Wikipedia has an extensive discussion of this expression. It says it is “…a principle with broad application that is not intended to be strictly accurate or reliable for every situation. It is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination.” It is compared to heuristic, which is a similar concept used in math, psychology, and computer science.  The origin is uncertain, but likely comes several possible instances where the thumb is used to make an imprecise but convenient measurement. There is less evidence that the expression originated from prohibition of using a stick wider than the thumb to beat a wife.  Early colonial law often specifically rejected the concept. However, “…the phrase and connection gained currency in 1982, when the U.S. Commission on Civil Rights issued a report on wife abuse, titled ‘Under the Rule of Thumb’.”