Sampling Bias Examples in Math

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

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

Concept Recap

Sampling bias occurs when the method of selecting a sample systematically over- or under-represents certain groups relative to their actual proportion in the population.

A biased sample gives you a skewed picture of the population — like judging average student height by only surveying the basketball team.

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: Sampling bias is when the selection method systematically over- or under-represents certain groups in the population.

Common stuck point: The procedure for sampling bias is the easy part; the trap is believing a large sample cures bias. Asking "Did the selection method systematically include or exclude certain groups?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Did the selection method systematically include or exclude certain groups?

Worked Examples

Example 1

medium
A magazine surveys its readers about political preferences and finds 75% support Policy X. Explain why this result may not represent the general population, identifying the bias type.

Answer

Voluntary response and convenience bias make readers non-representative of the general population.

First step

1
The sample is magazine readers — a self-selected, non-representative group

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Example 2

hard
A researcher studies income by sampling shopping mall visitors on weekday afternoons. Identify at least three sources of sampling bias and explain the likely direction of each bias.

Example 3

medium
A magazine survey of 50,00050{,}000 readers in 19361936 predicted a landslide for Landon — but Roosevelt won. What bias caused this?

Example 4

medium
Why is a survey conducted at the food court at 11 11\,am likely biased for estimating teenager opinions?

Example 5

hard
Why does asking sensitive questions face-to-face introduce bias?

Example 6

medium
Why is stratified random sampling better than simple random for estimating opinions across a population with very different subgroups?

Example 7

hard
Survivorship bias — explain with the WWII bomber example.

Example 8

challenge
A polling firm reports a 'margin of error of ±3%\pm 3\%' but uses a self-selected online panel. Is the margin reliable?

Practice Problems

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

Example 1

easy
A school wants to know students' favorite lunch option. They survey only students in the cafeteria at noon. Identify the bias and suggest a better sampling method.

Example 2

hard
An online survey about internet usage gets 10,000 responses. Explain why this large sample does not eliminate bias, and identify what type of people are systematically excluded.

Example 3

easy
A phone poll calls only landline numbers. Which group is most likely underrepresented?

Example 4

easy
A survey is posted only on a gaming website. Is the sample biased for estimating national TV-watching habits?

Example 5

easy
True or false: a biased sample of 1,000,000 is more trustworthy than an unbiased sample of 100.

Example 6

easy
A study mails surveys and uses only the responses that come back. What bias is this?

Example 7

easy
To estimate average student height, a researcher surveys only the basketball team. Biased?

Example 8

easy
A mall exit survey is conducted at 2pm on a weekday. Which group is underrepresented?

Example 9

easy
Which sampling method best avoids bias: (a) ask friends, (b) random selection from a full population list?

Example 10

easy
A website asks 'Do you love our product?' with only a 'Yes' button. What bias does this design create?

Example 11

medium
A city has 60% renters and 40% owners, but a survey of 500 reaches 90% owners. The estimate of 'support for rent control' will likely be biased in which direction, given owners oppose it?

Example 12

medium
A poll oversamples landline users (older). To correct estimates, what general technique reweights groups back to true proportions?

Example 13

medium
A study recruits volunteers for a 'new diet.' Highly motivated people volunteer. Why might results overstate the diet's effect?

Example 14

medium
An online opt-in poll says '70% oppose the policy.' Why can't we trust this as the national figure?

Example 15

medium
A survey reaches 1000 people but 800 hang up. The 200 who stay differ in patience. What bias dominates, and does n=200n=200 fix it?

Example 16

medium
Researchers sample households by knocking on doors only at houses with cars in the driveway. Who is excluded?

Example 17

medium
A pre-election poll in 1936 mailed to car and phone owners predicted the wrong winner. What single lesson does this teach?

Example 18

medium
If true support is 50% but a method always over-counts supporters by reaching them more often, the estimate's expected value is above 50%. Is this bias or random error?

Example 19

medium
A researcher samples names from an old phone book to study internet usage. Why might this introduce bias today?

Example 20

challenge
A sample's expected estimate is p^=p+0.08\hat{p}=p+0.08 regardless of nn, with standard error 0.5/n0.5/\sqrt{n}. At what nn does random error finally shrink below the bias?

Example 21

challenge
Two designs: A has bias 0.100.10, SE 0.010.01; B has bias 00, SE 0.060.06. Using total error bias2+SE2\sqrt{\text{bias}^2+\text{SE}^2}, which is more accurate?

Example 22

challenge
A survey reaches group X with probability 0.90.9 and group Y with 0.30.3. X favors a measure 20%, Y favors 80%, and the population is half X, half Y. Find the true rate and the (unweighted) sampled rate.

Example 23

easy
A teacher surveys only students who volunteered to respond. What type of bias is this?

Example 24

easy
A reporter interviews people only outside one coffee shop. What type of bias is most likely?

Example 25

medium
A company emails 10,00010{,}000 customers a satisfaction survey; 500500 respond. What bias is likely?

Example 26

medium
A study of average household income samples only homeowners. What bias is created?

Example 27

easy
True or false: a sample of 11 million is unbiased simply because it is large.

Example 28

medium
A school surveys parents using only English-language forms. What bias might result?

Example 29

medium
An online poll about smartphone usage is biased. Why?

Example 30

medium
A pollster asks 'Don't you agree that taxes are too high?'. What bias is this?

Example 31

medium
A radio show asks listeners to call in about gun control. What two biases dominate?

Example 32

medium
How could the school better survey lunch preferences than asking only in the cafeteria?

Example 33

hard
Researchers want to estimate a city's drug-use rate. House visits at 10 10\,am produce 5%5\%. Anonymous online survey produces 15%15\%. Which is closer to true and why?

Example 34

medium
A survey of grocery shoppers about prices is conducted only at premium organic stores. Bias?

Example 35

medium
A survey on movie preferences is mailed to homes; only the 10%10\% who return it are counted. Is bias likely?

Example 36

medium
A weight-loss program advertises 'success' using testimonials from satisfied customers. Bias?

Example 37

easy
A study of national TV-watching habits surveys only people on a streaming-only platform. Bias?

Example 38

hard
A medical trial uses only volunteers. Why might results not generalize to the patient population?
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Background Knowledge

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

data abstract