Statistical Sampling – A Level Maths Year 1

1 Populations and Samples

Every statistical investigation starts with a question about a group of people or things. Before you can collect any data, you have to decide who or what you are going to measure.

Term	Meaning	Example
Population	The whole set of items you are interested in.	All $1200$ students at a school
Census	Data collected from every member of the population.	Asking every student
Sample	A subset of the population, used to draw conclusions about the whole.	$60$ of the students
Sampling unit	One individual member that could be chosen.	A single student
Sampling frame	A list of all the sampling units, from which the sample is actually chosen.	The school register
Statistic	A number calculated from the sample.	The mean height of the $60$

A sample is only useful if it is representative of the population — if the people in it are, as a group, like the people who are not in it. The sampling method is how you try to guarantee that.

	Advantages	Disadvantages
Census	Completely accurate picture of the population (apart from measurement error); no sampling variation.	Time-consuming and expensive; difficult to process large amounts of data; impossible if testing destroys the item.
Sample	Cheaper and quicker; less data to process; the only option when testing is destructive.	Results vary from sample to sample; the sample may not be representative; small sub-groups may be missed.

Worked Example 1 — Census or sample?

For each situation, say whether a census or a sample is more appropriate. (a) A manufacturer tests the lifetime of its light bulbs. (b) A teacher records the test marks of her class of $28$. (c) A council asks residents of a city of $300\,000$ about bin collections.

①(a) Sample — testing lifetime destroys the bulb, so a census would leave nothing to sell.

②(b) Census — the population is small and the data are already available.

③(c) Sample — a census would be very expensive and slow, and a well-chosen sample of a few hundred gives a good picture.

2 Random Sampling Methods

In a random method, chance decides who is chosen, so the person choosing cannot influence the result. Random methods need a sampling frame.

Simple random sampling

Every sample of size $n$ has an equal chance of being chosen

Number every member of the sampling frame, e.g. $1$ to $N$.
Use a random number generator (calculator, computer or table) to pick numbers between $1$ and $N$.
Ignore repeats, and keep going until you have $n$ different numbers.
Select the members with those numbers. (Equivalently: names in a hat — the lottery method.)

Systematic sampling

Choose every $k$th member of an ordered list, where $k = \dfrac{N}{n}$, starting from a random member among the first $k$

Stratified sampling

Split the population into groups (strata); sample each stratum in proportion to its size:
number from a stratum $= \dfrac{\text{size of stratum}}{\text{size of population}} \times \text{sample size}$

Within each stratum the members are then chosen by simple random sampling.

Worked Example 2 — Simple random sampling

Describe how to take a simple random sample of $30$ employees from a company of $450$.

①Obtain a list of all $450$ employees (the sampling frame) and number them $1$ to $450$.

②Use a calculator's random integer function to generate numbers from $1$ to $450$.

③Discard any repeats until $30$ different numbers have been obtained.

④The employees with those numbers form the sample.

Full marks for "describe" usually need all three ideas: list and number, random numbers, ignore repeats.

Worked Example 3 — Systematic sampling

A quality controller wants a sample of $80$ from the $2400$ cans filled on a production line in a day. Describe a systematic sample.

①Sampling interval $k = \dfrac{2400}{80} = 30$.

②Choose a random starting number from $1$ to $30$ — say $17$.

③Take cans $17, 47, 77, 107, \ldots$, i.e. every $30$th can, up to can $2387$.

If the list has a pattern that repeats every $30$ items (for example a fault on one filling nozzle out of $30$), a systematic sample could hit it every time or miss it completely.

Worked Example 4 — Stratified sampling

A sports club has $84$ junior, $156$ adult and $60$ senior members. A stratified sample of $25$ is required. How many of each type should be chosen?

①Total membership: $84 + 156 + 60 = 300$.

②Juniors: $\dfrac{84}{300} \times 25 = 7$

③Adults: $\dfrac{156}{300} \times 25 = 13$

④Seniors: $\dfrac{60}{300} \times 25 = 5$ (check: $7 + 13 + 5 = 25$ ✓)

⑤Number the members of each group and select $7$, $13$ and $5$ of them by simple random sampling.

Worked Example 5 — Stratified sampling with rounding

A college has $412$ full-time, $270$ part-time and $118$ evening students. Find the numbers for a stratified sample of $50$.

