Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given finite elements X1,⋯,Xn∈M, there is a finite decomposition of the identity into integer N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXiEj=τ(Xi)Ej for all j=1,⋯,N and i=1,⋯,n. Equivalently, there is a unitary operator U∈M such that 1N∑j=0N−1U⁎jXiUj=τ(Xi)I for i=1,⋯,n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors. As the first application, we show that all elements of trace zero in a type II1 factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [6]. As the second application, we prove that any self-adjoint element in a type II1 factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if (M,τ) is a finite factor, X∈M, then there exists a normal operator N∈M and a nilpotent operator K such that X=N+K.