Two Ways to Gather Data About a Population
In Defining the Population of Interest, you set out who or what a statistical question is about. In Deciding Which Variables to Measure, you identified what information to record for each unit. Now consider a different planning question: should data be gathered from every member of that population, or from only some of its members?
A census is an attempt to collect data from every member of a defined population. A sample is a subset of that population from which data are collected. The choice affects how much time, money, staff, and access an investigation requires. Neither approach is automatically best; the practical situation and the question matter.
The word “seeks” matters. Planning a census means trying to include the whole population, but it does not guarantee that every member will respond or that every measurement will be recorded correctly. For example, a city might send a questionnaire to every household and still receive no answer from some households. A census plan describes whom the investigators intend to contact; it does not promise perfect information.
A sample can be useful because a smaller group generally requires fewer resources to contact and measure. However, it describes only the units actually included. How well the sample can help answer a question about the larger population depends on how it was selected and how the data were collected. Those details will be developed in later tutorials. For now, keep the basic distinction clear: a census aims to cover everyone in the population, while a sample covers a subset.
When Is a Census Practical?
A census may be a sensible choice when the population is small, the units are easy to reach, and collecting data from everyone is affordable. For instance, a school club could ask all 24 current members which meeting time they prefer. The group is small enough that contacting everyone may be manageable, and the club may want each member’s preference before setting a schedule.
A census can also be attractive when the goal is to make a decision about each individual unit, rather than simply to describe an overall pattern. If a community workshop must check every borrowed safety helmet before it is used, measuring only a subset would leave unchecked helmets. In that setting, the purpose calls for inspecting each item.
But a census becomes difficult when the population is large, spread out, expensive to reach, or hard to define and locate. It may also be too slow when a decision is needed quickly. Even when contacting every member is possible, the added effort may not be worthwhile if a smaller data collection can answer the question adequately.
- Size and access: How many population members are there, and how difficult are they to find or contact?
- Cost and staff: Can the available budget and personnel support contacting and measuring everyone?
- Time: Is there enough time to collect and check all the data before a decision is needed?
- Purpose: Does the question require information about each unit, or is a description of the overall population sufficient?
- Measurement: Would measuring every unit be possible without damaging, using up, or changing what is being measured?
These considerations are not a mechanical scorecard. A population’s size alone does not determine the answer. A large group may be easy to contact through existing records, while a smaller group may be difficult to reach. A good plan weighs the investigation’s goal against the real demands of collecting the data.
Why a Sample May Be More Feasible
A sample can reduce the number of units that need to be contacted, observed, or measured. This may lower costs and shorten data-collection time. It can also make a project possible when testing every unit is unsafe, destructive, or simply impractical. For example, a manufacturer testing how long a particular type of battery lasts cannot use every battery in the product line for a test that runs each battery until it stops working. The tested batteries would no longer be available for sale.
A smaller data collection does not mean the population should be described carelessly. The sample still needs to come from the population relevant to the question, and the measurement plan should still define the units, variables, and procedures. A sample of easily available volunteers, for instance, is not automatically a useful stand-in for everyone the question is about. Later tutorials will examine ways to select samples and assess the conclusions they support.
The goal is not necessarily to gather as many observations as possible. It is to gather information that is suitable for the question within the available constraints. A sample that is too difficult or costly to collect may not be practical; a census that cannot be completed may be less useful than a feasible, well-planned sample.
Worked Example: Choosing for a Small Club
Worked Example: Choosing for a Small Club
A neighborhood photography club has 24 active members. Its organizers want to choose between two meeting times for the next month. They plan to ask each person which time they prefer, using the same two options for everyone.
Consider a census. The population of interest is the 24 active club members. Asking all of them would be a census because the organizers would seek a response from every member. Suppose each message costs $0.20 to send and each member needs one message. The direct messaging cost would be \(24 \times \$0.20=\$4.80\). If the club can contact all 24 members easily and has time to wait for replies, a census is feasible.
Consider a sample. If the organizers asked 8 of the 24 members, they would collect data from a sample. That would require fewer messages: \(8 \times \$0.20=\$1.60\), saving \(\$4.80-\$1.60=\$3.20\). But the savings are small, and the club may want every member’s preference to set a schedule that affects everyone.
Conclusion. Given the small population, easy access, and modest cost, asking all 24 members is a reasonable choice. The census plan still needs a way to record nonresponses; sending a message to all members does not guarantee that all 24 will reply.
Worked Example: A Citywide Question Under a Deadline
Worked Example: A Citywide Question Under a Deadline
A city transportation office wants to describe how residents rate the clarity of signs at bus stops. Its defined population is 18,000 residents who used a city bus during a specified month. The office has 10 staff members and two weeks for data collection. Each completed interview takes about 6 minutes, and the office estimates that one staff member can complete 50 interviews per workday.
