Track 6 · Sampling and the Central Limit Theorem · Lesson 1 of 8

Sampling methods

Describe simple random, stratified, cluster and systematic sampling, carry each one out, and choose the design that fits a situation.

30 minIntermediate7 graded exercisesFree

An HR team wants to know how engaged its 1 400 employees are, but interviewing everyone would take months. Which 150 people should it talk to? The way the sample is chosen decides whether the answer can be trusted — and how precise it will be.

A population is the whole group we want to learn about; a sample is the part we actually measure. To pick a sample we need a sampling frame: a list (or map) of the units we can choose from, such as the HR system's list of employees.

A probability sample gives every unit in the frame a known, non-zero chance of selection, chosen by a random mechanism rather than by a person's judgement. That randomness is what lets us measure how far a sample estimate is likely to be from the truth. Convenience samples (whoever is easiest to reach) and voluntary-response samples (whoever chooses to answer) are not probability samples.

Design	How units are chosen	Strength	Watch out for
Simple random sample (SRS)	Every set of $n$ units is equally likely (e.g. random numbers from the full list)	Simplest to analyse; the basis of most formulas	Needs a complete list; can by chance under-represent small groups
Stratified	Split the population into groups (strata) that differ, then take an SRS inside every stratum	Guarantees every group appears; more precise when strata differ	Needs the stratum of every unit in advance
Cluster	Split the population into natural groups (clusters), pick some clusters at random, measure everyone in them	Cheap when units are spread out (schools, villages, branches)	Less precise when units in a cluster are alike
Systematic	Random start $r$ between 1 and $k$ , then every $k$ -th unit on the list	Easy in the field (every 10th customer)	A list with a repeating pattern can line up with $k$

Four probability sampling designs

A population of 200 units arranged in four regions (strata) with blocks of 10 neighbouring units (clusters). Draw samples with each design and compare how much the estimates vary from draw to draw.

n_h = n \times \frac{N_h}{N} \qquad\qquad k = \left\lfloor \frac{N}{n} \right\rfloor

Proportional allocation (left) and the systematic sampling interval (right)

$N$: the population size (all units in the frame)
$N_h$: the number of population units in stratum $h$
$n$: the total sample size
$n_h$: the sample size given to stratum $h$ (rounded so the $n_h$ add up to $n$ )
$k$: the sampling interval: take every $k$ -th unit after a random start between 1 and $k$

Worked example

Allocating 150 interviews across six departments

The company in the (synthetic) engagement_population dataset has 1,400 employees in six departments. The HR team wants a stratified sample of 150 with proportional allocation.

Shares. Operations has 420 of the 1,400 employees, a share of $420/1400 = 0.300$ . Its allocation is $150 \times 0.300 = 45.00$ .
All departments. The same calculation gives Operations 45.00, Customer Service 35.36, Sales 27.86, Technology 22.50, Finance 12.86, Human Resources 6.43.
Round so the total is 150. Take the whole parts first, then give the remaining interviews to the departments with the largest fractional parts (the largest-remainder rule).
Draw. Inside each department, choose that many employees by simple random sampling.

Allocation: Operations 45, Customer Service 35, Sales 28, Technology 23, Finance 13, Human Resources 6 — a total of 150, with every department represented in proportion to its size.

Worked example

A systematic sample from a list

Choose a systematic sample of 70 employees from an alphabetical list of 1 400.

The interval is $k = 1400 / 70 = 20$ .
Pick a random start between 1 and 20 — here the random start was 5.
Take employees 5, 25, 45, 65, … , 1385: every 20th name.

Each employee has a 1-in-20 chance, but only 20 different samples are possible, so the list order matters: an alphabetical list is harmless, a list sorted by a repeating pattern is not.

Quick check

A national health survey randomly picks 60 towns, then interviews every household in those towns. Which design is this?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

0 of 7 exercises correct (not saved — sign in to keep your progress)

Exercise 1 · Guided, difficulty 1 of 3

Set up a systematic sample

Not started

A store wants a systematic sample of 100 receipts from the 3,094 receipts printed last month, numbered 1 to 3,094.

What sampling interval $k$ should it use? (Use $k = \lfloor N/n \rfloor$ , the whole part of $N/n$ .)
The random start drawn between 1 and $k$ is 13. What is the number of the 5th receipt in the sample?

Exercise 2 · Practice, difficulty 1 of 3

Name the design

Not started

Classify each sampling plan.

A bank numbers all 8 000 current-account holders and uses a random number generator to choose 400 of them.

A university splits students by faculty and randomly selects 5% of the students in each faculty.

An inspector randomly chooses 12 of a factory's 90 pallets and tests every item on those pallets.

A call centre reviews call number 7 of the day and then every 25th call after it.

A news website asks readers to vote in an online poll about a new tax.

A researcher interviews the first 50 shoppers who walk past her at a mall entrance.

A school district randomly picks 8 of its 40 schools and surveys every teacher in those schools.

Exercise 3 · Application, difficulty 2 of 3

Proportional allocation from the headcount table

Not started

HR plans a stratified sample of 250 employees from the six departments in the engagement_population dataset, using proportional allocation by headcount.

Before rounding, how many interviews should Technology receive? Give your answer to 2 decimal places.

Data: engagement_population · Open in the lab to explore and calculate.

Exercise 4 · Interpretation, difficulty 2 of 3

Choose a design for a reading survey

Not started

An education ministry wants to estimate the average reading score of Grade 4 pupils across a large country. There is a list of the 6 000 schools but no national list of pupils, and sending testers to a school is expensive, while testing extra pupils once there is cheap. Which design fits best?

Exercise 5 · Practice, difficulty 2 of 3

A periodic list

Not started

A café records daily sales for a whole year (365 days, starting on a Monday). The manager takes a systematic sample with interval $k = 7$ and random start 3 to estimate average daily sales. What is the problem?

Exercise 6 · Practice, difficulty 1 of 3

Drawing a stratified sample

Not started

Put the steps for drawing a proportional stratified random sample in order.

Exercise 7 · Interpretation, difficulty 3 of 3

Why stratify?

Not started

In the sampling widget the four regions have very different average values. In two or three sentences, explain why a stratified sample gives estimates that vary less from draw to draw than a simple random sample of the same size, and when stratifying would not help much.

What you can now do

Explain what makes a sample a probability sample, and why convenience and voluntary samples are not
Carry out simple random, stratified, cluster and systematic selection
Allocate a stratified sample proportionally to stratum sizes
Choose a design from the frame available, the costs and how groups differ