Reading a Small Clinic's Waiting-Time Data
A walk-in clinic logs waiting times (in minutes) for 11 patients on a Monday morning: 8, 12, 12, 15, 18, 22, 25, 25, 25, 40, 68. The clinic manager reports the 'average wait' to the regional board as 24.5 minutes, computed as the mean. A patient group argues the typical experience is worse than that. The manager also notes that two patients waited over 35 minutes and wants to set a target that no more than 10% of patients exceed. The clinic sees roughly 55 patients per week, and the manager wonders whether the Monday sample is representative.
1. Compute the mean, median, and mode of the waiting times, and explain which measure best represents a typical patient.
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Sum = 8+12+12+15+18+22+25+25+25+40+68.
Group: 8+12=20; +12=32; +15=47; +18=65; +22=87; +25=112; +25=137; +25=162; +40=202; +68=270.
Mean = 270/11 = 24.545..., about 24.5 minutes, matching the manager.
Median: with 11 values the median is the 6th in order. Ordered: 8, 12, 12, 15, 18, 22, 25, 25, 25, 40, 68. The 6th value is 22 minutes.
Mode = 25 minutes (appears three times).
The mean is pulled upward by the 68-minute outlier: without it the mean would be 202/10 = 20.2 minutes. The median (22) and mode (25) both sit below the mean, so the patient group has a point: the mean overstates the typical wait, but the median still shows most patients waiting over 20 minutes. The median is the fairest single summary here because the data are skewed.
2. The manager wants a target such that no more than 10% of patients exceed it. Using the Monday data as a guide, what threshold is defensible, and what is the limitation?
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10% of 11 patients is 1.1 patients, so at most 1 patient may exceed the threshold. Sort descending: 68, 40, 25, ... The second-highest value is 40, so setting the target at 40 minutes means exactly 1 of 11 patients (the 68-minute case) exceeds it, which is 1/11 = 9.1%, within 10%.
Setting the target at 25 minutes would fail: 40 and 68 exceed it, i.e. 2/11 = 18.2%.
So a 40-minute threshold is the tightest that satisfies the rule on this sample.
Limitation: with only 11 observations, one patient is 9.1% of the sample, so the target is extremely sensitive to a single case. A single additional long wait would push the exceedance to 2/12 = 16.7%. The threshold should be confirmed on a larger sample before being adopted as a board target.
3. The clinic sees about 55 patients per week. If the Monday distribution held, estimate how many patients per week would exceed 25 minutes, and assess whether the 11-patient sample is adequate.
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Count above 25 minutes in the sample: 40 and 68, so 2 of 11 = 18.18%.
Expected per week = 0.1818 x 55 = 10.0 patients, so about 10 patients per week wait more than 25 minutes.
Adequacy: 11 patients is about 20% of a week's volume, and the sample is a single Monday morning, which may differ systematically from other days (Mondays often carry weekend backlog). The estimate of 18.18% has wide uncertainty: with 2 of 11, a rough 95% interval spans roughly 5% to 40%, which at 55 patients per week is anywhere from about 3 to 22 patients.
The judgement call is that the Monday figure is a reasonable planning signal but not a basis for a firm weekly commitment; the clinic should pool at least four weeks of data before quoting a number to the board.