Reporting an "Average" Wait Time With an Outlier Present
A walk-in clinic posts a sign reading "Our average wait is under 21 minutes." To check the claim, a reviewer records the wait times of nine consecutive patients on one Tuesday: 12, 15, 15, 18, 20, 22, 25, 30, and 68 minutes. The 68-minute patient arrived as a walk-in during a staff lunch break. The clinic manager defends the sign by pointing out that "the middle number is 20." The reviewer argues the mean is 25 and the sign is false. Both people are working from the same nine numbers and neither has made an arithmetic error.
1. Compute the mean, median, mode, and range of the nine waits.
Read the worked answer
Sum: 12 + 15 + 15 + 18 + 20 + 22 + 25 + 30 + 68 = 225.
Mean: 225 ÷ 9 = 25 minutes.
Median: the data are already ordered and n = 9, so the median is the 5th value = 20 minutes.
Mode: 15 appears twice, more than any other value → 15 minutes.
Range: 68 − 12 = 56 minutes.
The mean sits 5 minutes above the median because one value (68) lies 43 minutes beyond the next-highest value (30). That single observation drags the mean upward while leaving the median untouched.
2. Which measure best represents a typical wait here, and is the sign defensible?
Read the worked answer
The median (20 minutes) is the better description of a typical wait, because 8 of the 9 patients waited 30 minutes or less and the 68-minute case was a documented one-off staffing gap. The mean is distorted by that single outlier.
The sign is technically defensible only if it specifies which average it means. As written, "average" most often reads as the mean, and the mean is 25 - over the stated threshold. A sign saying "median wait: 20 minutes" would be accurate and honest; a bare "average under 21 minutes" invites the reader to assume the mean and is misleading. The judgement call is not which number is right, but whether the label matches the number being quoted.
3. Suppose the 68 is discovered to be a data-entry error and the true value is 18. Recompute the mean and state whether the sign becomes accurate - and whether correcting the number is legitimate.
Read the worked answer
New sum: 225 − 68 + 18 = 175. New mean: 175 ÷ 9 ≈ 19.4 minutes.
Now both the mean (19.4) and the median (18, since the ordered list is 12, 15, 15, 18, 18, 20, 22, 25, 30) fall below 21, so "average wait under 21 minutes" is true under either reading.
The correction is legitimate only if there is independent evidence - a timestamp log, the patient's record - that 68 was a transcription error. Deleting or altering a value because it makes the statistic inconvenient is fabrication, not cleaning. The right sequence is: verify the source record first, correct it if wrong, and report the corrected figure with a note that it was amended.