Mean bounds after deleting bounded observations (${\bar{y}}_{\mathrm{remaining}}\in [(N\bar{y}-db)/(N-d),(N\bar{y}-da)/(N-d)]$)

Suppose $N>d\ge 0$ observations lie in $[a,b]$ and have sample mean $\bar{y}$. Delete any $d$ observations. The remaining sample mean obeysIndeed, 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.