The paper studies controllability of an aggregate demand response system, i.e., the amount of the change of the total electric consumption in response to the change of the electric price, for real-time pricing (RTP). In order to quantify the controllability, this paper defines the controllability index as the lowest occurrence probability of the total electric consumption when the best possible the electric price is chosen. Then the paper formulates the problem which finds the consumer group maximizing the controllability index. The controllability problem becomes hard to solve as the number of consumers increases. To give a solution of the controllability problem, the article approximates the controllability index by the generalized central limit theorem. Using the approximated controllability index, the controllability problem can be reduced to a problem for solving nonlinear equations. Since the number of variables of the equations is independent of the number of consumers, an approximate solution of the controllability problem is obtained by numerically solving the equations.