Practice Representativeness in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick 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.
Showing a random 20 of 50 problems.
Example 1
easyA school is freshman, sophomore, junior, senior. A sample of should include how many sophomores to match?
Example 2
mediumA sample matches the population on race and age but was drawn only from one city. On what variable is it likely unrepresentative?
Example 3
easyPopulation: 25% kids, 75% adults. Sample: 50% kids, 50% adults. Representative on age?
Example 4
hardA sample matches the population on race and age but not on income. Should the researchers report it as 'representative'?
Example 5
mediumPopulation: left-handed. A sample of should include about how many left-handed people?
Example 6
mediumA stratified sample takes 30% of each region exactly matching regional population shares. Why is this more representative than simple random sampling for region?
Example 7
mediumPopulation is adults, children. Sample is /. Is the sample representative on age?
Example 8
easyTrue or false: post-stratification weighting can fix every form of unrepresentativeness.
Example 9
challengeWhy is achieving representativeness on the dependent variable (the outcome being measured) generally impossible?
Example 10
mediumA sample matches the population on gender and age. A survey about income still seems off. What likely went wrong?
Example 11
easyA 1000-person sample matches the population on income but not on region. Is it fully representative?
Example 12
mediumWhat is one limitation of stratified sampling for achieving representativeness?
Example 13
easyA national survey adjusts results so each region's share matches the census. What technique is this?
Example 14
challengeUsing 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 15
easyA 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 16
mediumA pollster wants a sample representative of a 55% urban, 45% rural country, sample size 200. How many urban respondents are needed?
Example 17
mediumTwo samples match the population mean income exactly but one has a very different income spread. Are both representative of the income distribution?
Example 18
challengeA 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 19
mediumPopulation income distribution: low, middle, high. A sample of should have how many of each?
Example 20
mediumSample shares: 70% urban (true 55%). To reweight, what weight does each urban respondent get?