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: Representative samples allow valid generalization from sample to population.

Common stuck point: Large samples aren't automatically representative—they can be large AND biased.

Sense of Study hint: Compare the key characteristics of your sample (age, gender, location) with the population. Do the proportions roughly match?

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.

Solution

  1. 1
    Current sample: 50 adults (50%), 50 children (50%) — does NOT match 60%/40% city distribution
  2. 2
    For representativeness: sample should reflect 60% adults, 40% children
  3. 3
    Proportionally representative sample of 100: 100 \times 0.60 = 60 adults; 100 \times 0.40 = 40 children
  4. 4
    Method: stratified random sampling — randomly select from each group (stratum) proportional to its size

Answer

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

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.
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Related Concepts

Sampling Bias

Background Knowledge

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

sampling bias