The mode is the most critical measure of central tendency here, as it indicates the single most frequently purchased shoe size.

Ordering inventory based solely on the mean shoe size (e.g., size $9.34$) is impractical because shoe sizes are discrete, and the retailer cannot order size $9.34$.

The median shoe size is useful for dividing the customer base in half, but it does not necessarily represent the size that sells the most.

The retailer should only order shoes in sizes that are strictly greater than the mean.