We propose a new way of quantifying a team's strength of schedule for NCAA basketball. This strength of a schedule is defined as the number of games a team on the borderline of the annual national tournament would expect to win if they played that schedule. This gives a direct way of quantifying how well different teams have done relative to the schedules they have played. Our motivation for constructing this strength of schedule is to help inform the choice of teams given bids to the national tournament: teams who have won more games than their strength of schedule have strongest evidence that they deserve such a bid. Estimating the strength of schedules is possible through fitting a simple statistical model to the results of all regular season matches. We are able to quantify the uncertainty in these estimates, which helps differentiate between teams with clear evidence for selection and those on the borderline. We apply our method to data from the 2007/08 and 2008/09 season. Our results suggest that St. Mary's warranted a bid to the 2009 tournament, at the expense of Wisconsin; and that both Arizona State and Nebraska warranted bids to the 2008 tournament instead of South Alabama and St. Joseph's.