Statistics
Statistical Sampling
Year 12 · Year 13
- ✓By the end of this lesson students will be able to define and distinguish between a population and a sample.
- ✓By the end of this lesson students will be able to understand the advantages and disadvantages of sampling.
- ✓By the end of this lesson students will be able to describe and apply various sampling methods, including random, systematic, stratified, and quota sampling.
- ✓By the end of this lesson students will be able to identify potential sources of bias in sampling.
Key concepts
The entire group of individuals or items about which information is desired. It is the complete set of data from which a sample is taken.
A subset of the population that is selected for study. It is intended to be representative of the population.
A study that involves collecting data from every single member of the population.
A list of all the items or individuals in the population from which a sample is drawn. It is essential for random sampling methods.
A method where every member of the population has an equal chance of being selected for the sample. This helps to minimise bias and allows for statistical inference.
Selecting items from a population at regular intervals from an ordered list. The starting point is usually chosen randomly.
Dividing the population into distinct subgroups (strata) based on shared characteristics, then taking a random sample from each stratum in proportion to its size in the population. This ensures representation from all key subgroups.
A non-random method where interviewers are given a quota to fill for different categories (e.g., age, gender). They select individuals until their quota is met. It is often used in market research.
Selecting individuals who are available at the time of the study and fit the criteria. It is often easy and inexpensive but highly prone to bias as it is unlikely to be representative.
A systematic error in a sampling method that causes the sample to not be representative of the population, leading to inaccurate conclusions.
Key facts to remember
- 1A population is the entire group of interest; a sample is a subset of the population.
- 2A census collects data from every member of the population, providing complete information but often being costly and time-consuming.
- 3A sampling frame is a list of all members of the population, necessary for many random sampling methods.
- 4Random sampling methods (e.g., simple random, systematic, stratified) aim to give every member of the population an equal chance of selection, reducing bias and allowing for generalisation.
- 5Stratified sampling divides the population into distinct subgroups (strata) and samples proportionally from each, ensuring representation.
- 6Systematic sampling selects items at regular intervals from an ordered list, with a random starting point.
- 7Non-random sampling methods (e.g., quota, opportunity) are often quicker and cheaper but are prone to bias and may not be representative.
- 8Bias is a systematic error that causes a sample to not accurately reflect the population.
Worked examples
Example 1
A school wants to find out the average height of its Year 12 students. Identify the population, a possible sample, and explain how a census would be conducted.
Answer
Population: All Year 12 students in the school. Sample: A subset of Year 12 students, for example, 30 randomly selected Year 12 students. Census: Measuring the height of every single Year 12 student in the school.
A sample is usually taken when a census is impractical due to time, cost, or logistical constraints.
Example 2
A college has 1200 students: 700 male and 500 female. A sample of 60 students is required, stratified by gender. Calculate how many male and female students should be in the sample.
Answer
The sample should consist of 35 male students and 25 female students.
Stratified sampling ensures that each subgroup is represented in the sample in proportion to its size in the population, reducing sampling bias.
Example 3
A factory produces 2000 items per day. A quality control manager wants to inspect 50 items using systematic sampling. Describe how the manager would select the sample.
Answer
The sampling interval is 40. The manager would randomly select a number between 1 and 40 (e.g., 15). Then, they would select the 15th item, 55th item, 95th item, and so on, inspecting every 40th item until a sample of 50 items is obtained.
The initial random selection of the starting point is crucial for systematic sampling to be considered a random method.
Common mistakes
- ✗Confusing the population with a sample, or failing to clearly define them in context.
- ✗Not understanding that a sampling frame is a prerequisite for true random sampling methods.
- ✗Failing to use proportional allocation when performing stratified sampling, leading to an unrepresentative sample.
- ✗Not randomising the starting point in systematic sampling, which can introduce bias.
- ✗Assuming a sample is representative without considering the potential biases introduced by the chosen sampling method.
Exam tips
- ★Clearly define the population and sample in context for any given problem, using precise language.
- ★When asked to describe a sampling method, provide enough detail for someone to replicate it (e.g., 'use a random number generator', 'select every k-th item from an ordered list').
- ★Always justify the choice of sampling method by discussing its advantages and disadvantages in the specific context of the problem.
- ★Be prepared to discuss potential sources of bias for different sampling methods and suggest ways to mitigate them.
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