Screen Time Claim: Mean, Median and Sampling
A Year 11 statistics class surveys screen time on a school night for two groups. Group A is a random sample of 30 students from the year group; Group B is 30 students who volunteered from the digital technology club. The summaries are: Group A mean 3.2 hours, median 2.9 hours, range 6.1 hours, interquartile range 1.8 hours. Group B mean 4.1 hours, median 3.4 hours, range 9.8 hours, interquartile range 3.6 hours. Group B includes one student who reported 11.5 hours; the next highest value is 6.7 hours. A histogram of Group A is roughly symmetric. A histogram of Group B has a long right tail. The school newsletter writes: 'Technology club students spend 28% more time on screens than other students.' A parent challenges the claim, saying the median should be used because the mean is affected by the 11.5-hour value. A student replies that removing the outlier is the fairest fix.
1. Calculate the percentage differences in screen time using the means and using the medians. Which calculation supports the newsletter's 28% figure?
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Using means: Difference = 4.1 - 3.2 = 0.9 hours. Percentage difference = 0.9 / 3.2 x 100% = 28.125%, which rounds to about 28%.
Using medians: Difference = 3.4 - 2.9 = 0.5 hours. Percentage difference = 0.5 / 2.9 x 100% = 17.24%, which rounds to about 17%.
The newsletter's 28% figure comes from comparing the means. That is arithmetically correct but it is not the only reasonable comparison, because Group B is skewed by the 11.5-hour value.
2. Explain the effect of the 11.5-hour value on Group B's mean and median. Is removing it a fair fix?
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The 11.5-hour value pulls the mean upward because the mean uses the size of every value. The median is resistant to extreme values because it depends on position, not size.
Estimate the effect on the mean: Group B's total is 30 x 4.1 = 123 hours. Removing 11.5 gives 111.5 hours over 29 students, so the new mean is 111.5 / 29 = 3.84 hours, not 4.1. The mean is still above Group A's mean of 3.2, but the difference falls from 0.9 to about 0.64 hours, or about 20% of 3.2.
Removing a value is not automatically fair. If 11.5 hours is a genuine report, deleting it hides real variation. It should only be removed if there is evidence it is an error or not part of the population being studied. A better response is to report the median and the interquartile range, and to mention the outlier rather than silently delete it.
3. Recommend what the newsletter should report. Include the sampling issue in your answer.
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The newsletter should not make a definite claim that technology club students spend 28% more time on screens than other students.
For Group B, the median of 3.4 hours is a better 'typical' value than the mean because the distribution is right-skewed. For Group A, the mean and median are closer (3.2 and 2.9), and the distribution is roughly symmetric. A fair summary would say that in these samples the technology club group had a higher median screen time (3.4 h versus 2.9 h) and much greater spread (IQR 3.6 h versus 1.8 h), with one very high value of 11.5 h.
The sampling issue is more serious than the choice of average. Group A is a random sample of the year group, but Group B is a volunteer sample from one club. Club members may differ from other students in ways related to screen time, and volunteers may not represent all club members. The comparison therefore describes these two groups only; it cannot support a general percentage claim about all technology club students versus all other students.