The SPA will hold the new Police Service of Scotland and its chief constable to account from April 1.
All 12 members have been appointed for four years and will take up posts this month, with the SPA having the option of appointing another two in due course.
Its initial tasks will be to consult on a strategic plan for how Scotland will be policed, and agree how the public money spent on policing is allocated.
According to the Scottish Government, all appointments were made on merit, with political activity playing no part in the selection process. However, all appointees' political activity had to be made public.
The members include Morag McLaughlin, head of both the High Court Unit at the Crown Office and Policy for the Crown, as well as area procurator-fiscal for Grampian and Lothian and Borders; David Hume, the recently retired chief executive of Borders Council; former Strathclyde Police Authority chairman and Glasgow City Council treasurer Paul Rooney; his counterpart at Lothian and Borders Police Board Iain Whyte; and Jeane Freeman, a board member of the Scottish Police Services Authority and a member of the Judicial Appointments Board for Scotland.