Choosing Mean or Median for a Skewed Sibling Count
Ms. Rivera's class surveyed 24 students about the number of siblings they have. Results: 0 siblings, 6 students; 1 sibling, 8 students; 2 siblings, 7 students; 3 siblings, 2 students; 4 siblings, 1 student. The class calculates a mean of about 1.33, a median of 1, and a mode of 1. During discussion, one student says, 'The mean says the typical student has 1.33 siblings, which is impossible.' Another says, 'We should use the median because it is smaller.' A third says, 'A graph could make the difference between 1 sibling and 2 siblings look huge.' Ms. Rivera wants students to choose summary measures based on the shape of the distribution and the question being asked, not on which number is easiest.
1. Verify the mean, median, and mode from the raw counts. Then explain which measure best summarizes the typical number of siblings and why.
Read the worked answer
Total siblings = 0(6) + 1(8) + 2(7) + 3(2) + 4(1) = 0 + 8 + 14 + 6 + 4 = 32. Mean = 32/24 ≈ 1.33. With 24 values, the median is the average of the 12th and 13th values in order. The six 0s occupy positions 1–6; the eight 1s occupy positions 7–14, so both the 12th and 13th values are 1, and the median is 1. The mode is also 1, the most frequent value. The mean is not a claim that any student has 1.33 siblings; it is the fair share if the 32 siblings were redistributed equally. Because the distribution is right-skewed by the one student with 4 siblings, the mean is pulled above the median. The median is the better measure of a typical student's sibling count, while the mean is useful for questions about total siblings per student or projected totals. The median is not better because it is smaller; it is better because it resists the pull of the extreme value.
2. Critique the third student's claim about a misleading graph. Describe a specific graphing choice that could create that impression and how to redesign it.
Read the worked answer
A truncated vertical axis is the clearest way to exaggerate the 1-versus-2 difference. If a bar graph starts the y-axis at 6 instead of 0, the bar for 7 students (2 siblings) may look only slightly taller than the bar for 8 students (1 sibling), depending on scaling; more strikingly, starting at 7 can make the 8 bar look enormous relative to the 7 bar. The redesign should start the y-axis at 0, use equal intervals, and label counts clearly. A dot plot with one dot per student avoids the issue because each dot has the same value. Students should also report the actual counts and percentages: 8/24 ≈ 33% have 1 sibling, 7/24 ≈ 29% have 2, and only 3/24 = 12.5% have 3 or 4. A graph can emphasize a real difference, but it should not change the ratio between bar heights.