Two days of waiting times: mean, median and how many nurses
A clinic records the waiting times in minutes for nine patients seen on Monday: 12, 15, 18, 20, 22, 25, 28, 31, 96. The 96-minute wait followed a single emergency that stopped the clinic for over an hour. On Tuesday, nine patients waited 24, 25, 26, 27, 28, 29, 30, 31 and 32 minutes.
On Wednesday the clinic expects 45 patients across a 4-hour session. Arrivals are uneven: 40% of patients arrive in the busiest hour. The clinic standard is one nurse for every 6 patients concurrently waiting, and the mean wait must stay at or below 30 minutes.
1. Calculate the mean and median waiting time for both days. Which measure better describes the typical patient's experience, and why?
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Monday. Sum = 12+15+18+20+22+25+28+31+96 = 267.
- mean = 267 ÷ 9 = 29.7 min
- median = 5th of 9 ordered values = 22 min
Tuesday. The values are symmetric around 28, so sum = 9 × 28 = 252.
- mean = 252 ÷ 9 = 28.0 min
- median = 28 min
Which is more representative. On Tuesday mean and median agree, so the day is consistent. On Monday the two diverge sharply: the mean (29.7) is dragged up by one 96-minute value, while eight of the nine patients waited 31 minutes or less and the middle patient waited only 22 minutes. For describing the typical patient, the median is the better summary on Monday because it resists a single extreme observation; the mean describes the total burden on the clinic (nurse-hours, complaints) rather than the typical experience.
The two days are almost identical by mean (29.7 vs 28.0) yet quite different in character - which is the point of looking at both measures.
2. A manager says Monday is compliant because the mean of 29.7 is under 30. Evaluate that claim, and assess whether removing the 96-minute outlier is legitimate.
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The claim is technically true but substantively wrong. The standard is met by 0.3 minutes, i.e. a margin of about 1%, and that margin depends entirely on one data point. With a mean this close to the threshold, the measure gives no protection: a single emergency can absorb almost the whole allowance, so the clinic can be "compliant" while one patient waits over an hour and a half.
Effect of removing the outlier. Recomputing without the 96:
- sum = 267 − 96 = 171; mean = 171 ÷ 8 = 21.4 min
- median of the eight remaining values = (20 + 22) ÷ 2 = 21.0 min
Is removal legitimate? It depends on the question being asked.
- For understanding routine flow, temporarily excluding a documented one-off interruption is defensible, provided the exclusion is stated and the event is analysed separately.
- For accountability, service standards and patient complaints, it is not defensible - the patient who waited 96 minutes experienced exactly the harm the standard exists to prevent.
The better fix is to stop relying on a mean at all: report the median and the 90th percentile, so that both the typical patient and the tail are visible, and count separately any waits above a hard ceiling.
3. Using the Wednesday forecast, calculate the minimum number of nurses needed in the busiest hour, and say what happens if waits run at 45 minutes instead of 30.
Read the worked answer
Patients in the busiest hour. 40% of 45 = 18 patients.
Average number waiting at any instant. A patient is in the queue for the mean waiting time, so the average number concurrently waiting follows from Little's law:
- number waiting ≈ arrivals per hour × wait in hours
- = 18 × (30 ÷ 60) = 9 patients
Nurses. One nurse per 6 waiting: 9 ÷ 6 = 1.5, and staff cannot be fractional, so 2 nurses minimum in the busiest hour.
If waits stretch to 45 minutes.
- 18 × (45 ÷ 60) = 13.5 patients waiting on average
- 13.5 ÷ 6 = 2.25 → 3 nurses
Judgement. Staffing exactly at the calculated minimum leaves no buffer: because arrivals are random rather than evenly spaced, the average is the middle of a distribution, and half the time the queue will be above it. Two nurses is the arithmetic floor; three gives the resilience to absorb a normal spike and to prevent the 30-minute mean from drifting into a 45-minute reality. The calculation should be used to argue for the buffer, not to justify the floor.