No.  Material density has very little to do with atomic weight.  It's more related to molecular arrangement.
Oh, and tungsten is significantly more dense than lead, even though it has a lower atomic weight.

In the case of metals, density is related both to atomic mass (formerly called atomic weight) and atomic arrangement. In the case of salts (called ceramics by Material Science) instead of atoms they have molecules, that are formed by at least two atoms but can be many more.
In this case, molecular masses and arrangements are valid. 
The atomic (or molecular) arrangement is described by the "unit cells", that are the way, or geometric figures, in which the atoms (or molecules) organize themselves. To see how the unit cells look like (there are 14), enter Google and write "unit cells".

Let's give an example: iron. Knowing its atomic mass, atomic radius and its unit cell (body centered cube at room temperature), you can calculate theoretically its density, that will be almost equal to the actual one. Reason of the very little difference is that solid materials are not absolutely compact, they have empty spaces not occupied by atoms or molecules, called "vacancies" by Material Scientists. 
Giovanni S. Crisi
Sao Paulo - Brazil