  Consider an incoming sequence of vectors, all belonging to an unknown
subspace $\operatorname{S}$, and each with many missing entries. In order to
estimate $\operatorname{S}$, it is common to partition the data into blocks and
iteratively update the estimate of $\operatorname{S}$ with each new incoming
measurement block.
  In this paper, we investigate a rather basic question: Is it possible to
identify $\operatorname{S}$ by averaging the column span of the partially
observed incoming measurement blocks on the Grassmannian?
  We show that in general the span of the incoming blocks is in fact a biased
estimator of $\operatorname{S}$ when data suffers from erasures, and we find an
upper bound for this bias. We reach this conclusion by examining the defining
optimization program for the Fr\'{e}chet expectation on the Grassmannian, and
with the aid of a sharp perturbation bound and standard large deviation
results.
