Sampling Methods Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sampling Methods.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Systematic approaches for selecting a subset of individuals from a population. The main probability methods are: simple random sample (SRS), stratified random sample, cluster sample, and systematic sample. Convenience sampling is a non-probability method that is generally biased.

You want to know the average GPA of 10,000 students. You can't ask everyone, so you pick a sample. How you pick matters enormously: grab the first 50 students you see in the cafeteria (convenience—biased), or give every student a number and use a random number generator to pick 50 (SRS—unbiased). Stratified sampling is like making sure you get proportional numbers from each grade level. Cluster sampling picks entire groups (like randomly selecting 5 classrooms and surveying everyone in them).

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Sampling methods are the systematic ways to choose a subset of a population — SRS, stratified, cluster, systematic — versus biased convenience sampling.

Common stuck point: The procedure for sampling methods is the easy part; the trap is equating a large sample with a representative one. Asking "Is the focus on the RULE for choosing who enters the sample (rather than how to assign treatments or compute a statistic)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the focus on the RULE for choosing who enters the sample (rather than how to assign treatments or compute a statistic)?

Worked Examples

Example 1

medium
Describe four sampling methods: simple random, stratified, cluster, and systematic. Compare their advantages and disadvantages.

Answer

Four methods: SRS (purest), stratified (best for subgroups), cluster (cheapest), systematic (easiest).

First step

1
Simple Random: every member has equal chance; randomly select n from N; advantage: unbiased; disadvantage: difficult/expensive for large populations

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Example 2

hard
A school has 500 students: 200 freshmen, 150 sophomores, 100 juniors, 50 seniors. Design a proportionally stratified sample of 50 students.

Example 3

medium
A district has 600 students: 300 in elementary, 200 in middle, 100 in high. Design a proportional stratified sample of 60.

Example 4

medium
A school has 1000 students: 60% female, 40% male. A stratified sample of 50 by sex should contain how many of each?

Example 5

medium
Population of 2000 has 800 women, 1200 men. A stratified sample of 100 proportional to sex has how many men?

Example 6

hard
A list of 1000 houses is ordered by street, so every 10th house is on a corner (different from middle houses). A systematic sample of k=10k=10 would bias what?

Example 7

hard
A polling firm uses random-digit dialing of landlines. Why might this introduce bias today?

Example 8

challenge
The 1936 Literary Digest poll predicted Landon would beat Roosevelt based on 2.4 million respondents. Roosevelt won in a landslide. Identify the TWO main sampling errors.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A researcher wants to study opinions of 10,000 employees across 50 departments. She randomly selects 5 departments and surveys all employees in those departments. What sampling method is this, and what are its limitations?

Example 2

hard
A researcher randomly selects every 10th name from an alphabetical list of 1000 employees. Explain why this is systematic sampling, calculate the starting point needed, and describe a potential bias if the list is alphabetized by department.

Example 3

easy
You number all 1000010000 students and use a random generator to pick 200200, each equally likely. Which sampling method is this?

Example 4

easy
You divide students into grade levels, then randomly sample within each grade. Which method is this?

Example 5

easy
You randomly pick 55 entire classrooms and survey everyone in them. Which method is this?

Example 6

easy
You survey every 1010th person on an alphabetical list. Which method is this?

Example 7

easy
You survey the first 5050 people you happen to meet at the mall. Which method is this, and is it biased?

Example 8

easy
Which is a non-probability sampling method generally prone to bias?

Example 9

easy
True or false: a biased sample of 1000010000 is generally more trustworthy than an unbiased sample of 100100.

Example 10

easy
Which sampling method guarantees that each defined subgroup is represented in the sample?

Example 11

medium
A researcher splits a city into neighborhoods, randomly picks a few neighborhoods, and surveys everyone there. Stratified or cluster?

Example 12

medium
A polling firm divides voters into age strata and samples proportionally within each. Why does this typically beat a plain SRS for the same nn?

Example 13

medium
A factory inspects every 5050th item. If a malfunction makes every 5050th item defective, what problem arises?

Example 14

medium
A website posts an online poll anyone can answer. Why is this likely biased, and what is it called?

Example 15

medium
To estimate average household income, which method best ensures both rich and poor neighborhoods are represented?

Example 16

medium
A teacher wants an SRS of 44 from 2020 students using a random number table. Describe the key property the result must have.

Example 17

medium
A survey reaches people only via landline phones. Even with random dialing, what kind of error does this introduce?

Example 18

challenge
Explain the core difference between stratified and cluster sampling using how each handles the subgroups it forms.

Example 19

challenge
Why does a well-designed probability sample of 10001000 beat a self-selected sample of 10000001000000 for estimating a national proportion?

Example 20

challenge
A pollster wants narrow margins cheaply. Compare why stratified sampling improves precision while cluster sampling often reduces it, for the same nn.

Example 21

medium
A company lists employees by department, then randomly samples within every department. Stratified or cluster?

Example 22

medium
A quality team randomly selects 33 shipping crates and inspects every item in those crates. Stratified or cluster?

Example 23

easy
A school randomly numbers all 2000 students and uses a generator to pick 100. Method?

Example 24

easy
A pollster surveys 50 friends in their dorm. Method? Likely biased?

Example 25

easy
A study randomly chooses 4 of 50 high schools and surveys ALL students in each. Method?

Example 26

medium
A list has 800 customers. A researcher wants a systematic sample of 40. What is kk?

Example 27

medium
A magazine sends a survey to all subscribers; only 5% respond. What bias is this?

Example 28

medium
True or false: a larger SRS is always more accurate than a smaller one of the same population.

Example 29

hard
A city has 4 boroughs of equal size, but residents within a borough are very similar to each other. Why is cluster sampling LESS precise than SRS here?

Example 30

hard
A school has classes of widely varying sizes. Why might cluster sampling of classes give estimates of student attitudes that are biased toward larger classes if you don't reweight?

Example 31

medium
A teacher pulls names from a hat to pick 5 of 30 students. Method?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

sampling biasrepresentativenessrandomness