Population vs Sample Examples in Statistics

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 Statistics.

Concept Recap

In statistics, the population is the entire group of individuals or items you want to study, while the sample is the smaller subset you actually collect data from. We use sample statistics to estimate unknown population parameters.

You want to know the average height of ALL teenagers in your country (population), but you can't measure everyone. So you measure 1000 teenagers (sample) and use that to estimate the whole.

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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: Population vs Sample starts by matching the sample and collection method to the population named in the question.

Common stuck point: Students often know a procedure related to population vs sample but skip the recognition step: Do I know the population, the sample, and the method used to choose or measure the cases? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Do I know the population, the sample, and the method used to choose or measure the cases?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Sampling Bias Mistakes

Worked Examples

Example 1

medium
A study reports: 'Among the 250250 patients in our trial, average blood pressure dropped 88 mmHg.' Identify (a) the population, (b) the sample, (c) the statistic.

Answer

(a) all patients of the type studied; (b) the 250250 patients in the trial; (c) the average drop of 88 mmHg.

First step

1
The population is the broader group the study wants to draw conclusions about (similar patients).

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Example 2

medium
A factory makes 50,00050{,}000 lightbulbs per day. Quality control tests 200200 of them and finds 44 defective. Identify the statistic and what it estimates.

Example 3

medium
A researcher writes μ=5.2\mu = 5.2 and xˉ=5.1\bar{x} = 5.1. Explain in one sentence what each symbol stands for.

Example 4

hard
A poll uses random digit dialing of landline phones. They want to estimate the proportion of all U.S. adults who support a policy. What is the population, and what is the sampling frame, and why are they different?

Example 5

hard
An online survey asks volunteers to rate a movie. 5,0005{,}000 people respond. Why is this a sample of all moviegoers, but a biased one?

Example 6

challenge
A drug-trial enrolls 500500 volunteers from one city. The researchers report 'the drug reduces blood pressure by 66 mmHg on average.' Discuss the scope of inference: to whom can this statistic legitimately generalize?

Example 7

medium
A teacher wants to estimate the mean math score for all 12001200 students in her district. She samples 8080 students, finds xˉ=74\bar{x} = 74. State the parameter, statistic, and population/sample sizes.

Example 8

hard
A school has 500500 seniors. The school polls 5050 randomly chosen seniors and finds 3535 plan to attend college. Estimate the population proportion and explain why 35/5035/50 is the best point estimate.

Example 9

challenge
A scientist samples 2020 trees from a forest of 50005000 and measures heights with mean xˉ=14.2\bar{x} = 14.2 m. She wants to claim the forest's mean tree height. List two assumptions that must hold for xˉ\bar{x} to be a credible estimate of the population μ\mu.

Example 10

easy
A researcher wants to know the average height of all 16-year-olds in the UK. She measures 500 randomly selected 16-year-olds. Identify the population and sample.

Example 11

medium
Distinguish between a parameter and a statistic. Give an example of each.

Practice Problems

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

Example 1

easy
'We want average height of all 8th graders in the district; we measured 150 of them.' What is the population?

Example 2

easy
In the same study, what is the sample?

Example 3

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

Example 4

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

Example 5

easy
True or false: in most studies we can measure the entire population directly.

Example 6

easy
A company has 5,000 employees. HR surveys 250. Which number is the sample size?

Example 7

easy
'42% of surveyed voters support the measure.' Is 42% a statistic or a parameter?

Example 8

easy
Why can a well-chosen sample of 1,000 represent a population of millions?

Example 9

medium
A researcher reports the mean test score of the 200 students she sampled as 78 and says 'the population mean is exactly 78.' What error did she make?

Example 10

medium
Two studies estimate the same population mean: study A samples 100, study B samples 2,500 (both random). Which sample mean is expected to be closer to the true mean, and why?

Example 11

medium
A magazine surveys its own subscribers to estimate the reading habits of 'all adults in the country.' Identify the population the survey can actually describe versus the population claimed.

Example 12

medium
In a quality check, 30 of 6,000 light bulbs are tested and 2 are defective. Estimate the population defect rate and state whether it is a statistic or parameter.

Example 13

medium
A scientist measures every fish in one isolated pond to study that pond's fish. Is this a sample or a census of that population?

Example 14

medium
A poll of 1,500 randomly chosen adults finds 60% favor a policy, reported with a margin of error. Why is a margin of error needed at all if the sampling was random?

Example 15

medium
A teacher claims her 30-student class average of 85 proves 'all students in the school average 85.' Identify the population, the sample, and the flaw.

Example 16

medium
A national survey wants to estimate the proportion of left-handed people. It samples 2,000 people randomly and finds 220 left-handed. Express both the sample proportion (statistic) and explain what unknown it estimates.

Example 17

challenge
A researcher samples 500 from a population and computes a mean of 50. She then realizes 100 of those 500 were accidentally measured twice (so really only 400 distinct people, 100 counted twice). Does the duplication change which group is the population? What is the true sample size of distinct individuals?

Example 18

challenge
To estimate the mean of a population of 10,000, a researcher can either census all 10,000 or randomly sample 1,000. The census has data-entry errors on 5% of records; the random sample is entered carefully with no errors. Argue which may give a more accurate estimate of the true mean.

Example 19

challenge
A population has exactly two values: 1,000 people scored 0 and 1,000 people scored 100, so the parameter mean is 50. A random sample of size 2 is drawn. List the possible sample means and explain why the sample mean is still an unbiased estimator of 50.

Example 20

medium
A factory samples 50 of 10,000 bolts and finds the mean length is 20.1 mm. Is 20.1 mm a statistic or a parameter, and what is it estimating?

Example 21

easy
A polling firm wants to know the opinion of all 12,00012{,}000 employees at a company. They survey 400400 of them. What is the population?

