The Survey With a Missing Group
A class surveys 200 students about favorite after-school activity: Sports, Arts, or Clubs. The results: Sports 90, Arts 70, Clubs 30, and 10 students did not respond. A student builds a circle graph using only the 190 respondents and reports "Sports is 47%." The teacher's rubric expects percentages of the whole surveyed group (200), and the principal will read the report. A second issue: the 10 non-respondents are not random - they are mostly students who leave early for jobs. The teacher wants students to reason about the choice of denominator and about how missing data affects claims, not just recompute percentages.
1. Compute Sports' share using both denominators (190 and 200), and explain which is appropriate for the principal's report and why.
Read the worked answer
Using respondents only (190): 90/190 ≈ 47.4%. Using the whole surveyed group (200): 90/200 = 45.0%.
For a report to the principal about "favorite activity among students surveyed," the denominator should match the claim. If the claim is "of the 200 students surveyed, 45% chose Sports," then 200 is correct, and the 10 non-respondents are honestly represented as "no response." Using 190 silently drops the non-respondents and inflates every category's share. The circle graph must either include a "no response" slice (10/200 = 5%) or explicitly state the denominator is respondents only.
The key judgment: the denominator is part of the claim, not a computational detail. Changing 190 to 200 changes Sports from 47% to 45%, which is a meaningful difference in a report and must be disclosed.
2. Explain how the non-random nature of the 10 non-respondents could bias conclusions, and describe one way the class could report the data more honestly.
Read the worked answer
Because the 10 non-respondents are mostly students who leave early for jobs, they are not a random subset - they differ systematically from respondents (likely different activity patterns, possibly more likely to work or have less time for clubs). Treating them as if they simply "didn't matter" can bias the estimate of the favorite activity for the whole student body. The 45% figure is only valid as a description of the 200 surveyed if the missingness is unrelated to the variable; here it plausibly is related.
An honest report would: (1) state the response rate (190/200 = 95%) and the reason for non-response; (2) present results as "among the 190 who responded" rather than implying they represent all 200; (3) avoid generalizing to the whole school without noting the bias. A stronger design would follow up with the 10 students or offer an alternate collection method (e.g., a short form) so their voices are included. The lesson is that a percentage is only as trustworthy as its denominator and its missing-data mechanism.