Survey Wait Times: The Mean That Overstates
A community nonprofit ran a 240-person survey about wait times at a food pantry. The main field, response_time_seconds, is recorded automatically; 12 responses are blank (5%). The distribution is right-skewed: median 38 seconds, mean 57 seconds, and five values above 900 seconds that look like people who left the browser open. The survey link was posted on a lobby kiosk (180 responses) and social media (60 responses), so the sample is a convenience sample, not a random one. The program director wants a single 'average wait' for a funder report, suggests filling the 12 blanks with the mean so the sample stays at 240, and asks for a 95% confidence interval. You have Python's statistics and math modules, no imputation library, and 30 minutes before the report is due.
1. Recommend which summary statistics to report for response_time_seconds and explain why the mean is risky here.
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Report the median and interquartile range as the primary summary, with the sample size and missing count. The mean is risky because the distribution is right-skewed: a few very large values pull it upward, so the mean of 57 seconds describes almost no typical respondent when the median is 38 seconds. A mean computed with the five values above 900 seconds will overstate the typical wait. I would also report a trimmed mean or the mean with and without flagged extremes as a sensitivity check, but not as the headline. Use statistics.median and statistics.quantiles for the IQR; report n=228 complete cases and 12 blanks. This choice matches the shape of the data and the funder's practical question: what wait does a typical visitor experience?
2. Decide how to treat the 12 blanks and five extreme values. Give a defensible rule and show its effect on the headline number.
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I would not mean-impute the 12 blanks: mean imputation shrinks variance and pulls missing values toward the center, which hides the uncertainty the director is asking about. I would report complete-case statistics (n=228) and disclose the 5% missingness. For the five values above 900 seconds, I would flag them as probable idle browser tabs, not silently delete them. Primary analysis: report the median and IQR with the flagged values included, because the median is robust; sensitivity analysis: recompute after excluding the flagged values and after capping them at 900 seconds. If the median stays near 38 seconds and the mean changes materially, the headline should be the median. The effect to show the director is a small table: median with all values, median without flagged values, mean with all values, mean without flagged values. That makes the treatment transparent instead of hidden inside one number.
3. Write the one-sentence finding for the funder and one caveat, using the numbers you would report.
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Finding: 'Based on 228 complete responses, the median reported wait was about 38 seconds (IQR to be computed), while the mean of 57 seconds was pulled upward by five values above 900 seconds that likely reflect idle browser tabs.' Caveat: 'This is a convenience sample from a lobby kiosk and social media, so the 95% confidence interval should not be interpreted as a random-sample margin of error; the 12 blank responses also mean the true typical wait could differ.' The sentence uses the robust statistic and names the distortion instead of hiding it, and the caveat prevents the funder from overgeneralizing the interval.