Research Methods guide, Unit 2, step 6 of 16Guide overview Next step

Step 6: Sampling and Sample Size

By the EconTutorials team. 13-minute read with 6 practice questions.

MatchesMethodology chapter: participants and sampling · survey design · see also sampling methods in statistics

In 30 seconds

The population is everyone your question is about; the sample is the people you actually study; the sampling frame is the list you draw the sample from.
Probability sampling (simple random, systematic, stratified, cluster) gives everyone a known chance of selection and supports generalisation.
Non-probability sampling (convenience, quota, purposive, snowball) is quicker but limits how far you can generalise; purposive sampling is standard in qualitative work.
To estimate a proportion within ±5 percentage points at 95% confidence you need about 385 responses from a large population; fewer from a small one.
Plan for non-response: if you expect 40% to reply, invite about 2.5 times the sample you need.

Video lesson coming soonA short animated lesson for this step will appear here.

Population, frame and sample

Key termsThe population is the whole group your conclusions are about. The sampling frame is the list from which you choose (a student register, an email list). The sample is the subset you study. Sampling error is the chance difference between a sample result and the population value; sampling bias is a systematic difference caused by how the sample was chosen.

In the part-time jobs study the population is the 2,000 full-time undergraduates at one university, and the frame is the student email list. If the frame misses some students (for example those who never read university email), the sample may be biased however carefully you draw it.

Probability sampling

Method	How	Strength
Simple random	Every member has an equal chance; draw with a random-number generator	Unbiased; the basis of most statistics
Systematic	Pick every $ k $th name from a list after a random start	Easy to do; fine unless the list has a repeating pattern
Stratified	Split into groups (strata), then sample randomly within each	Guarantees each group is represented; more precise
Cluster	Randomly choose whole groups (classes, schools), then study everyone in them	Cheap when the population is spread out; less precise

Worked example

The university has 800 first-years, 700 second-years and 500 third-years. Draw a proportional stratified sample of 200.

Step 1Sampling fraction: $ 200 / 2{,}000 = 0.10 $.

Step 2First-years: $ 0.10 \times 800 = 80 $. Second-years: $ 0.10 \times 700 = 70 $. Third-years: $ 0.10 \times 500 = 50 $.

Step 3Check: $ 80 + 70 + 50 = 200 $. Draw each group randomly from its own list.

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Non-probability sampling

Method	How	Typical use
Convenience	Whoever is easy to reach	Pilots; quick student projects (with clear limits)
Quota	Fill set numbers per group, but not randomly	Market research; when no frame exists
Purposive	Choose people because they hold the information you need	Qualitative interviews
Snowball	Participants recruit others like them	Hard-to-reach groups

For qualitative research, non-probability sampling is not a weakness: you are choosing information-rich cases, not trying to estimate a population percentage. A purposive sample of working students who differ in hours and job type is exactly right for interviews (Step 9).

How big should the sample be?

For estimating a proportion, the sample size needed for a margin of error $ E $ at 95% confidence is

\[ n_0 = \frac{z^2\,p(1-p)}{E^2}, \qquad z = 1.96. \]

If you have no idea what $ p $ is, use $ p = 0.5 $, which gives the largest (safest) answer.

Worked example

You want to estimate the share of students with a part-time job to within ±5 percentage points, at 95% confidence. The population is 2,000 students and you expect 40% to respond.

Step 1Large-population size: $ n_0 = 1.96^2 \times 0.5 \times 0.5 / 0.05^2 = 384.2 $, so 385.

Step 2Finite population correction: $ n = \dfrac{n_0}{1 + (n_0 – 1)/N} = \dfrac{384.2}{1 + 383.2/2{,}000} = 322.4 $, so 323 responses.

Step 3Allow for non-response: $ 323 / 0.40 = 807.5 $, so invite about 808 students.

Step 4For ±3 points instead, $ n_0 = 1.96^2 \times 0.25 / 0.03^2 = 1{,}067.1 $, so 1,068 before correction. Halving the margin roughly quadruples the sample.

In your dissertationReport your target sample, how you calculated it, how many you invited, how many responded and the response rate. If you could not reach the target, say so and widen the margin of error you report, rather than hiding it.

For comparing groups or testing an effect, sample size depends on the size of the effect you want to detect; this is called power analysis, covered with the tests in Step 12. For interviews, samples are smaller: many studies reach data saturation, the point where new interviews add no new themes, somewhere around 10 to 20 interviews for a focused question.

Common mistakeBelieving a bigger sample fixes bias. A survey posted on one social media page with 2,000 replies is still a self-selected, biased sample. Size reduces random sampling error; only a good sampling method reduces bias.

Practise until it sticks: Research Methods Made Simple Practice Questions has 1,004 questions, each with a worked answer.Get the practice book · $9

Practise: 6 questions

Try each one on paper first, then check the worked answer.

EasyQ1. Name the method: (a) every 10th name on a list after a random start; (b) randomly choosing 5 tutorial groups and surveying everyone in them. Show worked answer

(a) Systematic sampling. (b) Cluster sampling.

EasyQ2. What is a sampling frame? Show worked answer

The list or source from which the sample is actually drawn, such as a register, a membership list or an email list.

MediumQ3. A school has 600 boys and 400 girls. Draw a proportional stratified sample of 50. Show worked answer

Fraction $ 50/1{,}000 = 0.05 $. Boys $ 0.05 \times 600 = 30 $; girls $ 0.05 \times 400 = 20 $.

MediumQ4. Previous studies suggest about 30% of students work. What sample gives ±5 points at 95% confidence (large population)? Show worked answer

$ n_0 = 1.96^2 \times 0.3 \times 0.7 / 0.05^2 = 322.7 $, so 323. Using a prior estimate below 0.5 reduces the sample needed from 385.

MediumQ5. Why is purposive sampling appropriate for interviews but not for estimating what percentage of students work? Show worked answer

Interviews aim for depth and a range of experiences, so choosing information-rich participants is the point. Estimating a percentage needs a sample that represents the population; purposive choice gives unknown selection chances and so a biased estimate.

ExamQ6. A student needs 323 responses and expects a 25% response rate. How many should be invited, and give two ways to raise the response rate. [4] Show worked answer

$ 323 / 0.25 = 1{,}292 $ invitations [2]. Ways to raise response: send reminders; keep the survey short and mobile-friendly; explain why it matters; offer a small prize draw; send it at a quiet time of term (any two) [2].

Check yourself

1. Which method guarantees each year group is represented?

Stratified sampling samples separately within each group.

2. Halving the margin of error roughly

$ E $ is squared in the denominator, so $ n $ rises with $ 1/E^2 $.

3. A larger sample mainly reduces

Size reduces chance error; bias comes from how the sample is chosen.

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The short version

Define the population, find a frame, then choose a sampling method.
Probability methods support generalisation; non-probability methods suit pilots and qualitative work.
±5 points at 95% confidence needs about 385 from a large population; correct for small populations and inflate for non-response.
Bigger is not unbiased: method matters more than size.

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Next: step 7 of 16Measurement, Validity and Reliability