= Mean bounds after deleting bounded observations
{title2=$\bar y_{\rm remaining}\in[(N\bar y-db)/(N-d),(N\bar y-da)/(N-d)]$}
= Deleted-observation mean bounds
{synonym}
Suppose $N>d\ge0$ observations lie in $[a,b]$ and have <sample mean> $\bar y$. Delete any $d$ observations. The remaining <sample mean> obeys
$$
\frac{N\bar y-db}{N-d}\le\bar y_{\mathrm{remaining}}\le\frac{N\bar y-da}{N-d}.
$$
Indeed, the removed sum is between $da$ and $db$; subtract it from the original sum $N\bar y$ and divide by the remaining positive count. If only a rounded mean in $[\ell,u]$ is known, a valid enclosing <interval> is $[(N\ell-db)/(N-d),(Nu-da)/(N-d)]$, optionally intersected with $[a,b]$. These deterministic bounds do not require <missing at random>. They constrain a complete-case summary without claiming that complete-case selection is unbiased for a <target population>.
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