Threshold biasing of a Gaussian random field gives a linear amplification of the reduced two point correlation function at large distances. We show that for standard cosmological models this does not translate into a linear amplification of the power spectrum (PS) at small k. For standard CDM type models this means that the ``turn-over'' at small k of the original PS disappears in the PS of the biased field for the physically relevant range of the threshold parameter \nu. In real space this difference is manifest in the asymptotic behaviour of the normalised mass variance in spheres of radius R, which changes from the ``super-homogeneous'' behaviour \sigma^2(R) \sim R^{-4} to a Poisson-like behaviour \sigma^2 {\nu}(R) \sim R^{-3}. This qualitative change results from the intrinsic stochasticity of the threshold sampling. While our quantitative results are specific to the simplest threshold biasing model, we argue that our qualitative conclusions should be valid generically for any biasing mechanism involving a scale-dependent amplification of the correlation function. One implication is that the real-space correlation function will be a better instrument to probe for the underlying Harrison Zeldovich spectrum in the distribution of visible matter, as the characteristic asymptotic negative power-law \xi (r) \sim -r^{-4} tail is undistorted by biasing.