  The average distance from a node to all other nodes in a graph, or from a
query point in a metric space to a set of points, is a fundamental quantity in
data analysis. The inverse of the average distance, known as the (classic)
closeness centrality of a node, is a popular importance measure in the study of
social networks. We develop novel structural insights on the sparsifiability of
the distance relation via weighted sampling. Based on that, we present highly
practical algorithms with strong statistical guarantees for fundamental
problems. We show that the average distance (and hence the centrality) for all
nodes in a graph can be estimated using $O(\epsilon^{-2})$ single-source
distance computations. For a set $V$ of $n$ points in a metric space, we show
that after preprocessing which uses $O(n)$ distance computations we can compute
a weighted sample $S\subset V$ of size $O(\epsilon^{-2})$ such that the average
distance from any query point $v$ to $V$ can be estimated from the distances
from $v$ to $S$. Finally, we show that for a set of points $V$ in a metric
space, we can estimate the average pairwise distance using $O(n+\epsilon^{-2})$
distance computations. The estimate is based on a weighted sample of
$O(\epsilon^{-2})$ pairs of points, which is computed using $O(n)$ distance
computations. Our estimates are unbiased with normalized mean square error
(NRMSE) of at most $\epsilon$. Increasing the sample size by a $O(\log n)$
factor ensures that the probability that the relative error exceeds $\epsilon$
is polynomially small.
