  A $t$-digest is a compact data structure that allows estimates of quantiles
which increased accuracy near $q = 0$ or $q=1$. This is done by clustering
samples from $\mathbb R$ subject to a constraint that the number of points
associated with any particular centroid is constrained so that the so-called
$k$-size of the centroid is always $\le 1$. The $k$-size is defined using a
scale function that maps quantile $q$ to index $k$. This paper provides bounds
on the sizes of $t$-digests created using any of four known scale functions.
