Clinic Wait Times: Mean, Median, and Sampling Scope
A community health clinic advertises that patients 'typically wait less than 30 minutes.' To check the claim, a volunteer records the wait times, in minutes, of 20 consecutive patients on a Monday morning: 12, 15, 18, 20, 22, 24, 25, 26, 28, 30, 31, 33, 35, 36, 38, 40, 42, 45, 48, 95. The 95-minute case involved a patient who arrived without an appointment during a staff shortage. The clinic sees about 140 patients per week, with Mondays accounting for 30% of visits. The volunteer plans to present the mean as evidence that the clinic meets its claim. The clinic manager argues that the median is more appropriate because the data are skewed. A patient advocacy group wants to know the probability that a randomly selected patient waits more than 30 minutes. The volunteer also wonders whether the Monday sample can support a claim about the whole week.
1. Compute the mean and median. Explain how the 95-minute observation affects each measure, and state whether either supports the advertisement.
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Sum = 663; n = 20; mean = 663/20 = 33.15 minutes. Sorted data place the median between the 10th and 11th values: (30 + 31)/2 = 30.5 minutes. The 95-minute observation contributes a deviation of 95 − 33.15 = 61.85 minutes and pulls the mean upward. Without it, sum = 568, n = 19, mean = 29.89; the median of the remaining 19 values is the 10th value, 30. So the outlier raises the mean by about 3.26 minutes but barely changes the median. Neither the mean nor the median is below 30, so the advertisement's 'less than 30 minutes' claim is not supported by this sample, regardless of which measure is used.
2. Estimate the probability that a randomly selected patient waits more than 30 minutes. What is the main limitation of that estimate?
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Count values strictly greater than 30: 31, 33, 35, 36, 38, 40, 42, 45, 48, 95 = 10 of 20, so the empirical probability is 10/20 = 0.50, or 50%. If the clinic means '30 minutes or more,' include the value 30, giving 11/20 = 55%. With only 20 observations, uncertainty is large: a 95% confidence interval for a proportion near 0.5 with n = 20 is roughly 0.27 to 0.73. The data are therefore consistent with anything from about a quarter to three-quarters of patients waiting over 30 minutes. The volunteer should report the count and sample size, not a single percentage presented as precise.
3. Can the Monday-morning sample support a claim about the clinic's whole week? Recommend a better sampling plan.
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No. The sample is 20 consecutive Monday-morning patients, while Monday is only 30% of weekly visits; weekend and evening patients are absent, and consecutive sampling may miss patients who left without being seen. A whole-week claim needs a random or stratified sample across days and times, with a defined denominator and a consistent definition of wait time. The 95-minute case should be flagged but not automatically deleted; it is real data and should be analyzed with a sensitivity check. The manager is right that the median is more resistant to skew, but using the median alone does not rescue the claim or fix the sampling problem.