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IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability — Topic 4.12
📋 Learning Intentions
- Identify the key features of a well-designed survey, including question types, sampling frame, and potential sources of bias.
- Distinguish between reliability and validity in the context of data collection instruments and statistical studies.
- Describe the main sampling methods — simple random, systematic, stratified, quota, and convenience — and evaluate their appropriateness for a given context.
- Evaluate the impact of bias, non-response, and poor question design on the conclusions drawn from a statistical study.
- Justify the choice of a sampling method or survey design in a given real-world context, with reference to validity and reliability.
- Comment on the ethical considerations and limitations of data collection methods, including privacy and informed consent.
Real-World Introduction
In 2016, several major polling organisations predicted a different outcome for the UK Brexit referendum than what actually occurred. The polls were not fabricated — they used real respondents. The problem was in how those respondents were selected and how the questions were framed. A survey is only as trustworthy as its design.
Every statistical conclusion — whether about consumer preferences, health outcomes, or election results — depends entirely on the quality of the data collected. A poorly worded question, a biased sample, or an unrepresentative sampling frame can invalidate an entire study, regardless of how sophisticated the subsequent analysis is.
🌍 Why This Matters Medical clinical trials, government census design, and market research all depend on valid and reliable data collection. The World Health Organization's disease surveillance systems, for example, use stratified sampling across regions to ensure that health data accurately represents diverse populations — a direct application of the principles in this topic.
Core Concepts & Definitions
Before any statistical analysis can begin, data must be collected. The way data is gathered determines whether conclusions drawn from it are meaningful. Two fundamental questions must be asked of any data collection process: does it measure what it claims to measure, and would it produce consistent results if repeated?
🔑 Key Definition — Validity A data collection instrument or study is valid if it accurately measures what it is intended to measure. A survey question asking "How often do you exercise?" when the study aims to measure cardiovascular fitness is measuring a proxy, not the target variable directly — this is a validity concern.
Validity is about accuracy of measurement. A bathroom scale that consistently reads 2 kg too high is not valid for measuring true body mass, even if it gives the same reading every time. In survey design, validity is threatened by ambiguous