Café Sales Week: Choosing a Representative Value
A café records cups sold each day for one week:
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Cups | 210 | 185 | 240 | 260 | 390 | 480 | 355 |
The owner is planning next week's staffing and says, "A typical day is about 300 cups, so three workers who each handle 80 cups will cover any day." The staff is scheduled identically Monday through Friday, with extra help added only on weekends. Nothing about the café's hours or menu is changing next week, but the owner expects Saturday sales to rise 15% because of a nearby event, while Sunday sales are expected to fall 10% afterward.
1. Compute the mean and the median for the week, then evaluate the owner's claim that "a typical day is about 300 cups."
Read the worked answer
Total: 210 + 185 + 240 + 260 + 390 + 480 + 355 = 2,120 cups.
Mean: 2,120 ÷ 7 ≈ 302.9 cups.
Median: ordered values are 185, 210, 240, 260, 355, 390, 480 → the 4th value is 260 cups.
The two summaries disagree by about 43 cups. The mean sits near 300 only because the weekend days (390, 480, 355) pull it upward; on five of seven days the café sells fewer than 300 cups, and half the days are at or below 260. So "about 300" is a fair description of the week's average but a poor description of a typical day, and it would overstate weekday demand by roughly 40–60 cups. For staffing a weekday, the five weekday values (210, 185, 240, 260, 390) give a median of 240 and a mean of 257.
2. The owner plans weekday staffing for three workers at 80 cups each. Using the daily data, is that plan adequate? Support your answer with the specific days that fail.
Read the worked answer
Three workers × 80 cups = 240 cups of capacity.
Compare each weekday:
- Mon 210 ✔ (30 spare)
- Tue 185 ✔ (55 spare)
- Wed 240 ✔ exactly at capacity, no slack for a rush
- Thu 260 ✘ short by 20
- Fri 390 ✘ short by 150
Two of five weekdays exceed capacity, and Wednesday has zero margin. Friday needs 390 ÷ 80 = 4.875 → 5 workers; Thursday needs 3.25 → 4 workers. A plan built on the 240-cup median would fail on the two highest-volume weekdays, which are precisely the days when a shortfall is most visible. The plan should be set from the upper weekdays, not the middle of the distribution.
3. Apply the expected changes (Saturday +15%, Sunday −10%) and report the new weekly total and the percent change from the original week.
Read the worked answer
Saturday: 480 × 1.15 = 552 (an increase of 72).
Sunday: 355 × 0.90 = 319.5 (a decrease of 35.5).
New weekly total: 2,120 − 480 − 355 + 552 + 319.5 = 2,120 − 835 + 871.5 = 2,156.5 cups.
Change: 2,156.5 − 2,120 = 36.5 cups → 36.5 ÷ 2,120 ≈ 0.0172, about +1.7%.
The judgement point is that a 15% gain on the largest day and a 10% loss on a mid-size day nearly cancel: the net effect is small even though both individual swings are large. The new day counts (Sat 552, Sun 319.5) also shift the busiest day further out, so weekend staffing should be revisited even though the weekly total barely moved.