Example 22

easy
In the same study (polling 400 of 12,000 employees), what is the sample?

Example 23

easy
A biologist wants to estimate the average weight of all salmon in a river. She catches 8080 salmon and weighs them. Identify the sample size nn.

Example 24

medium
A school has 1,5001{,}500 students. The principal asks 6060 of them about lunch options. The proportion who prefer salad in those 6060 is 0.450.45. Is 0.450.45 a parameter or a statistic?

Example 25

medium
A city has 2,0002{,}000 small businesses. A researcher contacts every single one of them and records revenue. What kind of study is this?

Example 26

medium
Which symbol denotes the population standard deviation, and which denotes the sample standard deviation?

Example 27

medium
A reporter claims '62%62\% of Americans support the bill' based on a survey of 1,2001{,}200 adults. Is 62%62\% the parameter pp or the statistic p^\hat{p}?

Example 28

medium
A teacher wants to know the average homework time for all 300300 kids in the grade. He only asks his own class of 2828 students. Why might his sample be biased?

Example 29

medium
A population has true mean μ=72\mu = 72. Two random samples of size n=30n = 30 give sample means xˉ1=70.5\bar{x}_1 = 70.5 and xˉ2=73.8\bar{x}_2 = 73.8. What is the name for the natural difference between samples?

Example 30

hard
A study of teen sleep at one high school concludes 'teens get 7.27.2 hours of sleep on average.' Why might this generalization be misleading?

Example 31

hard
A study measures every patient in a hospital ward (n = 40) and reports their mean recovery time. Is this μ\mu or xˉ\bar{x}?

Example 32

hard
Two researchers report p^1=0.40\hat{p}_1 = 0.40 from n1=50n_1 = 50 and p^2=0.42\hat{p}_2 = 0.42 from n2=2000n_2 = 2000. Which statistic is likely to be closer to the true parameter pp? Why?

Example 33

medium
A study writes: 'population proportion p=0.30p = 0.30'. Is this realistic in practice?

Example 34

easy
A scientist tags 200200 deer in a forest and later samples 5050 deer, finding 1010 tagged. The full deer count in that forest is unknown. Identify the population.

Example 35

easy
A wildlife biologist wants to estimate the average weight of all deer in a national park; she catches and weighs 4040 of them. What is the population?

Example 36

easy
Which of these is a parameter: the average height of all 10001000 students in a school, or the average height of 5050 surveyed students?

Example 37

easy
A toy company tests 2525 randomly chosen toys from a production run of 40004000. What is the sample size?

Example 38

easy
Identify whether each is a population or a sample: 'all flights at JFK on Jan 1, 2025' and '50 flights pulled from JFK records on Jan 1, 2025.'

Example 39

easy
A company has 80008000 customers. A satisfaction survey reaches 600600 of them. Identify the sampling fraction.

Example 40

medium
A market researcher writes: 'The mean satisfaction in our 300300-person sample is 7.27.2, so the mean satisfaction of all 40,00040{,}000 customers is 7.27.2.' Diagnose the issue.

Example 41

medium
A census measures every person in a country. Is the resulting mean a parameter or a statistic?

Example 42

medium
True or false: in general, larger samples give more accurate estimates of population parameters, all else equal.

Example 43

medium
An online news site polls its own readers about national policy. What is the population the poll can credibly describe, and what is the population the headline likely claims?

Example 44

medium
A sample of 5050 from a population of 10,00010{,}000 has mean xˉ=12\bar{x} = 12 and SD s=3s = 3. Which of these are statistics: 1212, 33, 5050, 10,00010{,}000?

Example 45

medium
Why is it dangerous to confuse a sample statistic with a population parameter when making business decisions?

Example 46

medium
A pollster says, 'Among the 1,2001{,}200 sampled voters, 58%58\% support Proposition A.' Is 58%58\% a parameter or a statistic? What does it estimate?

Example 47

hard
Two factories make light bulbs. Factory A has a sample mean lifetime xˉA=990\bar{x}_A = 990 hr from n=100n=100. Factory B has xˉB=1010\bar{x}_B = 1010 hr from n=20n=20. Can we be confident Factory B's bulbs last longer on average? Explain briefly.

Example 48

hard
A national survey samples 1,0001{,}000 adults; sample mean income is $58,000\$58{,}000. The TRUE population mean is $60,000\$60{,}000. Identify xˉ\bar{x}, μ\mu, and the sampling error of this estimate.

Example 49

hard
Why can a representative sample of 10001000 adults estimate national opinion roughly as well in a country of 1010 million as in a country of 100100 million?

Example 50

hard
In a class of 4040 students, the teacher measures every student's grade. Is computing the mean of those 4040 grades estimating a parameter or computing one?

Example 51

hard
A journalist writes: 'We polled 300300 tech workers; the mean salary in tech is $140,000\$140{,}000.' Critique the wording from a population-vs-sample standpoint.

Example 52

challenge
Suppose μ\mu is unknown and we draw a simple random sample of size nn. The sample mean Xˉ\bar{X} has expectation E[Xˉ]=μE[\bar{X}] = \mu. What does this property say about Xˉ\bar{X}?

Example 53

challenge
A national exam is taken by all eligible students; districts publish district-level mean scores. Can these district means be considered parameters for the district populations and statistics for the national population?

Example 54

easy
A factory produces 10,000 light bulbs per day. Quality control tests 100 randomly chosen bulbs. Identify the population, sample, and explain why sampling is used.

Example 55

easy
A website wants to know the average time spent on the site by all visitors. It studies 1,200 randomly selected visits from last month. Identify the population, the sample, and the parameter of interest.
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Background Knowledge

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

data collectionstat sample space