The more important thing is the quality and properties of the soil, since you need enough area to spread the load out. Most strip footings (20" to 24" wide) for a house are over-sized because of prescriptive code requirements and the need to have some standardized footing construction systems for all homes. Usually, the footing thickness is immaterial unless it is a very wide footing, where it would have to be thickened slightly.
In many areas on decent soils, a 20" wide 8" thick unreinforced concrete footing is allowed for a 12" thick wall bearing on it, but 24" may be commonly used. - No big savings on the cheap materials (concrete) since the forms are re-used for many jobs and it just a dump and place job with minimal finishing. It is the cost of layout, leveling, forming and stripping that does not vary too much for slightly over-sized standard footing widths.
Dick pretty well covered it. Two factors play into this....Soil type/bearing capacity and the structure itself. You need to determine the applicable code in your area, which will establish minimum requirements. A rule of thumb is twice the wall thickness for the footing width. Thickness of the footing varies, as does reinforcement. Your local building inspection office will possibly be able to help you answer the question, as would an engineer.
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