In the book "The Case for Mars", cited in this answer, there is a study cited that states that 20-50 REM radiation occurs due to cosmic rays in deep space. It seems likely that the atmosphere of Mars would reduce that somewhat. By how much will the atmosphere of Mars reduce the radiation?
where H is the scale-height of the gas. For the composition of Mars atmosphere, H = 11.1 km. The sea-level density for Mars is N(0) = 0.00020 g/cm3. (For Earth: H=8.5 km, N(0) = 0.012 g/cm3)
The amount of surface shielding for radiation arriving from 90° direction (vertical) is given by evaluating the integral of N(z), which results in:
Equivalent Shielding = $N(0)H$
Mars Equivalent Shielding = 0.020 kg/m3 x 11.1 km = 22 gm/cm2
(Earth Equivalent Shielding = 1.2 kg/m3 x 8.5 km = 1,020 gm/cm2)
(onboard ISS: 10 g/cm2)
(Apollo space suite: 0.1 g/cm2)
Assume R (Earth) = 6378 kilometers, R (Mars = 3374 km)
It's worth noting that a bare body on Mars' surface receives half the radiation of an astronaut onboard ISS.
It's also worth noting that to get equivalent shielding of the whole Earth atmosphere by using a water shielding, only 10 meters thickness is required (given that water density is 1000 kg m3, compared to 1.2 kg/m3 for Earth atmosphere).