Choosing Mean vs Median for a Stakeholder Report
Session durations (minutes) for 12 users of a mobile app:
1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 45, 120
A product manager wants "the average session length" for a board slide.
- Compute the mean and the median. Which would you put on the slide, and how would you phrase it?
- The finance team separately asks for total minutes of engagement per 1,000 users to size a server bill. Which statistic do they need, and why is it a different answer?
- Two weeks later the 120-minute session is discovered to be a user who left the app open overnight. Recompute both statistics without it. What does the change tell you about each measure?
1. Mean vs median for "typical"
Sum = 1+1+2+2+3+3+4+5+6+8+45+120 = 200, so the mean is 200/12 \approx 16.7 minutes. Sorted, the 6th and 7th values are 3 and 4, so the median is 3.5 minutes.
The distribution is severely right-skewed: ten of twelve sessions are under 10 minutes and two are enormous. The mean of 16.7 describes nobody — no user had a session anywhere near it. For a "how long does a typical user stay?" slide, report the median (3.5 min) and, ideally, a couple of percentiles ("half of sessions are under 3.5 minutes; 90% are under about 45"). If a single number is demanded, say "median session is 3.5 minutes; the mean is 16.7, pulled up by a small number of very long sessions".
2. Totals need the mean
Total engagement = n \times \bar{x}. Per 1,000 users at this rate that is 1{,}000 \times 16.7 \approx 16{,}700 minutes. The median cannot be used here — 1{,}000 \times 3.5 = 3{,}500 would underestimate the load by almost 5×. The long sessions are real load on the servers even though they are atypical, so a capacity question is inherently a question about the sum, and the mean is the sum divided by n. The two teams asked genuinely different questions, so they get different statistics.
3. Removing the artefact
Without 120: sum = 80, n = 11, mean \approx 7.3; sorted middle value (6th of 11) is 3, so median = 3.
The mean fell from 16.7 to 7.3 — it more than halved because of one row. The median moved from 3.5 to 3.0. The mean has a breakdown point of zero (a single arbitrarily bad value can move it arbitrarily far); the median tolerates up to half the data being corrupted. That is exactly why the median belongs on the "typical user" slide: it is stable under the data-quality problems you have not found yet. It is also why the finance estimate should be caveated — the total is only as trustworthy as the tail, and the tail is where the errors live.
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