Choosing a Typical Value from Skewed Practice Data
Ms. Lam's Grade 6 class is comparing weekly math-practice minutes on a learning app. Fifteen students reported these weekly totals: 30, 40, 40, 45, 50, 50, 50, 60, 60, 70, 75, 80, 90, 120, and 240. The app's promotional page says, 'Our average student practices more than 70 minutes per week,' and shows a smiling student. The student council will give a prize to the class with the highest typical weekly practice. One student says the class should report the mean because it is the largest measure. Another says the 240-minute value should be deleted because it makes the class look uneven. The class must choose one measure to describe a typical student and write one accurate sentence for the school newsletter. They also want to respond to the app company's claim without misleading readers.
1. Calculate the mean, median, and mode of the 15 weekly practice times. Show enough work that another student could check your answer.
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Mean: add all 15 values. 30 + 40 + 40 + 45 + 50 + 50 + 50 + 60 + 60 + 70 + 75 + 80 + 90 + 120 + 240 = 1,100. Mean = 1,100 ÷ 15 = 73.33..., or about 73.3 minutes. Median: with 15 values in order, the median is the 8th value. Counting in, the 8th value is 60 minutes. Mode: 50 minutes appears three times, more than any other value, so the mode is 50 minutes. The three measures tell different stories: the mean is pulled upward by the 240, the median is 60, and the most common value is 50. The class should report more than one measure if it wants to be transparent.
2. Which measure best represents a typical student in this class? Justify your choice using the distribution, including the effect of the 240-minute value.
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The median, 60 minutes, best represents a typical student in this class. The 240-minute value is far above the rest: the next highest is 120, and 14 of the 15 values are at or below 120. That single value pulls the mean from what most students did up to about 73.3. The median is resistant to this outlier because it depends on position, not size; half the class practiced 60 minutes or less and half practiced 60 minutes or more. The mode, 50 minutes, is also typical, but it ignores the spread and the students who practiced more. A fair newsletter sentence would say the median was 60 minutes and note the mean was about 73, rather than calling either one 'the average' without explanation.
3. Evaluate the app company's claim 'Our average student practices more than 70 minutes per week.' Is it mathematically supported? What wording would make it fairer?
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The claim 'Our average student practices more than 70 minutes per week' is mathematically possible only if 'average' means the mean of this 15-student sample: the mean is about 73.3. But the median is 60, so a typical student did not practice more than 70 minutes. The claim is also unsupported as a statement about all app users because the data come from one class and we are not told how the 15 students were selected. The 240-minute report may be accurate or may be an error. A fairer statement would be: 'In one Grade 6 class of 15 students, the mean weekly practice time was about 73 minutes, but the median was 60 minutes; one student reported 240 minutes.' That wording gives the sample, the measure, and the outlier.