Surveys and statistical investigations

A school wants to understand travel time for all 120 Year 9 students. It measures 12 selected students. Who is the population?

Use the sampling lab. Inspect the observed records, change the run, compare five runs, then try the morning-only method. The data are invented for this investigation

What is the main limitation of morning-only sampling?

Start with a question and two variables

Ask: how does travel time vary with the usual travel method for this school’s Year 9 students this term? Define who, what and when before collecting data

1. 1Categorical variable Travel method names groups: walk, bus, car or another specified method. These are nominal categories, with no natural ranking
2. 2Numerical variable Travel time in minutes is a measurement. Number of journeys is a discrete count; satisfaction categories such as poor/fair/good are ordinal
3. 3Population and census Name the target population. A census attempts to record every unit; missing responses can still limit a planned census

A travel-mode code uses 1 = walk, 2 = bus and 3 = car. What type of variable is it?

Which survey item best measures a single opinion without leading the respondent?

Sampling and response are different stages

A complete sampling frame is a list that covers the target population. Random selection can reduce selection bias, but it does not guarantee a representative realised sample

1. 1Simple random sample Select a fixed-size sample uniformly from the full list, without replacing a selected unit. Every possible subset of that size is equally likely
2. 2Stratified sample Split into relevant groups and select randomly within each. For proportionate allocation, multiply the planned sample size by the group’s share of the population
3. 3Response and measurement Record who was selected and who responded, use consistent units and wording, and check possible nonresponse or measurement bias

Two full-list random samples of the same size give different means. What follows?

A morning-only sample increases from 8 to 24 visits. Does this fix the omitted afternoon visits?

Recorded minutes

\(\displaystyle 4,6,8,10,12,20\)

A random sample has travel times 4, 6, 8, 10, 12 and 20 minutes. Its total is 60 minutes. What is its mean?

Recorded minutes

\(\displaystyle 4,6,8,10,12,20\)

The same ordered sample is 4, 6, 8, 10, 12, 20 minutes. What is its median?

A random sample of six Year 9 students at one school this term has a mean travel time of 10 minutes and a median of 9 minutes. How should these sample measures be reported for all Year 9 students at that school this term?

Population groups

\(\displaystyle 60,20\)

Planned sample

\(\displaystyle n=16\)

A population has 60 morning and 20 afternoon visits. For a proportionate stratified sample of 16, how many morning visits should be selected?

Everyone is invited, but only people with strong opinions tend to reply. Which limitation remains?

You want to compare how many sampled students use each travel method. Which display and reason fit?

You want to show the shape of many recorded travel-time measurements. Which display is suitable?

Compare the same counts with two vertical baselines. The data stay at 12 and 15; only the presentation changes

What happens when the baseline changes to 10?

The stacked count chart shows morning: 12 bus and 8 walk; afternoon: 20 bus and 20 walk. How many afternoon responses were recorded?

The percentage chart has equal-height bars. Can you conclude that both sessions had the same number of responses?

Which question does this line chart help answer?

In week 3 the stacked area has a top boundary of 27 loans, and fiction contributes 12. How many are nonfiction loans?

Judge the representation and audience

A display can show correct numbers while encouraging an exaggerated conclusion. Check the question it answers, its units, denominator, scale, grouping and omitted information

1. 1Infographics If icon height and width both double, its area becomes four times as large. Do not use that image to imply only a doubling without explaining the scale
2. 2Fair comparisons Use common units and sensible shared scales. Percentages compare shares; counts describe numbers. Do not hide sample sizes or select only convenient periods
3. 3Accessibility Use readable labels and legends. A data table can supplement a graph, especially where precise values matter

A report states that bus users have a higher average travel time. It was an observational survey. Which conclusion is warranted?

Trace claims back to their source

For a report from digital media or elsewhere, find its original data and methods before accepting a population claim

1. 1Source checklist Who collected the data, when, for what population and purpose? What were the sample size, frame, selection method, response rate and question wording?
2. 2Variables and summaries Identify at least one categorical and one numerical variable. Check units and whether average means mean or median
3. 3Reasonable inference Link each claim to the relevant data, display and statistic. Distinguish unsupported from demonstrably contradicted. Record what remains unknown

A polished online chart gives no source, dates or sampling method. What should you do?

Conduct your own investigation

Use an everyday question with a numerical measure and a category, for example task-completion time under two clearly described conditions. Use anonymous observations and obtain appropriate classroom permission. Do not collect names, health details or other sensitive information

1. 1Plan before collecting Write the question, target population, period, variables, units, sampling frame and method. Explain why the method can address the question
2. 2Collect and document Record actual observations, how each unit was selected and any missing responses. Alternatively choose a genuine secondary data source and record its title, dates, definitions, coverage and provenance. Do not present invented values as collected data
3. 3Keep the evidence Keep an anonymous row per observation. Check units, impossible values and duplicates. Retain valid unusual observations; correct documented errors rather than deleting inconvenient values

Analyse and communicate

Choose and justify suitable displays. Compare numerical distributions and categories, calculate relevant summaries, and relate them to the original question

1. 1Analysis Use mean, median and range with shape/outliers where relevant. For group comparisons, use like measures, units and scales; report group sample sizes
2. 2Conclusion State the observed evidence and a conditional population estimate where justified. Discuss sampling variation, coverage, nonresponse, measurement and other plausible explanations
3. 3Evaluate Explain what the evidence can and cannot support, how a better investigation could reduce limitations, and why a larger sample alone is not a cure for bias

Which report is the strongest?

Submit and review your investigation

Finish your own survey or secondary-data investigation. Ask a teacher or peer to review it using the checklist below

1. 1Evidence retained Question, population, dated source or collection log, anonymous raw data, variables and units, sampling/responding counts and method
2. 2Reasoning visible Justified displays, correct summaries, contextual comparison, uncertainty and evidence-strength discussion
3. 3Evaluation explicit Explain possible biases and one justified refinement. Separate supported findings from assumptions, unsupported claims and causal speculation