Our observations made on dry‐sandstone ultrasonic velocity data relate to the variation in velocity (or modulus) with effective stress, and the ability to predict a velocity for a rock under one effective pressure when it is known only under a different effective pressure. We find that the sensitivity of elastic moduli, and velocities, to effective hydrostatic stress increases with decreasing porosity. Specifically, we calculate the difference between an elastic modulus, [Formula: see text], of a sample of porosity ϕ at effective pressure [Formula: see text] and the same modulus, [Formula: see text], at effective pressure [Formula: see text]. If this difference, [Formula: see text], is plotted versus porosity for a suite of samples, then the scatter of ΔM is close to zero as porosity approaches the critical porosity value, and reaches its maximum as porosity approaches zero. The dependence of this scatter on porosity is close to linear. Critical porosity here is the porosity above which rock can exist only as a suspension—between 36% and 40% for sandstones. This stress‐sensitivity pattern of grain‐supported sandstones (clay content below 0.35) practically does not depend on clay content. In practical terms, the uncertainty of determining elastic moduli at a higher effective stress from the measurements at a lower effective stress is small at high porosity and increases with decreasing porosity. We explain this effect by using a combination of two heuristic models—the critical porosity model and the modified solid model. The former is based on the observation that the elastic‐modulus‐versus‐porosity relation can be approximated by a straight line that connects two points in the modulus‐porosity plane: the modulus of the solid phase at zero porosity and zero at critical porosity. The second one reflects the fact that at constant effective stress, low‐porosity sandstones (even with small amounts of clay) exhibit large variability in elastic moduli. We attribute this variability to compliant cracks that hardly affect porosity but strongly affect the stiffness. The above qualitative observation helps to quantitatively constrain P‐ and S‐wave velocities at varying stresses from a single measurement at a fixed stress. We also show that there are distinctive linear relations between Poisson’s ratios (ν) of sandstone measured at two different stresses. For example, in consolidated medium‐porosity sandstones [Formula: see text], where the subscripts indicate hydrostatic stress in MPa. Linear functions can also be used to relate the changes (with hydrostatic stress) in shear moduli to those in compressional moduli. For example, [Formula: see text], where [Formula: see text] is shear modulus and [Formula: see text] is compressional modulus, both in GPa, and the subscripts indicate stress in MPa.