Mean, Median, and an Extreme Value in Screen-Time Data
A high school statistics class collects daily recreational screen-time data (minutes) from 15 students: 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 480. One value, 480, came from a student who included a day when the family watched a long movie together. The class is asked, 'What is a typical daily screen time for this group?' Student A reports the mean, about 105 minutes, and says it uses all the data. Student B reports the median, 80 minutes, and says it is more typical because the 480 is an outlier. Student C says the outlier should be deleted because it is not a school day, and then the mean becomes about 78 minutes. The teacher must decide how to frame the discussion and whether Student C's deletion is defensible. The class has studied mean, median, and outliers but not formal outlier tests.
1. Student A's mean is about 105 minutes, and Student B's median is 80 minutes. Which is the better summary of 'typical' for this data set, and what does each measure emphasize?
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With the 480 included, the mean is 1580 ÷ 15 ≈ 105.3 minutes and the median is the 8th ordered value, 80 minutes. For 'typical daily screen time,' the median is the better single summary because 14 of the 15 values are at or below 120 minutes; the mean is pulled upward by the 480. The mean still has a legitimate meaning: it is the total screen time divided equally among the 15 students, so it answers a different question about the group's total. If the question is 'What is a representative day?' the median is more resistant to the extreme value. The teacher should make the purpose of the summary explicit rather than declaring one measure universally correct.
2. Student C wants to delete the 480 because it is not a school day. Under what conditions would deleting it be defensible, and when would it be inappropriate? Propose a better classroom move.
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Deleting the 480 is defensible only if the variable was defined in advance as school-day recreational screen time and the 480 observation was collected under a different definition, making it a measurement or protocol error. Here the class collected 'daily' data, and the 480 is a real reported value from a day that included a long movie. Removing it because it is inconvenient would bias the summary and teach students to discard valid data. A better move is to keep the full data set, report both mean and median, and discuss why they differ. If students want a school-day estimate, they should define that subset before analyzing it and report both analyses transparently.
3. The class has not studied formal outlier tests. Design a short activity using the 1.5×IQR rule that helps students decide whether 480 is unusual, and interpret the result.
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Order the 15 values: 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 480. The median is 80. Q1 is the median of the lower seven values, 60; Q3 is the median of the upper seven values, 100. IQR = 100 - 60 = 40. Upper fence = 100 + 1.5(40) = 160; lower fence = 60 - 1.5(40) = 0. Since 480 > 160, it is flagged as a potential outlier; no value is below 0. Students can mark the fences on a box plot. The correct interpretation is 'unusual relative to the spread,' not 'automatically delete.' The next question should be contextual: is 480 a valid daily total, a data-entry error, or a different kind of day? That is the judgement the rule cannot make.