Population vs Sample Examples in Math

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

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 population is the entire group you want to study. A sample is a smaller subset of that population that you actually collect data from.

You cannot taste every cookie in the bakery to check quality — you taste a few (sample) and draw conclusions about the whole batch (population).

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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: A population is every member you want a conclusion about; a sample is the smaller subset you actually measure to stand in for it.

Common stuck point: The procedure for population vs sample is the easy part; the trap is calling the data you collected 'the population' just because it is all the data you have. Asking "Is this number describing every single member I care about, or only the subset I actually collected data from?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this number describing every single member I care about, or only the subset I actually collected data from?

Worked Examples

Example 1

easy
Identify population and sample: A teacher records test scores from 4 of her 28 students.

Answer

Population: 28 students; Sample: 4 students\text{Population: 28 students; Sample: 4 students}

First step

1
Find the whole group the question targets: the 28 students.

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Example 2

medium
A factory produces 50,000 bolts per day. Inspectors test 250. They report a defect rate of 1.6%. Identify each: population, sample, parameter, statistic.

Example 3

medium
A city has 40% homeowners. A truly representative sample of 250 residents should contain about how many homeowners?

Example 4

hard
You want the population mean weekly screen time of US teens. You only have data on 12 of your cousins. List two reasons this sample is poor.

Example 5

hard
A pollster wants to predict who will win a city's mayoral election (population: all city voters). She asks 500 random voters; 270 say Candidate A. Compute the sample proportion p^\hat{p}.

Example 6

challenge
In one election, exit polls of 1,500 voters predict 51% for Candidate B, but the final result among 2 million voters is 49%. Identify each number's role and explain the gap.

Practice Problems

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

Example 1

easy
A researcher studies all 500 students in a school by surveying 50 of them. What are the 50?

Example 2

easy
In that study, what are all 500 students called?

Example 3

easy
Why study a sample instead of the whole population?

Example 4

easy
A value computed from a sample (like the sample mean) is called a what?

Example 5

easy
A value describing the whole population (like the true mean) is called a what?

Example 6

easy
Is asking only your friends a good sample of the whole school?

Example 7

easy
To represent a whole town, should you sample only people at a gym?

Example 8

easy
Population: all light bulbs made today. A sample of 20 is tested. What is the population?

Example 9

medium
A poll of 200 voters finds 55% support a measure. Is 55% a statistic or a parameter?

Example 10

medium
Which is better: a random sample of 100 or a biased sample of 1000?

Example 11

medium
A school has 60% girls. A representative sample of 50 should have about how many girls?

Example 12

medium
To estimate average height of a city, you measure 500 residents. What is the sample?

Example 13

medium
Why does a larger random sample usually give a better estimate?

Example 14

medium
A factory checks every 100th item off the line. Population vs sample?

Example 15

medium
A sample mean is xˉ=72\bar{x}=72; the true population mean is unknown. What symbol denotes the population mean?

Example 16

challenge
A sample of 40 has mean 75. Does the population necessarily have mean 75?

Example 17

challenge
Two samples from the same population give means 70 and 74. Why differ?

Example 18

challenge
You want to generalize to ALL teens but sample only a coding-camp. What's the threat to validity?

Example 19

medium
A TV rating samples 5000 households to represent millions. The 5000 are the ___?

Example 20

medium
A census measures EVERY person in a country. Is that a sample or the population?

Example 21

easy
A scientist wants to know the average weight of all redwood trees in California. She measures 80 trees. What is the population?

Example 22

easy
You ask 30 customers at a coffee shop their favorite drink. What is the sample size?

Example 23

easy
You read every page of a 300-page book to count words on each. Sample or population?

Example 24

easy
Which symbol denotes a sample mean?

Example 25

easy
A bakery makes 1000 cookies a day; a quality tester eats 5 at random. The 5 are a ___.

Example 26

medium
A college has 12,000 students. A survey of 400 finds 62% own a car. Which number is the parameter and which is the statistic?

Example 27

medium
To estimate the average bedtime of US 8th graders, a researcher polls students at one middle school in Texas. What is the population vs the sample, and is there a bias risk?

Example 28

medium
A sample of 200 has mean weight 68 kg; the population mean is unknown. Write the sample mean and the population mean using proper symbols.

Example 29

medium
A school's principal wants to know average daily steps of all 800 students. She uses every student's fitness-tracker data. Sample or census?

Example 30

medium
A national poll calls 1,000 random adults. What is the population, and what is the sample size?

Example 31

medium
A sample mean is xˉ=12.4\bar{x}=12.4 and the population mean is μ=12.0\mu=12.0. The difference xˉ−μ=0.4\bar{x}-\mu=0.4 is called what?

Example 32

medium
A magazine surveys readers via mail; 8% respond. Why is the resulting sample likely biased?

Example 33

medium
In a sample of 50, the mean test score is 78. Does this prove the population mean is 78?

Example 34

hard
You want the average height of all 9th graders in your district. You measure the volleyball team. Identify the bias.

Example 35

hard
A sample of 100 estimates a population proportion as p^=0.30\hat{p}=0.30. Which symbol denotes the unknown population proportion?

Example 36

hard
A random sample gives xˉ=84\bar{x}=84. A second random sample from the same population gives xˉ=86\bar{x}=86. Why aren't they equal?

Example 37

hard
Is a sample of 50 people from a population of 100,000 'too small'? Explain in one sentence.

Example 38

hard
In notation: sample size of a sample is denoted by nn. What letter usually denotes population size?

Example 39

hard
A sample mean of 200 students' heights is 165 cm. A friend says, 'So the average for all students nationally is exactly 165 cm.' What is wrong with that claim?

Example 40

challenge
A researcher claims 'My sample of 10,000 customers is so large that bias doesn't matter.' Why is this reasoning flawed?
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Background Knowledge

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

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