  Let $K/k$ be an extension of number fields. We describe theoretical results
and computational methods for calculating the obstruction to the Hasse norm
principle for $K/k$ and the defect of weak approximation for the norm one torus
$R^1_{K/k} \mathbb{G}_m$. We apply our techniques to give explicit and
computable formulae for the obstruction to the Hasse norm principle and the
defect of weak approximation when the normal closure of $K/k$ has symmetric or
alternating Galois group.
