We use a sample of 83 high-resolution cosmological zoom-in simulations and a semi-analytic model (SAM) to study the stochasticity of galaxy formation in haloes ranging from dwarf to Milky Way masses. Our simulated galaxies reproduce the observed inefficiency of galaxy formation as expressed through the stellar, gas and baryonic Tully-Fisher relations. For HI velocities in the range (70 less than or similar to V less than or similar to 220 km s(-1)), the scatter is just 0.08 to 0.14 dex, consistent with the observed intrinsic scatter at these scales. At low velocities (20 less than or similar to V less than or similar to 70 km s(-1)), the simulated scatter is 0.2-0.25 dex, which could be tested with future observations. The scatter in the stellar mass versus dark halo velocity relation is constant for 30 less than or similar to V less than or similar to 180 km s(-1), and smaller (similar or equal to 0.17 dex) when using the maximum circular velocity of the dark-matter-only simulation, V-max(DMO), compared to the virial velocity (V-200 or V-200(DMO)). The scatter in stellar mass is correlated with halo concentration, and is minimized when using a circular velocity at a fixed fraction of the virial radius similar or equal to 0.4R(200) or with V-alpha = V-200(DMO)(V-max(DMO) / V-200(DMO))(alpha) with alpha similar or equal to 0.7, consistent with constraints from halo clustering. Using the SAM we show the correlation between halo formation time and concentration is essential in order to reproduce this result. This uniformity in galaxy formation efficiency we see in our hydrodynamical simulations and an SAM proves the simplicity and self-regulating nature of galaxy formation in a Lambda cold dark matter (Lambda CDM) universe.