mathematics

Sampling design

Sampling design is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sampling design rather than just read about it. In short: In the theory of finite population sampling, a sampling design specifies for every possible sample its probability of being drawn. Mathematical formulation Mathematically, a sampling design is denoted by the function P ( S ) {\displaystyle P(S)} which gives the probability of drawing a sample S . {\displaystyle S.} An example of a sampling design During Bernoulli sampling, P ( S ) {\displaystyle P(S)} is given by P…

Key takeaways

Sampling design belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sampling design to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sampling design from memory before moving on to harder problems.

Reference excerpt

In the theory of finite population sampling, a sampling design specifies for every possible sample its probability of being drawn.

Mathematical formulation Mathematically, a sampling design is denoted by the function P ( S ) {\displaystyle P(S)} which gives the probability of drawing a sample S . {\displaystyle S.}

An example of a sampling design During Bernoulli sampling, P ( S ) {\displaystyle P(S)} is given by

P ( S ) = q N sample ( S ) × ( 1 − q ) ( N pop − N sample ( S ) ) {\displaystyle P(S)=q^{N_{\text{sample}}(S)}\times (1-q)^{(N_{\text{pop}}-N_{\text{sample}}(S))}}

where for each element q {\displaystyle q} is the probability of being included in the sample and N sample ( S ) {\displaystyle N_{\text{sample}}(S)} is the total number of elements in the sample S {\displaystyle S} and N pop {\displaystyle N_{\text{pop}}} is the total number of elements in the population (before sampling commenced).

Sample design for managerial research In business research, companies must often generate samples of customers, clients, employees, and so forth to gather their opinions. Sample design is also a critical component of marketing research and employee research for many organizations. During sample design, firms must answer questions such as:

What is the relevant population, sampling frame, and sampling unit? What is the appropriate margin of error that should be achieved? How should sampling error and non-sampling error be assessed and balanced? These issues require very careful consideration, and good commentaries are provided in several sources.

See also Bernoulli sampling Sampling probability Sampling (statistics)

References

Further reading Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, ISBN 0-387-40620-4

Worked examples

Example 1 — a first encounter with Sampling design

Start with the simplest possible case. Write down what Sampling design claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sampling design before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sampling design ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sampling design

In research: Sampling design appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sampling design in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sampling design is common in secondary-school and first-year university syllabi. It links to neighbouring topics Sampling (statistics), so understanding it makes those chapters shorter.
In everyday life: Look for Sampling design outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Sampling design in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sampling design means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sampling design out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sampling design in simple terms?

In the theory of finite population sampling, a sampling design specifies for every possible sample its probability of being drawn. Mathematical formulation Mathematically, a sampling design is denoted by the function P ( S ) {\displaystyle P(S)} which gives the probability of drawing a sample S . {…

Why does Sampling design matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sampling design?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sampling design.