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Statistical Sampling
Statistical Sampling
This lesson covers the key concepts of statistical sampling as required by the A-Level Mathematics specification. Sampling is the process of selecting a subset of individuals from a population in order to make inferences about the whole. Understanding different sampling methods and their implications is essential for statistical analysis.
Populations and Samples
A population is the entire set of items or individuals that are of interest in a statistical investigation. A census collects data from every member of the population, while a sample collects data from a subset.
| Term | Definition |
|---|---|
| Population | The whole set of items that are of interest |
| Sample | A subset of the population selected for study |
| Census | A survey that collects data from every member of the population |
| Sampling frame | A list of all members of the population from which a sample can be drawn |
| Sampling unit | Each individual member of the population that can be sampled |
Advantages and Disadvantages
| Method | Advantages | Disadvantages |
|---|---|---|
| Census | Completely accurate, no bias | Time-consuming, expensive, impractical for large populations |
| Sample | Cheaper, quicker, feasible for large populations | May not be representative, subject to sampling error |
Exam Tip: When asked to compare a census and a sample, always give at least one advantage and one disadvantage of each. Explain why a sample is more practical for large populations.
Random Sampling Methods
Simple Random Sampling
Every member of the population has an equal chance of being selected. This is achieved by assigning each member a number and using a random number generator or lottery method.
Steps:
- Obtain a complete sampling frame (numbered list of all population members).
- Use a random number generator to select the required number of items.
- Select the corresponding members from the sampling frame.
Advantages: Free from bias, easy to implement with a suitable sampling frame. Disadvantages: Requires a complete sampling frame, may not be practical for very large populations.
Systematic Sampling
Select every (k)-th item from the sampling frame, where (k = \frac{\text{population size}}{\text{sample size}}).
Steps:
- Calculate the sampling interval (k).
- Choose a random starting point between 1 and (k).
- Select every (k)-th item from the starting point.
Example: For a population of 500 and sample size of 50, (k = \frac{500}{50} = 10). If the random start is 7, select the 7th, 17th, 27th, ... members.
Advantages: Simple to use, evenly spread across the population. Disadvantages: Can introduce bias if there is a periodic pattern in the data.
Stratified Sampling
The population is divided into distinct strata (groups), and a proportional random sample is taken from each stratum.
Formula for the number from each stratum:
[ \text{Number from stratum} = \frac{\text{stratum size}}{\text{population size}} \times \text{total sample size} ]
Example: A school has 600 students in Year 12 and 400 in Year 13. For a sample of 50:
- Year 12: (\frac{600}{1000} \times 50 = 30) students
- Year 13: (\frac{400}{1000} \times 50 = 20) students
Advantages: Guarantees proportional representation of each group. Disadvantages: Requires knowledge of the population structure, the strata must be clearly defined.
Non-Random Sampling Methods
Quota Sampling
The interviewer selects a specified number of individuals from each group, but the choice of individuals is not random.
Advantages: Quick and cheap, no sampling frame required. Disadvantages: Prone to bias as the interviewer chooses who to include.
Opportunity (Convenience) Sampling
The sample is taken from those who are available at the time of the study.
Advantages: Easy to carry out. Disadvantages: Unlikely to be representative, highly prone to bias.
Summary
- A population is the entire group of interest; a sample is a subset selected for study.
- Random sampling methods (simple random, systematic, stratified) reduce bias and allow valid inferences.
- Non-random methods (quota, opportunity) are easier but more prone to bias.
- Stratified sampling ensures proportional representation of subgroups.
- Always consider the sampling frame, practical constraints, and potential sources of bias when choosing a sampling method.
Exam Tip: Be prepared to calculate the number of items to sample from each stratum in stratified sampling. Show your working clearly — you will often need to round your answer and explain any adjustments.