Let's try to put some qualitative numbers in a crash accident:
You are riding downhill on a paved street and for whatever reason (pothole, small animal, a truck going by, a pedestrian...) you take a spill and hit the sidewalk with top right of your skull. The impact is strong enough to break the bones ...and that's it...You are a vegetable or a cadaver.
Let's assume that the contact surface was 2"x0.5" = 1 square inch (the edge of the sidewalk, crushing your skull).
Had you worn a helmet, the contact patch on your skull would be more like 3" x 2" = 6 in sq therefore the contact pressure would be 1/6th than that of the helmetless.
Moving on, in a collision the kinetic energy of your head K=1/2 mV2 which will increase slightly due to the addition of the helmet mass, has to be absorbed. The kinetic energy absorption is proportional to the load P(t)xΔt. The P(t) is a triangular more or less function and if Δt is the base of the triangle, the kinetic energy is proportional to the area of the triangle.
In the helmeted crash the time Δt increases, because it takes time to crush the styrofoam, before the load is transmitted to your skull. Let's say 3Δt. Since the kinetic energy is the same, this means that the load is further reduced by threefold (more or less independently of the sixfold reduction due to the contact patch). we are at 18-fold reduction of the load which tries to crush your skull.
I told a small lie before. The kinetic energy to be absorbed is not the same. There is energy consumed in crushing the styrofoam. I am not sure how much and how it can be incorporated in a quick qualitative calculation.
Anyway, reducing the load on the skull by 18 times (reducing it to 5.5% of the original) is good enough reason for me. YMMV.