion bin ˆ ξ i corresponds to the shell averaged correlation function where V i is the volume of the shell. The covariance of this estimate is given by the following expression, e.g., see (S´anchez et al. 2008) where is the volume averaged Bessel function. So the variance in ¯ ξ is σ 2 ¯ $_{ξ}$( r ) = C ˆ $_{ξ}$( i, i ) . We find that at scales of interest σ ¯ $_{ξ}$( r ) ≪ σ$_{ξ}$ ( r ) . From here, it is straightforward to compute the standard deviation in Δ D q using equation 15. We can describe a distribution of points to be homogeneous if the standard deviation of Δ D q is greater than Δ D $_{q}$. Note that this for- mulation gives us a unique scale: above this scale it is not possible to distinguish between the given point distribution and a homoge- neous distribution. In the cosmological context, we have bypassed the details of contribution to the variance arising from the survey geometry, survey size, etc. Clearly, if we were to take these con- tributions into account, the error in determination of Δ D q will be larger and hence we will recover a smaller scale of homogeneity (Sarkar et al. 2009). This change results not from any change in the real scale of homogeneity but because of the limitations of observa- tions. In our view, the real scale of homogeneity is one where these limitations do not matter. In the limit where we ignore these addi- tional sources of errors, we have the following general conclusions: • As long as non-linear correction are not important, the scale of homogeneity does not change with epoch. • In real space, the scale of homogeneity is independent of the tracer used, because the deviation Δ D q as well as the dispersion in this quantity scale in the same manner with bias. This follows from the fact that bias in correlation function can be approximated by a constant number at a given epoch at sufficiently large scales (Bagla 1998; Seljak 2000). • Redshift space distortions introduce some bias dependence in the scale of homogeneity. • As long as our assumption of q ¯ ξ ≪ 1 is valid, the scale of homogeneity is the same for all q . These points highlight the robust and unique nature of the scale of homogeneity defined in the manner proposed in this work. We now turn to more specific conclusions in the cosmological setting below. We have calculated σ $^{Δ}$D$_{q}$ and Δ D q using the power spectrum of a large N-body simulation as well as using the lin- ear power spectrum obtained from WMAP5 parameters. We have Figure 2. Variation of Δ D$_{q}$ and its predicted standard deviation with scale is shown in these plots. The top panel shows these for the Λ CDM simula- tion described in the text, the middle panel shows the same using the linearly evolved power spectrum and the lower panel again shows the same quantity with data from simulations patched with the linearly evolved power spec- trum at large scales. Δ D$_{q}$/ (0 . 5 qD ) is shown using a solid curve in each panel, the dashed line shows the dispersion σ Δ $_{D$_{q}$}$/ (0 . 5 qD ) and the dot- dashed line shows 2 σ Δ $_{D$_{q}$}$/ (0 . 5 qD ) . The intersection of Δ D$_{q}$/ (0 . 5 qD ) and σ Δ $_{D$_{q}$}$/ (0 . 5 qD ) is the scale beyond which we cannot distinguish be- tween the Λ CDM model and a homogeneous distribution. The scale of homogeneity 5 ˆ ξ i = 1 V i ∫$_{V}$$_{i}$ ξ ( r ) d $^{3}$r, (18) C ˆ $_{ξ}$( i, j ) = 1 V i V j ∫ d $^{3}$r ∫ d $^{3}$r $^{′}$C$_{ξ}$ ( r, r $^{′}$) = ∫ d k k 2 2 π 2 ¯ j 0 ( k, i ) ¯ j 0 ( k, j ) σ 2 $_{P}$( k ) , (19) ¯ j 0 ( k, i ) = 1 V i ∫$_{V}$$_{i}$ j 0 ( kr ) d $^{3}$r. (20) 4 SCALE OF HOMOGENEITY