Consider the QR factorization of a matrix \(A=QR\) and suppose that the factorization of the perturbed matrix is \(A+\Delta A=(Q+\Delta Q)(R+\Delta R)\). The perturbation analysis bounds \(\Delta Q\) and \(\Delta R\) in terms of \(\Delta A\). By using better, i.e., more elementwise individualized bounds for \(\Delta A\), and more accurate estimates for intermediate bounds, better results can be obtained than what was previously available in the literature. The perturbations considered are of the form \(|\Delta A|\leq\varepsilon C|A|D\) with for any matrix \(M\), the notation \(|M|\) means elementwise absolute values. The matrix \(C\) has elements \(c_{ij}\) with \(0\leq c_{ij}\leq 1\), \(D\) is a diagonal matrix with positive elements and \(\varepsilon\) depends on the size of the matrix \(A\). This type of perturbation was also considered by \textit{H. Zha} [SIAM J. Matrix Anal. Appl. 14, 1124-1131 (1993; Zbl 0787.65014)]. Precise analysis of the problem shows for example that a column pivoting strategy will improve the bounds for the condition number of \(R\), and this also tends to improve the condition of \(Q\). Besides the sharp estimates, also practical estimates for the condition are proposed.