Pooled-mean compatibility under rounding ($\overline{y}={\sum }_g(n_g/N){\overline{y}}_g$)

For a partition of a set of complete sample observations into groups of known sizes $n_g$ and total $N$, its sample mean is the weighted mean of the group sample means. Indeed, dividing ${\sum }_g{\sum }_{i\in g}{y}_i={\sum }_g{n}_g{\overline{y}}_g$ by $N$ proves the identity. If the printed group means $m_g$ each have absolute rounding error at most ${\delta }_g$, their compatible pooled mean lies inThis follows because all the weights are nonnegative. Equivalently, the pooled absolute rounding error is at most ${\sum }_g(n_g/N){\delta }_g$, by the triangle inequality. A printed overall mean must have a rounding interval intersecting this interval. Disjoint intervals prove incompatibility, rather than merely a difference of displayed decimals. This is a necessary consistency check for means of the same complete sample; it cannot silently equate a full-sample mean with a different complete-case mean or establish a unique repair to inconsistent output.