What the survey can carry
Keeping track of which group a percentage is a percentage of, four times running.
Notes and data
A residents' association sent a questionnaire about a proposed one-way system to all 1,200 households on its list.
320 households returned it.
Of those who returned it, 62 percent ticked "in favour".
The association's newsletter reports: "A clear majority of residents support the scheme."
Answers in order, explanations withheld, and a report on where the chain broke.
- 40.1Data interpretationFoundation
What proportion of the households sent a questionnaire returned it?
- AAbout 27 percent
- BAbout 62 percent
- CAbout 38 percent
- DAbout 16 percent
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About 27 percent
320 out of 1,200 is 0.267, so about 27 percent. The 62 percent belongs to a different group entirely — those who replied — which is the confusion the rest of this set is built on.
- 40.2Data interpretationStandard
Builds on: Question 1 established that 320, not 1,200, is the group the 62 percent describes.
How many households ticked "in favour"?
- A744
- B198
- C320
- D62
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198
62 percent of the 320 who replied is 198 households. Taking 62 percent of all 1,200 gives 744, which counts the opinions of 880 households that did not answer.
- 40.3Data interpretationChallenge
Builds on: The 198 from question 2 and the 1,200 from question 1. Neither figure alone gives this, and the last choice is wrong precisely because question 2 produced a count that can be divided.
What proportion of all 1,200 households on the list is known to be in favour?
- A62 percent
- BAbout 27 percent
- CAbout 17 percent
- DIt cannot be estimated at all.
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About 17 percent
198 out of 1,200 is 0.165, about 17 percent. That is what is known; the remaining 83 percent either said no or said nothing, and the two are not distinguishable from these figures.
- 40.4Command of evidenceChallenge
Builds on: All three earlier answers. The claim is not wrong about any number in the survey — it is wrong about which group the number describes, which only becomes visible once both denominators have been computed.
Which assessment of the newsletter's sentence is best supported?
- AIt is accurate: 62 percent is a clear majority.
- BIt is unsupported, because a majority of those who replied is not a majority of residents.
- CIt is unsupported, because the arithmetic behind the 62 percent is wrong.
- DIt is accurate, because non-responses can be assumed to follow the same split.
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It is unsupported, because a majority of those who replied is not a majority of residents.
The 62 percent is arithmetically fine and describes 320 households. "Residents" names all 1,200, and only about 17 percent of them are known to be in favour. The last choice states the assumption that would rescue the sentence, and a 27 percent response rate is exactly the situation in which that assumption is least safe.