Who Is the Question About?
In Writing a Statistical Question From a Context, you practiced naming the cases and the information recorded about them. The next step is to ask which cases the question wants to describe. Is it asking about every member of a defined group, or only about a subset of that group? That distinction helps identify whether the question concerns a population value or a sample description.
The words population and sample depend on the question’s scope. A population is the complete group of individuals the question is about. A sample is a subset of individuals taken from that population. A group of 10 packages, for example, could be the whole population if the question is about those 10 packages. The same 10 could be a sample if the question is about all packages produced during a shift.
For instance, a mean can be a parameter when it summarizes every member of the population named in the question. It can be a statistic when it summarizes a sample from a larger population. Likewise, a proportion can describe either a population or a sample, depending on the group being summarized.
A parameter is fixed for a particular population, even when its value is unknown. A statistic is calculated from observed sample data. If a different sample were selected, the statistic could be different. This is a preview of why sample statistics can help us learn about population parameters; the distinction itself does not tell us how accurate an estimate is.
Match the Question to Its Group
Do not decide that a number is a statistic merely because the data were collected by sampling. First read exactly what the question asks. If it asks for a summary of the observed sample, the answer is a statistic for that sample. If it asks for a summary of a larger population, a summary calculated from a subset is a statistic that may help address the question, while the population value is a parameter.
The group must be stated precisely enough to determine its membership. “Students at Northview High School this year” defines a different population from “students who completed the survey in Room 12 on Monday.” “Packages produced during the afternoon shift” is different from “the 10 packages checked.” Place, time, and any other boundary in the wording can change which individuals are included.
A data set can contain every individual in a small, clearly defined population. In that case, its summary is a population value for that group; no larger group should be assumed unless the question says so. If someone later wants to use the same data to describe a broader group, the observed summary becomes a statistic relative to that broader population. Always name the group before naming the number.
A Reliable Classification Process
Use the following steps whenever a question includes a count, mean, percentage, or other numerical summary. This is a way to read the question carefully, not a new calculation procedure.
Identify exactly who or what the question is about, including relevant place and time limits.
Decide whether data come from every member of the target group or only from a subset.
A number summarizing the whole target population is a parameter. A number summarizing a sample is a statistic.
Include the group in your explanation. If the wording allows two scopes, state both rather than guessing.
The process also helps separate two questions that sound similar: “What was the average among the people we surveyed?” and “What is the average among all people in the community?” The first asks for a summary of the surveyed group. The second asks about a broader population, even if only surveyed people are available in the data.
Worked Example: The Ten Packages Checked
A quality-control worker checks 10 packages from an afternoon production shift. Their weights, in grams, are 490, 495, 498, 499, 500, 501, 502, 505, 507, and 503. Consider two questions: “What was the mean weight of the 10 packages checked?” and “What was the mean weight of all packages produced during the afternoon shift?” Classify each target and calculate the observed mean.
State: For the first question, the target group is exactly the 10 packages checked. For the second, the target group is all packages produced during the afternoon shift. The checked packages are only a subset of that larger group.
Plan: Calculate the mean of the 10 observed weights. Then decide whether that mean summarizes the entire group named in each question or only a sample from it.
Do: The total of the 10 weights is \(5000\) grams. The sample-size count for these observed packages is \(10\), so their mean is
Conclude: For “the 10 packages checked,” 500 grams is the population mean for that defined group: all 10 packages included in the question have been measured. For “all packages produced during the afternoon shift,” the same 500 grams is a sample mean, or statistic, based on 10 packages. It is not automatically the mean of all packages in the shift.
The number did not change; the scope did. If the question is about the checked packages, those packages are the whole population for that question. If the question is about the entire shift’s production, they are a sample from a larger population. This is why a complete answer names the group the mean describes.
Sample Descriptions and Population Questions
A question can be about a sample without asking anyone to generalize beyond it. For example, “What proportion of surveyed residents chose the bus?” asks for a description of the surveyed residents. “What proportion of all residents in the district choose the bus?” asks about a population parameter. Data from a sample may be used to study that second question, but the target remains the full district population.
The same distinction applies to categorical variables. A proportion is a numerical summary: the number in a category divided by the group size. The calculation is the same whether the group is a sample or a population. Its role depends on the group summarized.
