Exercise 1: Why the Mean Alone Can Mislead, and Which Statistic Fits Revenue Projection — Possible Solution ==================================================================== WHAT THE CATERING-ORDER EXAMPLE SHOWS, PER THIS CHAPTER ------------------------------ Per this chapter, "that single order pulls the mean order size noticeably upward, even though most individual orders are much smaller — the mean is sensitive to extreme values because every value, including the largest, contributes to the sum. The median order size barely moves at all, because it only cares about which value sits in the middle position, not how large the largest one is." WHY REPORTING ONLY THE MEAN COULD MISLEAD ------------------------------ If a report states only "the average order size is $X" using the mean, and X is inflated by one unusually large catering order, a reader would reasonably conclude that a "typical" customer order is around that size — which, per this chapter, isn't true at all; most individual orders are considerably smaller. The mean, taken alone and without context, describes a number that no longer represents what a typical transaction actually looks like once even one extreme value is present in the data, which is exactly the misleading impression this chapter warns against. WHY THE MEDIAN AVOIDS THIS SPECIFIC DISTORTION ------------------------------ Because the median only depends on which value happens to occupy the middle position once the data is sorted, one unusually large order doesn't pull it upward at all — it would need enough large orders to actually shift which value sits in the middle before the median moved meaningfully. This is precisely why the median, not the mean, is the statistic that actually answers "what does a typical order look like." WHICH STATISTIC FITS REVENUE PROJECTION ------------------------------ Per this chapter, "a mean is the right choice for projecting total expected revenue." Projecting total revenue requires knowing the actual sum of everything sold, including the large catering order's own real contribution — deliberately excluding or down-weighting that order (which is effectively what the median does, since it ignores how large the extreme value actually is) would produce a projection that understates real expected revenue. The mean's own sensitivity to every value's magnitude, the same property that makes it a poor "typical order" descriptor, is exactly the property that makes it the correct choice when total magnitude is what actually matters. WHY THIS WORKS AS AN ANSWER ------------------------------ It explains precisely why the mean alone misrepresents a "typical" order using the chapter's own catering-order reasoning, and identifies the mean (not the median) as correct specifically for revenue projection, using the same underlying property (sensitivity to magnitude) to explain both the median's advantage in one context and the mean's advantage in the other.