①Total $= 800$.

②$\dfrac{412}{800} \times 50 = 25.75$, $\dfrac{270}{800} \times 50 = 16.875$, $\dfrac{118}{800} \times 50 = 7.375$

③Round: $26$, $17$ and $7$. Check: $26 + 17 + 7 = 50$ ✓

Always check the rounded numbers still add up to the sample size. If they do not, adjust the stratum whose decimal part was closest to $0.5$.

Cluster sampling (named in some specifications): the population is split into naturally occurring groups (clusters), such as schools or streets; a random selection of whole clusters is taken and everyone in them — or a random sample from them — is surveyed. It is cheap when the population is spread out geographically, but clusters may differ from one another, so it is less precise than a simple random sample of the same size.

3 Non-Random Sampling Methods

Sometimes no sampling frame exists, or there is no time to use one. Then non-random methods are used — they are quicker and cheaper, but the person collecting the data influences who is chosen.

Quota sampling

The interviewer is told how many people to find from each group (e.g. $10$ men under $30$), and chooses them personally until each quota is filled

Opportunity (convenience) sampling

The sample is whoever is available at the time and fits the criteria

Self-selected (volunteer) samples, such as online polls or phone-ins, are a form of opportunity sampling. The people who choose to respond often have strong views, so the results can be very unrepresentative.

Worked Example 6 — Quota versus stratified

A market researcher must interview $20$ women and $20$ men in a town centre. Name the method, and explain how it differs from stratified sampling.

①This is quota sampling: there are quotas for each group but no sampling frame.

②In stratified sampling the individuals within each group are chosen at random from a list. In quota sampling the interviewer chooses them.

③So the interviewer might, without meaning to, approach only people who look friendly or who are not in a hurry — a source of bias.

4 Advantages and Disadvantages

Method	Advantages	Disadvantages
Simple random	Free of selection bias; every member equally likely to be chosen.	Needs a sampling frame; can be slow for a large population; may by chance under-represent a group.
Systematic	Simple and quick to use; spreads the sample evenly through the list.	Needs a sampling frame; biased if the list has a repeating pattern.
Stratified	Each group is represented in the right proportion; results can be compared between strata.	Needs a frame for each stratum; strata must be clearly defined and non-overlapping.
Quota	No sampling frame needed; quick and cheap; groups represented in proportion.	Not random — interviewer choice can introduce bias; non-responses are simply replaced, hiding a source of bias.
Opportunity	Easiest, quickest and cheapest.	Very unlikely to be representative; highly dependent on who happens to be there.
Cluster	Practical and cheap for a geographically spread population.	Clusters may not be typical of the population; less precise.

Bias is a systematic tendency for a sample to over- or under-represent part of the population, so that conclusions are distorted in a consistent direction. A bigger sample does not cure bias — it just gives a more precise wrong answer. Random methods remove selection bias; larger random samples reduce the random variation between samples.

Worked Example 7 — Spotting bias

To estimate how much time students spend on social media, a student asks the first $30$ people who arrive at the school library one morning. Identify the sampling method and explain why the results may be biased.

①This is an opportunity sample.

②Students who arrive early at a library are likely to be more studious than average, so they may spend less time on social media.

③The sample is unlikely to be representative; the estimate is likely to be too low.

④Improvement: obtain the school roll and take a simple random (or stratified by year group) sample.

The large data set. Every exam board provides a large data set that you are expected to explore during the course, and some questions ask how you would sample from it. The same ideas apply: number the rows, then use random numbers (simple random) or every $k$th row (systematic). Familiarity with your board's data set can help you spot when an answer is unrealistic.

5 Quick Reference

Population

Everyone or everything of interest.

Sampling frame

The list the sample is chosen from.

Census

Accurate but costly; impossible if testing destroys.

Simple random

Number, random numbers, ignore repeats.

Systematic

Every $k$th, $k = N/n$, random start.

Stratified

$\dfrac{\text{stratum}}{\text{population}} \times n$, then random within.

Quota

Fill quotas; interviewer chooses; no frame.

Opportunity

Whoever is available; quick but biased.

Bias

Systematic distortion; not fixed by a bigger sample.

6 Practice Questions

Question 1

Explain the difference between a census and a sample, and give one advantage of each.

