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.

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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.

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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:

1. Obtain a complete sampling frame (numbered list of all population members).
2. Use a random number generator to select the required number of items.
3. 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 $k$k-th item from the sampling frame, where $k = \frac{\text{population size}}{\text{sample size}}$k=sample sizepopulation size​.

Steps:

1. Calculate the sampling interval $k$k.
2. Choose a random starting point between 1 and $k$k.
3. Select every $k$k-th item from the starting point.

Example: For a population of 500 and sample size of 50, $k = \frac{500}{50} = 10$k=50500​=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:

$\text{Number from stratum} = \frac{\text{stratum size}}{\text{population size}} \times \text{total sample size}$Number from stratum=population sizestratum size​×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$1000600​×50=30 students
• Year 13: $\frac{400}{1000} \times 50 = 20$1000400​×50=20 students

Advantages: Guarantees proportional representation of each group. Disadvantages: Requires knowledge of the population structure, the strata must be clearly defined.

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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.

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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.