Data and Displays · Challenge problem

An analyst has a histogram of $200$ whole-number values with bins $0$–$9$, $10$–$19$, $\ldots$, $90$–$99$ and estimates the mean using the bin midpoints $4.5, 14.5, \ldots, 94.5$. Whatever the counts are, what is the largest possible difference between this estimate and the true mean?