▶ Show solution

A census collects data from every member of the population; a sample collects data from only part of it.

Census: gives a complete, accurate picture of the population. Sample: cheaper and quicker, with less data to process.

Question 2

A dentist wants to survey her patients about appointment times. Suggest a suitable sampling frame, and identify the sampling units.

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Sampling frame: the practice's list of registered patients.

Sampling units: the individual patients.

Question 3

A systematic sample of $60$ is to be taken from a list of $1200$ voters. Describe how this should be done.

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Interval $k = \dfrac{1200}{60} = 20$.

Choose a random number $r$ from $1$ to $20$ (e.g. using a calculator), then select voters $r, r + 20, r + 40, \ldots$ until $60$ have been chosen.

Question 4

A factory employs $180$ day-shift and $120$ night-shift workers. Find the numbers from each shift in a stratified sample of $25$.

▶ Show solution

Total $300$.

Day: $\dfrac{180}{300} \times 25 = 15$. Night: $\dfrac{120}{300} \times 25 = 10$.

Question 5

A newspaper website asks readers to vote online on whether a new road should be built. Explain why the result may not reflect the views of the town's population.

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The sample is self-selected: only readers of that website who choose to vote are included.

People with strong opinions (often those opposed) are more likely to vote, and people who do not read the site are excluded entirely, so the sample is unlikely to be representative. One person might also vote several times.

Question 6

Give one advantage and one disadvantage of quota sampling compared with stratified sampling.

▶ Show solution

Advantage: no sampling frame is needed, so it is quicker and cheaper.

Disadvantage: it is not random — the interviewer chooses who to ask, which can introduce bias.

Question 7

A gardener has $96$ tomato plants in a greenhouse. Describe how to take a simple random sample of $12$ of them to measure.

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Label the plants $1$ to $96$ (e.g. with tags), forming a sampling frame.

Use a random number generator to produce integers from $1$ to $96$, ignoring repeats, until $12$ different numbers are obtained.

Measure the plants with those labels.

Question 8

A sixth form has $230$ students in Year 12, $190$ in Year 13 and $180$ on a one-year course. A stratified sample of $40$ is needed. How many should be chosen from each group?

▶ Show solution

Total $600$.

$\dfrac{230}{600} \times 40 = 15.33\ldots$, $\dfrac{190}{600} \times 40 = 12.66\ldots$, $\dfrac{180}{600} \times 40 = 12$

Round to $15$, $13$ and $12$. Check: $15 + 13 + 12 = 40$ ✓

Question 9

To find out how often adults in a town exercise, a researcher stands outside a gym on a Saturday morning and questions the first $50$ adults who walk past. (a) Name the sampling method. (b) Explain why the results are likely to be biased, and state the likely direction of the bias. (c) Suggest an improved method.

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(a) Opportunity sampling.

(b) People near a gym are more likely to be gym users, and Saturday morning excludes people who work then. The estimate of how often adults exercise is likely to be too high.

(c) Use the electoral register as a sampling frame and take a simple random sample (or one stratified by age group), contacting the chosen people directly.

Question 10

A council wants the views of the $6000$ adults living in four wards: A ($1800$), B ($1500$), C ($1650$) and D ($1050$).

(a) Give one reason why the council might not use a census. (b) Find the number from each ward in a stratified sample of $120$. (c) Describe how the adults from ward A would be selected. (d) Explain why stratifying by ward may give better results than a simple random sample of $120$. (e) Even with a well-designed method, give one reason why the results may still be biased.

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(a) Asking all $6000$ would be expensive and time-consuming, and processing that much data is difficult.

(b) A: $\dfrac{1800}{6000} \times 120 = 36$; B: $30$; C: $33$; D: $21$. Check: $36 + 30 + 33 + 21 = 120$ ✓

(c) Number the $1800$ adults on ward A's electoral list from $1$ to $1800$, generate random integers in that range, ignore repeats, and take the first $36$ different numbers.

(d) Views may differ between wards (for example, one ward may be more affected by a decision). Stratifying guarantees each ward is represented in proportion; a simple random sample could, by chance, over-represent one ward.

(e) Non-response: some selected adults will not reply, and those who do reply may differ from those who do not (for example, they may care more about the issue). Also, the electoral register may be incomplete or out of date.

Statistical Sampling · A Level Maths Year 1 · Created with MathJax