A composition of a positive integer n is a representation of n as an ordered sum of positive integers a₁ + a₂ + · · · + a_m = n. There are 2n−1 unrestricted compositions of n, which can be classified according to the number of inversions they contain. An inversion in a composition is a pair of summands {a_i, a_j} for which i < j and a_i > a_j. We consider compositions of n counted according to whether they contain an even number or an odd number of inversions, and include results on compositions where the first part occurs with odd multiplicity. The following statistics are also computed for compositions of n: total number of inversions, total sum of inversion sizes, and number of inversions of size d or more.