The 10-year-old boy charged with robbing a midtown bank last Feb. 25 was found guilty yesterday in Family Court in Manhattan. The boy, identified in court papers only as ''Robert M.,'' went up to a teller in the New York Bank for Savings, 1230 Avenue of the Americas, waved what appeared to be a gun, demanded money and was given $118.

Judge Peggy O. Davis rejected a defense contention that Robert had ''intended to play rather than to rob.'' It must now be determined whether Robert needs to be supervised. If so, he could be placed in a juvenile detention facility. If not, he would be released in the custody of his family. The Probation Department will conduct an investigation and report its findings in court on July 6.