What "Average Wait" Should Mean on the Clinic Sign
A walk-in clinic posts a sign reading "Average wait: 25 minutes." Last Tuesday's recorded waits for seven patients were 12, 15, 14, 13, 16, 90, and 15 minutes. The 90-minute entry is disputed: the triage nurse says that patient checked in at 4:40 p.m., left the waiting room for 70 minutes to pick up a child, and returned before being seen. The clinic manager wants one number for the lobby sign that will not mislead a typical patient, and a second number for the regional quality report, which is meant to capture the total waiting-room burden the clinic carries.
1. Compute the mean and median of the seven recorded values, then compute each again with the 90-minute entry removed.
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Order the data first: 12, 13, 14, 15, 15, 16, 90 → n = 7.
With all seven:
- Sum = 12+13+14+15+15+16+90 = 175
- Mean = 175 ÷ 7 = 25 minutes (this is the number on the sign)
- Median = 4th value = 15 minutes
Without the 90:
- Sum = 175 − 90 = 85; n = 6
- Mean = 85 ÷ 6 = 14.17 ≈ 14.2 minutes
- Median = average of 3rd and 4th values (14 and 15) = 14.5 minutes
What the arithmetic shows: removing one of seven values moves the mean by nearly 11 minutes and the median by half a minute. The posted 25 is not a rounding artifact - it is one data point doing most of the work.
2. Which number goes on the lobby sign and which goes in the quality report? Justify both with the purpose of each.
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Lobby sign → median, 15 minutes (14.5 if the 90 is excluded).
The sign's job is to set a realistic expectation for the next patient. The median is the middle of the actual distribution, so half the patients waited 15 minutes or less; the median barely moves when one extreme value appears. A sign reading 25 tells a patient to expect a wait that six of seven patients did not experience.
Quality report → mean, but only after deciding what the measure means. If the metric is "total waiting-room burden the clinic carries," the clinic does absorb all recorded minutes, and the mean (25, or 14.2 without the 90) is the right statistic because it is built from the total.
The catch that drives the choice: the 90 is probably not waiting-room time. The patient left for 70 minutes. If the metric is patient wait, exclude or correct the value and report 14.2. If the metric is occupancy burden, keep it. Decide the definition, then pick the statistic - not the reverse.
3. The manager proposes the target "average wait under 20 minutes." Using this data, show why the clinic can claim it is meeting the target or failing it depending on the statistic chosen.
Read the worked answer
Same data, opposite verdicts:
- Mean of all seven: 25 > 20 → failing the target.
- Mean without the disputed 90: 14.2 < 20 → meeting it.
- Median of all seven: 15 < 20 → meeting it.
- Proportion waiting 20 minutes or more: 1 of 7 = 14.3%; proportion waiting 16 minutes or less: 85.7%.
The judgment: "average" is undefined until someone names the statistic, and this dataset is engineered to expose that. One 90-minute entry - itself probably a recording error - flips the conclusion.
Recommendation: the target is unusable as written. Rewrite it as a defined metric, e.g., "median wait ≤ 20 minutes" plus a separate threshold such as "≤ 10% of patients wait longer than 30 minutes." That pair cannot be gamed by one outlier, and it forces the clinic to state whether the 70-minute absence counts as waiting. Note that any definition change must be applied to prior months too, or the trend line becomes meaningless.