Randy Duncan was born to Hearst and Louise Duncan in 1937. Randy moved from Osage to Mason City before finally attending Roosevelt High School in Des Moines, Iowa. Duncan was a highly regarded prospect in both football and basketball. He was a first team all-state guard on the Roosevelt basketball team that lost in the 1954 state championship game. Duncan was a first team all-state quarterback for Roosevelt, leading his high school to an undefeated season and a state title in 1954. His high school football teams only lost two games during Duncan's three years there.
Duncan graduated from high school after the 1954 fall semester, and he was heavily recruited after choosing to play football. He nearly went to Colorado but instead decided to attend the University of Iowa. Duncan has said that the only reason he went to Iowa was because of his friendship with Iowa assistant coach Bump Elliott.
Randy Duncan's college career got off to a slow start. As a mid-year graduate and due to freshman ineligibility, Duncan had to wait one and a half years to play, joining Iowa in the spring of 1955 but seeing his first action as a sophomore in the fall of 1956. Duncan became very discouraged over being constantly berated by Iowa coach Forest Evashevski and being clobbered in practice by Cal Jones. "Time after time, I was going to quit and transfer to Iowa State," Duncan has said.
But Duncan managed to win the backup quarterback job in 1956, playing behind Ken Ploen. In a non-conference game against Oregon State, Duncan led Iowa to two fourth-quarter touchdowns and a 14-13 win after Ploen was injured. Iowa qualified for the Rose Bowl that season, and the opponent was again Oregon State. Duncan played the second quarter following a Ploen injury and led Iowa to a touchdown in Iowa's 35-19 win in the 1957 Rose Bowl.
Duncan was named the starter and led the team in passing in 1957. He battled snow and sleet to throw a touchdown pass for the only score in Iowa's win over Northwestern, and he missed the end of Iowa's tie with Michigan due to leg cramps. Duncan scored four touchdowns, two passing and two rushing, against Minnesota, and two touchdowns against Notre Dame, one passing and one on an interception return as a defensive back. Iowa went 7-1-1 on the season, and Duncan was named first team All-Big Ten.
As a senior in 1958, Duncan led Iowa to one of its best seasons ever. After a surprising early season tie against Air Force, Iowa won five straight Big Ten games, clinching the Big Ten title as early as it had ever been clinched before. Duncan was terrific, helping Iowa lead the Big Ten in passing and the nation in total offense. He led the nation in completion percentage and passing yardage. His greatest game may have been in Iowa's lone loss in 1958, when he set a Big Ten record with 23 completions in 33 tries for 249 yards in a 38-28 loss to Ohio State. Duncan led Iowa to another Big Ten title and a 38-12 victory in the 1959 Rose Bowl. His lone touchdown pass in the Rose Bowl broke the school record for touchdown passes in a season, which had been held by Nile Kinnick in 1939.
Upon being voted Iowa's MVP, Duncan said, "There's nobody that knows any better than I do that this was all made possible by you guys here and the coaching staff behind me. I mean it. Just to be a part of this ball club was all that I really ever wanted."