The median (6 days). Your bonus penalty is |guess − actual| — absolute error, not squared
error, and not all-or-nothing. That’s exactly the loss the median minimizes.
Why not the mean? Squared error punishes being very far off disproportionately more than being a little off — a miss of 10 days costs 100 “squared-error units,” while a miss of 2 days costs only 4, a 25x difference for a 5x distance. To avoid ever being very wrong, minimizing squared error pulls your guess toward wherever the long tail is, even if the tail is rare — because a rare huge miss is catastrophically expensive under squaring. Your actual penalty (linear, absolute) has no such fear of being very wrong — 10 days off costs exactly 10x what 1 day off costs, no premium for extremity. Reporting the mean (8 days) would be optimizing for a penalty structure you don’t actually have; it overweights the long tail of nasty bugs relative to what you’re really being scored on.
Why not the mode? The mode (4 days) is the single most likely outcome, but “most likely” isn’t the same question as “minimizes expected distance.” Under 0/1 scoring — you only win if you guess the exact right number — betting everything on the single most probable outcome is optimal. But your bonus doesn’t require exactness; being off by 1 day only costs a little, so it pays to hedge toward the middle of the distribution’s mass rather than betting everything on its single peak.
The general rule, restated: a point estimate isn’t a neutral “best guess” — it’s the answer to “which single number minimizes my expected penalty, under the specific way I get penalized?” Change the penalty, and the “correct” point estimate changes with it, even though the underlying posterior never moved. Mean, median, and mode agree only when the distribution is symmetric (no skew, no long tail) — which is exactly why this distinction rarely bites you with simple bell-curve intuitions and regularly bites you with real-world skewed data: costs, wait times, response times, damages.
Where this goes: next lesson turns to intervals instead of single numbers — and a related confusion: a Bayesian credible interval and a classical confidence interval look almost identical on a page, but claim very different things.