Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 4: Statistics and Probability | Topic 4.1
📋 Learning Intentions
- Define the terms population, sample, census, parameter, and statistic in the context of a statistical investigation.
- Identify and describe the five main sampling techniques: simple random, systematic, stratified, quota, and convenience sampling.
- Evaluate the reliability and validity of a sampling method, identifying potential sources of bias.
- Identify outliers in a data set using the IQR fence method and interpret their effect on summary statistics.
- Distinguish between discrete and continuous data, and between qualitative and quantitative data.
- Justify the choice of sampling method for a given real-world context, considering practical constraints.
Real-World Introduction
Before every national election, polling organisations survey thousands of voters to predict the outcome. They cannot ask every citizen — that would take months and cost millions. Instead, they select a carefully chosen sample and use it to estimate what the entire population thinks. The accuracy of their prediction depends almost entirely on how that sample was chosen.
In 1936, the Literary Digest magazine polled over 2.4 million Americans and predicted that Alf Landon would defeat Franklin Roosevelt in a landslide. Roosevelt won by one of the largest margins in US history. The Digest had surveyed its own subscribers, car owners, and telephone users — all wealthier than average — producing a sample that was catastrophically unrepresentative of the voting population. The magazine folded within a year. This is the real cost of poor sampling.
🌍 Why This Matters Every statistical conclusion in medicine, economics, environmental science, and social research rests on a sample. Understanding how samples are chosen — and where they can go wrong — is the foundation of all applied statistics. IB examiners test this directly: students must name, describe, and critically evaluate sampling methods in context.
Core Concepts & Definitions
Every statistical investigation begins with a question about a group of interest. Before collecting any data, a researcher must be precise about exactly which group they are studying and what characteristic they want to measure. These decisions determine the entire structure of the investigation.
📖 Key Definitions — Population, Census, and Sample
- A population is the complete set of individuals or items that share a characteristic of interest to the investigator.
- A census collects data from every member of the population.
- A sample is a subset of the population selected for data collection.
- A parameter is a numerical value describing a characteristic of the population (e.g. population mean \( \mu \), population proportion \( p \)).
- A statistic is a numerical value calculated from sample data used to estimate a parameter (e.g. sample mean \( \bar{x} \), sample proportion \( \hat{p} \)).
A census is the gold standard — it leaves no uncertainty about the population. However, censuses are expensive, time-consuming, and sometimes destructive (testing every light bulb to failure leaves none to sell). A