Comparing two minor-injury units: which average and which spread?
A hospital trust compares waiting times (in minutes) for 11 patients at each of two minor-injury units, sampled during the same week.
Unit A: 12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 95
Unit B: 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31
A manager writes in a report: 'The mean wait at Unit A is 24.7 minutes and at Unit B is 25.9 minutes, so patients wait longer at B.' A nurse at Unit A replies that the report hides the real story.
1. Calculate the median and interquartile range for each unit and explain what the manager's comparison of the means has overlooked.
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With 11 values the median is the 6th value and each quartile is the median of the five values in that half.
| Unit A | Unit B | |
|---|---|---|
| Median | 18 | 26 |
| Lower quartile | 15 | 23 |
| Upper quartile | 22 | 29 |
| IQR | 7 | 6 |
| Range | 83 | 11 |
What the means hide: Unit A's mean of 24.7 is dragged upwards by the single 95-minute wait. That one value adds 95 to the total; without it the other ten total 177, giving a mean of 17.7. The median of 18 shows the typical patient at A waits 8 minutes less than the typical patient at B, and 10 of the 11 patients at A waited 24 minutes or less. Comparing means alone therefore gives the opposite conclusion to the one the data supports for a typical patient.
2. A patient must be seen within 40 minutes or miss a connecting bus. Which unit should the patient choose, and which measures best support your advice?
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The answer depends on which risk the patient cares about, and the statistics point in different directions:
- Typical wait: Unit A is clearly better - median 18 versus 26 - and even A's upper quartile (22) is below B's median. The IQRs (7 and 6) show both units are internally consistent around their own centres.
- Worst case: Unit B is safer. Its range is only 11 minutes, so every patient in the sample was seen within 31 minutes and the 40-minute deadline would never have been missed. Unit A's range of 83 shows one patient waited 95 minutes, so about 1 in 11 (roughly 9%) risk missing the bus.
Advice: if the patient only wants to be seen quickly on average, choose A, justified by the median and IQR. If missing the bus is costly and a guarantee matters more than the typical wait, choose B, justified by the range. The median and IQR describe the typical case; the range describes the risk of an extreme wait.
3. The manager proposes deleting the 95-minute value because it 'is not typical'. Discuss whether this is acceptable and state the effect on any statistics quoted.
Read the worked answer
A value should only be removed if there is a reason to believe it is not a genuine observation - for example it was a recording error, a duplicate entry, or the patient was not actually waiting (e.g. they left and returned). Being 'unusual' is not on its own a justification: a genuine 95-minute wait is exactly the kind of case a waiting-time report exists to reveal, and deleting it would flatter the unit.
The correct process is: check the original record, report the investigation, and if the value is genuine keep it and quote statistics that are resistant to it - the median and IQR - alongside the mean and the total range, so the reader sees both the typical wait and the extreme.
Effect if it were removed: Unit A's mean would fall from 24.7 to 17.7 (177 ÷ 10) and the range from 83 to 12, making Unit A look better than Unit B on every measure. The median would change only slightly (from 18 to 17.5).