Sampling Methods Notes | A-Level AQA Maths

01Sampling Methods

Data Types

This section distinguishes qualitative and quantitative data, and discrete and continuous variables.

Classify what the values represent

Qualitative data describes a category or characteristic. Quantitative data records a numerical count or measurement.

Discrete and continuous are subdivisions of quantitative data.

Two kinds of quantitative data

Discrete data takes separate, countable values. For example, the number of bicycles owned by a household could be $0,1,2,\ldots$ .

Continuous data can take any value within an interval. For example, a bicycle journey's duration is measured rather than counted.

Age depends on its meaning

Age recorded in completed years is discrete: the possible recorded values are whole numbers. Elapsed time since birth is continuous, even when a measurement is rounded.

02Sampling Methods

Populations & Samples

This section defines populations, samples and sampling frames, and links sample statistics to population parameters.

Identify the group being studied

Population

The population is the complete collection of individuals or items relevant to the investigation.

A sample is the smaller group drawn from that population to provide data.

Sampling frame

A sampling frame lists the population's members so that they can be identified for selection.

For a study of a company's employees, its employee register could provide the frame.

A statistic estimates a parameter

A population parameter is a numerical characteristic of the whole population; its value is often unknown. A sample statistic is calculated from the observations in a sample and can be used to estimate that parameter.

03Sampling Methods

Census & Sampling

This section compares collecting data from an entire population with collecting data from a sample.

All members or a selected part?

A census gathers information from every population member. Sampling gathers information from only a selected part of the population.

Census

Advantage: The entire population is covered, so there is no uncertainty from sampling only some members.

Limitations: Gathering and analysing all the observations can take considerable time and money. Destructive testing consumes the whole population.

Sample

Advantages: Usually faster and cheaper, with fewer observations to analyse.

Limitations: The selected members may give an unrepresentative picture of the population. The way they are chosen may introduce bias.

04Sampling Methods

Simple Random Sampling

This section explains equal-probability selection, numbering a sampling frame and choosing distinct members at random.

Give every possible sample an equal chance

In a simple random sample of size $n$ , every possible set of $n$ population members is equally likely to be chosen.

Use a complete sampling frame and assign each member a unique number.
Select $n$ different numbers using a random number generator, or draw numbered slips in a lottery.
Collect data from the members with those numbers.

This avoids deliberately favouring particular members during selection. It is practical for a small population or sample, but requires every member to be identifiable in a list. It cannot be applied in this way when no complete frame is available.

05Sampling Methods

Systematic Sampling

This section uses a fixed sampling interval, a random starting position and selection from an ordered list.

Select at regular intervals

Systematic sampling selects members at evenly spaced positions in a list. For a population of size $N$ and a sample of size $n$ , the interval is

k=\frac{N}{n}.

When $k$ is an integer, choose a starting position randomly from $1$ to $k$ , then select every $k$ th member after it.

First three selections, with a ten-position interval.

The method is convenient for a long list or a sequence of produced items. It still needs an identifiable frame; the order of that frame matters. A randomly ordered list avoids favouring a particular pattern in the selection.

06Sampling Methods

Stratified Sampling

This section allocates a proportional sample to non-overlapping groups and selects members randomly within each group.

Preserve the population's group proportions

A stratified sample divides the population into distinct, non-overlapping groups called strata. A random sample is then selected from each stratum in proportion to its size.

The groups should reflect a relevant characteristic, such as age range. Each member belongs to one stratum. To allocate a sample of size $n$ from a population of size $N$ , calculate

n_h=\frac{n}{N}N_h.

Here $N_h$ is the number of population members in the stratum and $n_h$ is its sample allocation.

Worked example: A college has $300$ students in three tutor groups. Select a stratified sample of $50$ students.

Group size	Sample allocation
A: $150$	$\frac{50}{300}\times150=25$
B: $90$	$\frac{50}{300}\times90=15$
C: $60$	$\frac{50}{300}\times60=10$

The allocations total $25+15+10=50$ . Use each group's register to choose its allocated students randomly.

If proportional targets are not whole numbers, use whole-person allocations that still total the required sample size. For three equally sized strata and a sample of $10$ , the target for each is $10/3$ . Allocations of $4,3,3$ total $10$ ; rounding every target separately to $3$ would leave one person short.

This ensures representation of the chosen groups and is useful when those groups differ. It requires suitable, non-overlapping strata and a way to identify members for random selection.

07Sampling Methods

Quota & Opportunity Sampling

This section distinguishes filling group quotas from selecting readily available members, and compares both with random sampling.

Two methods without random selection

Quota sampling

Set target numbers for groups in the population, then select suitable members until each target is reached.

Selection does not have to be random. If someone declines, choose another suitable member. Quotas can reflect the population's group proportions without requiring a complete list of names.

Opportunity sampling

Also called convenience sampling, this takes eligible members who are readily available.

It is quick and can be used without a complete sampling frame. However, the people or items easiest to access may not reflect the population as a whole.

Worked example: A library surveys its current users about opening hours. Each user belongs to one group according to their usual visiting period. The interviewers have these response targets and have started collecting answers:

Usual visits	Target	Collected	Still needed
Morning	$6$	$3$	$3$
Afternoon	$8$	$6$	$2$
Evening	$4$	$2$	$2$

The targets total $18$ responses. They need another $3+2+2=7$ . They approach available users in the relevant groups, choosing someone else from that group if a person declines, until each target is met.

This is quota sampling: the group counts are controlled, but the individuals are not chosen randomly. Collecting the first $18$ eligible responses without group targets would instead be opportunity sampling. The method is determined by the selection procedure, not just by the final number of responses.

Quota and stratified sampling can have similar group allocations. The decisive difference is how individuals are selected: stratified sampling requires random selection within the groups; quota sampling does not.

08Sampling Methods

Bias & Reliability

This section critiques a sample's representativeness, explains variation between samples and considers improvements to selection and size.

Ask whether the sample reflects the population

Bias arises when the selection systematically favours some members over others, so the sample may give a distorted picture of the population.

Random selection helps reduce selection bias. Quota sampling can be affected by who agrees to take part; opportunity sampling can over-represent the members easiest to reach.

Size and selection both matter

A sample directly describes only the members observed. Using it to draw conclusions about the population introduces uncertainty: two samples can produce different statistics and lead to different conclusions.

Taking a larger sample generally makes the conclusions more reliable. However, increasing the number of observations does not by itself make a non-random selection representative. Consider both sample size and selection method.

Make a contextual criticism

Explain which feature could weaken the conclusion: missing representation of a group, non-random selection or a very small sample. Then connect the proposed improvement to that feature.