One Big Catering Day: Mean, Median, and a Loan Application
A coffee shop records its daily sales revenue, in dollars, for 10 consecutive weekdays: 120, 135, 128, 142, 130, 300, 125, 138, 133, and 129. The $300 day was a one-time catering order for a local office, not a regular retail day. The owner is applying for a small-business loan and wants to report a single 'average daily sales' figure. A friend recommends the mean because it is higher; the shop's accountant recommends the median. The owner also has weekend sales that typically run about 15% higher than weekday sales but is tempted to leave them out because 'they make the numbers look too good.' The loan officer has asked for a representative figure and a brief explanation of any unusual days. The owner is unsure which average to use.
1. Compute the mean and median of the 10 weekday sales figures. Which measure better represents a typical weekday, and why?
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Sum = 120 + 135 + 128 + 142 + 130 + 300 + 125 + 138 + 133 + 129 = 1480, so mean = 1480/10 = $148. Sorted: 120, 125, 128, 129, 130, 133, 135, 138, 142, 300. The median is the average of the 5th and 6th values: (130 + 133)/2 = $131.50. Nine of the ten days are at or below $142, and the single $300 catering day adds about $30 to the mean relative to the other nine days: without it, the mean is 1180/9 ≈ $131.11. The median, $131.50, better represents a typical retail weekday because it is resistant to the one-time outlier; the mean is inflated by an atypical event.
2. The owner is tempted to report only the mean of $148 on the loan application. Evaluate this choice statistically and ethically, and suggest a more informative way to report the figure.
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The mean is a correct calculation, but reporting it alone as 'average daily sales' overstates typical sustainable revenue: 9 of 10 days were below $142, and the mean is driven by a one-time catering order. A loan officer relying on $148 could overestimate routine cash flow. Ethically, omitting a known unusual event that materially changes the interpretation is misleading even if no number is false. A better report gives both the median (about $131.50) and the mean ($148), identifies the $300 catering day, and possibly gives the range or a five-number summary. The owner could also separate regular retail sales from catering revenue.
3. Suppose the owner adds 8 weekend days whose sales are each 15% higher than the weekday mean of $148. Compute the combined mean for the 18 days, and explain how the choice of days affects the reported 'average daily sales.'
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A weekend day would average 148 × 1.15 = 170.2 dollars. Combined total = 10(148) + 8(170.2) = 1480 + 1361.6 = 2841.6 over 18 days, so the combined mean is 2841.6/18 ≈ $157.87. Including higher weekend days raises the overall mean from $148 to about $158, while a weekday-only figure describes a different business pattern. For a loan application, the owner should state the period and mix of days (weekday-only, weekend-only, or all days) and use the median alongside the mean so the lender can judge typical and peak revenue. A 7-day average is not inherently more honest; it is just a different definition.