Block Householder QR algorithms can be expected to perform better than the standard counterpart on modern computer architectures. The WY representation is one of the successful examples. As far as we know, however, the low-degree polynomial of its loss of orthogonality never seems to have been rigorously studied. This paper analyzes the distance between the exact orthogonal matrix and its computed approximation. In particular, we use the recently developed probabilistic approach to obtain an upper bound proportional to  \(\sqrt{m}nu\) , where m and n denote the dimensions and u denotes the unit roundoff. Numerical experiments confirm the theoretical results.