The statistical investigation process and questionnaire design: HSC Maths Standard 1 Year 12
“S3.1 The statistical investigation process for a survey: identify the steps of a statistical investigation, design an effective questionnaire free from bias, and consider sampling, non-response and possible misinterpretation of questions and statistics”
A statistical investigation moves from question to plan, collection, analysis, interpretation and communication. Effective questionnaires use clear, unbiased, well-structured questions, are piloted first, and are given to a representative sample. Consider non-response and misinterpretation, and be alert to misleading statistics.
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What this dot point is asking
NESA's focus for this subtopic is the purpose and process of statistical investigation, including basic design principles, the complex nature of questionnaire design, and misconceptions in statistical representations and reasoning.
The answer
The statistical investigation process
- Pose a question about a problem.
- Plan how to collect data (population, sample, method, questions).
- Collect the data.
- Organise and analyse it (tables, graphs, summary statistics).
- Interpret the results and draw conclusions.
- Communicate findings, including limitations.
Designing an effective questionnaire
- Clear, simple language, one idea per question.
- Free from bias: no leading or loaded words.
- Appropriate question types: closed questions (tick boxes, rating scales) are easy to analyse; open questions give richer detail.
- Complete, non-overlapping answer options (for example 0 to 2, 3 to 5, 6 or more).
- Logical order and a reasonable length.
- Privacy: avoid unnecessary personal questions and explain how data will be used.
- Pilot it first and revise.
Sampling and responses
- A sample should represent the population. Random, systematic and stratified samples reduce bias; convenience and self-selected samples often increase it.
- Non-response and unexpected responses can distort results if non-responders differ from responders.
- Different people may interpret the same question differently, so test wording in a pilot.
Misleading statistics
Watch for small samples, unrepresentative samples, missing sample sizes, percentages without totals, misleading graphs (truncated axes), and confusing correlation with causation.
A school wants to know whether students would use a bike rack.
- Poor design: "Don't you think a bike rack is a great idea?" asked only of students in the cycling club.
- Problems: leading question and a biased sample.
- Improved: "If a secure bike rack was installed, how often would you ride to school? (Never, Sometimes, Most days, Every day)", asked of a random sample of 10 students from each year group.
- Pilot: tested with five students; one said "Sometimes" was unclear, so it was changed to "1 to 2 days a week".
Double-barrelled questions ("Is the canteen cheap and healthy?").
Overlapping options ("0 to 5", "5 to 10").
Generalising beyond the sample surveyed.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksIdentify the problem with each question and rewrite it: (a) 'Don't you agree the canteen food is too expensive?' (b) 'How often do you exercise?'Show worked solution →
(a) Leading (biased): it pushes people to agree. Rewrite: "How would you rate the price of canteen food? (Very cheap, Cheap, About right, Expensive, Very expensive)".
(b) Vague: "often" is unclear. Rewrite: "On how many days in the last week did you exercise for at least 30 minutes? (0, 1 to 2, 3 to 4, 5 or more)".
Marking guide: 1 mark per problem identified, 1 mark for improved rewrites.
core4 marksA council wants residents' views on a new skate park. Plan the investigation using the statistical investigation process.Show worked solution →
- Question: do residents support building a skate park at Central Oval?
- Plan: a short questionnaire with closed questions (support level, how often household members would use it, concerns); a stratified random sample of households from each suburb.
- Collect: online and paper surveys; pilot first with 10 residents.
- Analyse: percentages supporting, compared by suburb and age group, displayed in column graphs.
- Interpret: identify overall support and concerns (noise, parking), noting response rates.
- Communicate: a report to council with graphs and recommendations.
Marking guide: 1 mark per two correctly applied steps (up to 3), 1 mark for addressing sampling or bias.
exam5 marksA gym surveys its members on its website and reports that 95% of people are happy with the gym. Evaluate this claim.Show worked solution →
- Sampling
- only members who visit the website and choose to respond are included (a self-selected sample). Unhappy members who have stopped visiting may not respond, so the sample is not representative.
- Population
- the claim says "people", but only members were surveyed, and not all of them.
- Question design
- the wording is unknown; a leading question ("How much do you love our new equipment?") would inflate positive responses.
- Missing information
- the number of respondents and the response rate are not given; 95% of 20 respondents means little.
- Judgement
- the claim is not reliable. A random sample of all members, neutral questions and reporting the sample size would give a more trustworthy result.
Marking guide: 1 mark each for sampling, population, question design and missing information, 1 mark for a judgement.