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The 26-year-old Scot became the first Briton in 77 years to win the Wimbledon men's singles title with victory over Novak Djokovic in the summer.
British and Irish Lions player Leigh Halfpenny was runner-up at the ceremony in Leeds, with jockey AP McCoy third.
BBC Sports Personality of the Year 2015 is broadcast live from Belfast on Sunday, 20 December from 19:00 GMT on BBC One. Further coverage on BBC Sport's online platforms and Radio 5 live.
Wimbledon champion Andy Murray is named the 2013 BBC Sports Personality of the Year.