From a question to defensible evidence
Name who and what you want to understand, collect compatible data, and limit conclusions to what the method supports
- Sampling variability Different random samples can produce different estimates. More observations usually reduce variability, but not every larger sample is closer
- Coverage and selection A complete list matters. A random sample from an incomplete list is still missing excluded groups
- Stratification Sample within relevant groups. Proportional allocation uses their population sizes; equal numbers are not always appropriate
- Question and variables For these library visits this week, how does waiting time vary by session? Record session and waiting time in minutes for the same visit
- Method and source For primary data, define a consistent observation procedure. For secondary data, check who collected it, when, definitions, coverage and licence or access conditions
- Ethics and privacy Use an approved collection process, explain the purpose and voluntary participation, minimise identifying data and protect stored records. Do not publish identifiable student responses
Worked example
A target population has 60 morning and 180 afternoon visits. Plan a proportional random sample of 40 visits. Later, 28 of 40 invited participants respond in a separate survey. Interpret the response rate
- \(\displaystyle 40\times\frac{60}{240}=10\)Select 10 morning visits at random from the morning list
- \(\displaystyle 40\times\frac{180}{240}=30\)Select 30 afternoon visits at random from the afternoon list
- \(\displaystyle \frac{28}{40}\times100=70\%\)The response rate is 70%. Nonrespondents may differ; random invitations alone do not remove nonresponse bias
Common mistakes
Difficulty levels
Questions about this topic
Does random sampling guarantee a representative sample?+
No. It provides a defensible selection mechanism, but realised samples vary and response or measurement problems can remain
Will increasing sample size remove bias?+
Not by itself. A larger sample from an incomplete frame still excludes missing groups, and leading wording still influences answers
Is 24 a required statistical sample size?+
No. Some practice questions state 24 as that project’s minimum response requirement. Precision and adequacy depend on the purpose, method and variability
Is a census always error-free?+
No. An attempted census may have missing responses, and even complete coverage can contain measurement or recording errors
When is equal stratified allocation proportional?+
When the strata have equal population sizes. Otherwise proportional allocation follows each stratum’s share of the population
Are all axis breaks misleading?+
No. A clearly marked break may be appropriate, but an unmarked jump can distort interpretation. Bar lengths used to compare amounts normally need a zero baseline
Can a column graph show measured quantities?+
Yes, for example group means or totals. A histogram instead displays frequencies across numerical intervals of individual observations
What evidence completes an investigation?+
A clear question and plan, suitable paired data, documented collection or source, relevant analysis, an evidence-based conclusion and limitations. Completing these lesson questions alone does not assess that written work
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 5: Statistics