Representativeness Examples in Math

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

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

Concept Recap

A sample is representative if its characteristics (distribution of key variables) closely match those of the population it is meant to represent.

A representative sample is a miniature version of the population — every relevant group is included in the right proportions so the sample mirrors the whole.

Read the full concept explanation →

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: A sample is representative when its key characteristics match the population's in the right proportions.

Common stuck point: The procedure for representativeness is the easy part; the trap is assuming a big sample is representative. Asking "Do the sample's group proportions match the population's?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the sample's group proportions match the population's?

Worked Examples

Example 1

easy
A city is 60% adults and 40% children. A survey samples 50 adults and 50 children. Is this sample representative of the city's age distribution? Design a proportionally representative sample of 100 people.

Answer

Current sample is not representative. A representative sample of 100 needs 60 adults and 40 children.

First step

1
Current sample: 50 adults (50%), 50 children (50%) — does NOT match 60%/40% city distribution

Full solution

  1. 2
    For representativeness: sample should reflect 60% adults, 40% children
  2. 3
    Proportionally representative sample of 100: 100×0.60=60100 \times 0.60 = 60 adults; 100×0.40=40100 \times 0.40 = 40 children
  3. 4
    Method: stratified random sampling — randomly select from each group (stratum) proportional to its size
A representative sample mirrors the population's characteristics. Stratified sampling ensures each subgroup is represented in proportion to its population share, eliminating systematic under/over-representation of any group.

Example 2

medium
The Representativeness Heuristic: A person is described as quiet, enjoys books, and is very detail-oriented. Most people guess 'librarian' over 'farmer.' Explain why this can be a probabilistic error using base rates.

Example 3

medium
Population is 70%70\% adults, 30%30\% children. Sample is 50%50\%/50%50\%. Is the sample representative on age?

Example 4

medium
Why might a random sample of 2020 from a population of 10,00010{,}000 accidentally fail to be representative on age?

Example 5

hard
Population: 60%60\% urban, 40%40\% rural; sample 200200. The sample has 130130 urban and 7070 rural. Is it close to representative?

Example 6

medium
Why is a sample of 3030 friends a poor approximation to a representative sample of a school of 500500?

Example 7

hard
Population: 80%80\% buy product, 20%20\% don't. Marketing samples only buyers and reports 100%100\% satisfaction. Why misleading?

Example 8

hard
The representativeness heuristic: described as 'meticulous, shy, helpful', most guess 'librarian' over 'farmer'. Why is this often wrong?

Example 9

hard
Why must researchers state which population a sample claims to represent?

Example 10

medium
What is one limitation of stratified sampling for achieving representativeness?

Example 11

challenge
Why is achieving representativeness on the dependent variable (the outcome being measured) generally impossible?

Practice Problems

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

Example 1

easy
A sample of 10 people from a class of 30 is selected. List two methods to ensure the sample is representative, and explain the limitation of randomly selecting 10 friends.

Example 2

hard
A clinical trial recruits patients from a university hospital. Explain why results may not be representative of the general patient population, and identify at least two characteristics that may differ.

Example 3

easy
A sample is representative when its characteristics match those of the ___.

Example 4

easy
Population is 50% female. A sample is 50% female and matches age too. Is it representative on those variables?

Example 5

easy
True or false: any random sample is automatically representative of the population.

Example 6

easy
A 1000-person sample matches the population on income but not on region. Is it fully representative?

Example 7

easy
Which is a miniature version of the population in the right proportions: (a) representative sample, (b) convenience sample?

Example 8

easy
To represent a school of 60% freshmen and 40% seniors, a sample of 100 should include about how many seniors?

Example 9

easy
A large sample drawn from a biased frame is still ___.

Example 10

easy
Population: 25% kids, 75% adults. Sample: 50% kids, 50% adults. Representative on age?

Example 11

medium
A stratified sample takes 30% of each region exactly matching regional population shares. Why is this more representative than simple random sampling for region?

Example 12

medium
A sample matches the population on gender and age. A survey about income still seems off. What likely went wrong?

Example 13

medium
A pollster wants a sample representative of a 55% urban, 45% rural country, sample size 200. How many urban respondents are needed?

Example 14

medium
If a sample over-represents one group, statistics computed without correction will be biased toward that group. What does this say about representativeness vs accuracy?

Example 15

medium
Sample shares: 70% urban (true 55%). To reweight, what weight does each urban respondent get?

Example 16

medium
A class is 40% A-students, 60% B-students. A 'representative' study group of 10 should contain how many A-students, and what happens if you instead pick 8 A-students?

Example 17

medium
Two samples match the population mean income exactly but one has a very different income spread. Are both representative of the income distribution?

Example 18

medium
A town is 55% adults, 45% children. A sample of 200 has 150 adults. Is it representative on age, and what count would be?

Example 19

medium
A sample matches the population on race and age but was drawn only from one city. On what variable is it likely unrepresentative?

Example 20

challenge
A population has strata X (60%, mean 50) and Y (40%, mean 100). A sample is 80% X, 20% Y. Compute the true population mean and the (unweighted) sample mean.

Example 21

challenge
Using the previous setup (X 60%/mean 50, Y 40%/mean 100; sample 80% X/20% Y), find weights to recover the true mean and verify.

Example 22

challenge
Population is 30% group A. A 'representative' panel of size nn must include a whole number of A's equal to 0.30n0.30n. What is the smallest n>1n>1 giving an exact integer count of A's, and how many A's?

Example 23

easy
A town is 55%55\% female. A sample of 200200 should include about how many females to be representative on sex?

Example 24

easy
A school is 30%30\% freshman, 25%25\% sophomore, 25%25\% junior, 20%20\% senior. A sample of 4040 should include how many sophomores to match?

Example 25

medium
Population income distribution: 25%25\% low, 50%50\% middle, 25%25\% high. A sample of 400400 should have how many of each?

Example 26

easy
True or false: a sample is representative only if every member of the population is included.

Example 27

medium
A national poll samples only urban residents. Is it representative of the country?

Example 28

medium
A medical study includes only patients over 6565. Can it claim to be representative of all patients?

Example 29

medium
Population: 40%40\% Asian, 40%40\% Hispanic, 20%20\% other. Sample of 5050 should include how many each?

Example 30

easy
A national survey adjusts results so each region's share matches the census. What technique is this?

Example 31

medium
Population: 10%10\% left-handed. A sample of 300300 should include about how many left-handed people?

Example 32

medium
Population: 50%50\% Android, 50%50\% iOS. Sample 80%80\% iOS. Representative?

Example 33

medium
Population: 20%20\% low-income, 60%60\% middle, 20%20\% high. A sample of 500500 should include how many middle-income?

Example 34

medium
A 1:50 scale model neighborhood has the same proportion of trees, houses, and roads as the real one. Is it a representative model?

Example 35

easy
True or false: post-stratification weighting can fix every form of unrepresentativeness.

Example 36

hard
A sample matches the population on race and age but not on income. Should the researchers report it as 'representative'?
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Related Concepts

Sampling Bias

Background Knowledge

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

sampling bias