It is a general rule of thumb, still used, but regarded as inferior to HLD theory (Hydrophilic Lipophilic Difference), which takes many more factors into consideration.
[2] It was named after Wilder Dwight Bancroft, an American physical chemist, who proposed the rule in the 1910s.
What the Bancroft rule states is that contrary to common sense, what makes an emulsion oil-in-water or water-in-oil is not the relative percentages of oil or water, but which phase the emulsifier is more soluble in.
The hydrophilic-lipophilic balance (HLB) of a surfactant can be used in order to determine whether it's a good choice for the desired emulsion or not.
This empirical observation can be rationalized by considering the interfacial tension at the oil-surfactant and water-surfactant interfaces.