A slab on grade foundation requires the same minimum depth of excavation as a crawl. The depth of excavation for both types require the bottom of the footings to be supported by solid bearing ground and to be below the frost line. The frost line is determined by where you live with the development services department, of your local building inspection, setting the minimum depth.
I build houses here on the southern part of Vancouver Island where the local building code requires 18" of frost protection. This means the distance from the bottom of the footings to the backfill height must be a minimum of 18". Allowing for 8" of concrete foundation wall above the backfill height that translates into 2 foot concrete foundation walls bearing on 6" thick footings. The width of the footings is determined by the bearing capacity of the soil. (16" to 24" normally).
A slab on grade, as it's normally called, is when the interior of the foundation is backfilled with compacted free draining gravel or native free draining earth to height of 3 1/2" from the top of foundation. A 3 1/2" slab is poured and placed bringing the top of floor flush to the top of foundation.The plumbing and electrical service is placed in the compacted soil under the slab.
Usually a slab on grade is used for bsmt. entry homes where the living is above on a joist and plywood floor. A one level rancher on a slab is usually cheaper to build but walking and living on a slab can sometimes be uncomfortable unless in-floor-heating is installed.
A crawl space is more desireable for a one level home with a minimum, for access, in non-plumbing areas of 18", in plumbing areas and of 24" and a minumum of 36" if a force air furnance is installed in the crawl space. These crawl space heights are minimum requirements but if increased to 48", this area serves well for storage.
The crawl space exterior foundation walls are insulated with 2" rigid foam and the joist ends with R-20 fiberglass and sealed with a 6 mil poly vapor barrier.
There you have it.
Rod