Estimate the census workload. If each of the 18,000 residents could be contacted and interviewed, the total interview time would be \(18{,}000 \times 6=108{,}000\) minutes. Since \(108{,}000 \div 60=1{,}800\) hours, this is 1,800 hours of interview time alone. It does not include time to locate residents, arrange interviews, check records, or handle people who do not respond.
Estimate the practical capacity. Over 10 workdays, 10 staff members completing 50 interviews per day could complete \(10 \times 10 \times 50=5{,}000\) interviews. That capacity is much smaller than the 18,000 interviews a census would require. The office should not call a plan to interview at most 5,000 residents a census of all 18,000.
Conclusion. A sample is more feasible under the stated staffing and deadline. The office must still plan carefully how it will select residents and carry out interviews; the workload calculation alone does not establish that any particular sample will represent the population well. It does show why attempting to interview everyone is unrealistic for this investigation.
Worked Example: When Testing Uses Up the Item
Worked Example: When Testing Uses Up the Item
A small manufacturer produces 4,000 rechargeable lantern batteries in a production run. Engineers want to measure how many hours a battery operates before it no longer holds a charge. Their test runs each battery until it fails, and a tested battery cannot be sold afterward.
Compare the data-collection plans. A census would test all 4,000 batteries. This would use up \(4{,}000\) saleable batteries in testing. If each battery costs the manufacturer $12 to produce, the production cost tied up in the tested batteries would be \(4{,}000 \times \$12=\$48{,}000\). This calculation does not include staff time or equipment.
Suppose the engineers instead test 40 batteries. The production cost of those tested units would be \(40 \times \$12=\$480\). The 40 batteries form a sample from the production run, and testing only that subset preserves the other \(4{,}000-40=3{,}960\) batteries for sale.
Conclusion. Because the measurement process uses up each tested battery, a census has a substantial practical cost. A sample is the workable approach for this question. The engineers should be careful not to claim that the 40 tested batteries are exactly like every battery in the run without considering how those units were chosen; that selection issue affects how useful the sample is for describing the full production run.
A Census Is Not Automatically Error-Free
It is tempting to assume that a census must be more accurate than a sample because it includes everyone. A census does avoid the particular limitation of observing only a subset, but other problems can affect either approach. Some population members may be missed, some may not respond, and measurements may be recorded inconsistently or inaccurately.
For example, a school that asks every student to report how many minutes they spent reading yesterday has attempted a census if its population is all enrolled students. If some students do not answer, the collected data do not include everyone. If students interpret “reading” differently, even completed responses may not be comparable. Clear variable definitions, consistent questions, and careful data recording matter whether the plan is a census or a sample.
Likewise, a sample is not necessarily poor merely because it excludes many members of the population. Its usefulness depends on the question and on how the sample is selected and measured. Do not claim that a sample represents the population just because it contains several observations, and do not claim that a census has no limitations merely because it aims to include everyone.
Common Mistakes and AP Exam Tips
- Calling any large data collection a census. A census is defined by its goal of collecting data from every member of the specified population, not by having a large number of observations. State the population and whether all of its members are included.
- Confusing the population with the sample. The population is the full group the question concerns; the sample is the subset from which data are collected. Name both groups precisely when describing an investigation.
- Assuming that contacting everyone guarantees a complete census. A full-credit explanation distinguishes the attempt to contact all members from the responses actually obtained. Nonresponse can leave gaps even in a census plan.
- Choosing a census just because it sounds more accurate. Mention practical constraints such as cost, time, access, or destructive testing. A census can still have missing responses or measurement problems.
- Choosing a sample only because it is smaller. A sample may save resources, but the way its members are selected affects whether it can help answer a question about the population. Do not assume that a convenient subset is automatically informative.
- Ignoring what the investigation needs to decide. If the goal requires information about every individual unit, a sample may not meet that goal. If the goal is to describe a large population and a census is infeasible, a sample may be the practical option.
For a strong AP response, identify the population, state whether the plan collects data from everyone or a subset, and connect the choice to the situation. Explain relevant trade-offs rather than saying only that one option is “better.” For example: “A sample is more feasible because interviewing all 18,000 residents would require far more time than the office has; the sample would still need to be selected appropriately for the population of interest.”
Check Your Understanding
For each situation, identify whether a census or a sample is proposed and explain one practical consideration that affects the choice.
- A school with 36 members in its debate team asks each member which day works for practice. Is this a census or a sample, and what might make it feasible?
- A regional clinic wants to learn about appointment experiences among 25,000 patients but can conduct only 300 phone interviews this month. Which approach is feasible under the stated limit, and what does the interview count alone not tell you?
- A lab tests every light bulb in a batch by operating each one until it burns out. Explain why that census may be impractical.
- A town sends a questionnaire to every household, but some households do not return it. Explain why sending the questionnaire to everyone does not guarantee data from everyone.
- Give one reason a census may still have data-collection problems, even though it aims to include the whole population.