Worked Example: A Cafeteria Survey
A school has 400 students who ate lunch in the cafeteria on Thursday. A randomly selected group of 80 of them is asked whether they brought lunch from home; 48 say yes. Compare the question “What proportion of the 80 surveyed students brought lunch from home?” with “What proportion of the 400 Thursday cafeteria students brought lunch from home?”
State: The observed group is the 80 surveyed students. The first question asks about that exact group. The second asks about all 400 students in the defined Thursday cafeteria group.
Plan: Find the proportion of surveyed students who answered yes, then distinguish its role for each question.
Do: The sample proportion is
This is 60% of the 80 surveyed students.
Conclude: For the question about the 80 surveyed students, 0.60 is a population proportion for that narrowly defined group, because all 80 members of that group are represented in the calculation. For the question about all 400 Thursday cafeteria students, 0.60 is a sample statistic based on 80 of the 400 students. The population proportion for all 400 is a parameter; its value is not established just by calculating the sample proportion.
The word “survey” does not settle the classification. The first question describes everyone who responded. The second asks about a larger group, so the same observed proportion has a different role.
These examples also show why it is useful to separate the question from the available data. A question can ask about a population parameter even when the data consist of a sample. In that case, the sample statistic is the observed information, while the parameter is the population quantity of interest. Determining how to make a reliable inference from that statistic requires additional ideas that come later.
When the Same Data Have Two Roles
The population–sample distinction is always relative to a target group. A data set is not permanently a population or permanently a sample. It can be the complete group for one question and a subset for another. Make the scope explicit rather than relying on labels such as “survey data,” “class data,” or “the people we measured.”
Worked Example: Choosing a Club Activity
All 24 members of a school robotics club submit a vote for the club’s next activity. Seventeen choose a build-a-robot workshop. Consider the questions “What proportion of club members chose the workshop?” and “What proportion of all students at the school would choose the workshop?”
State: The observed group is the 24 robotics club members. For the first question, the target is those same 24 members. For the second, the target is the larger group of all students at the school.
Plan: Calculate the proportion of the 24 club members who chose the workshop. Then compare the target group in each question with the observed group.
Do: The proportion is
or about 70.83% when expressed as a percentage.
Conclude: For the question about the club members, 0.7083 is the population proportion for the 24-member club because every member voted. For the question about all students at the school, 0.7083 is a statistic describing the club members, not the population proportion for all students. The club’s votes alone do not give the value for the broader group.
Even if the club members were randomly chosen from the whole school—which they were not in this example—the group represented by the calculation would still be the 24 members. Whether a sample is suitable for learning about a particular population is a separate question from whether a number is a statistic or a parameter.
Common Mistakes and AP Exam Tips
- Calling every measured group a sample. If a question asks about exactly the 10 packages measured, those 10 are the full population for that question. Explain separately that they can be viewed as a sample from a larger production group.
- Calling a number a parameter just because it is a mean or proportion. A formula does not determine whether a value is a parameter or statistic. State the group the number summarizes.
- Assuming “population” always means everyone in a city, school, or country. A population is the whole group defined by the question. It can be as small as the members of one club or the 10 objects being described.
- Confusing the target with the observed data. A question can ask about all students even when a survey includes only some of them. Say that the population value is the target and the sample statistic is what the observed data provide.
- Treating an unknown parameter as if it does not exist. A parameter is a fixed population summary whether or not its value has been measured or is known.
- Leaving the scope implicit. A full-credit response identifies the group and classifies the value in relation to it. For example: “The mean is 500 grams for the 10 checked packages, so it is the population mean for that group; relative to all packages from the shift, it is a sample mean.”
When a question’s wording is ambiguous, do not force one answer without explanation. State the group that would make the value a parameter, and the broader group that would make the same value a statistic. Clear scope is more important than a label used without context.
Check Your Understanding
For each question, identify the target group and decide whether the requested summary is a population parameter or a sample statistic. Explain any ambiguity.
- A coach records the lap times of all 12 runners on the relay team and asks for the team’s mean lap time. What group does the mean describe?
- A randomly selected 60 of 300 households report how many bicycles they own. Is the mean number for the 60 households a parameter or statistic for a question about all 300 households?
- A question asks, “What proportion of the 18 volunteers chose the river cleanup?” All 18 volunteers responded. Classify the proportion for that question.
- A technician weighs 15 light bulbs from a day’s production. Explain how their mean weight can be a population value for one question and a statistic for another.
- Explain why “the survey percentage” is not enough information to decide whether a percentage is a parameter or a statistic. What group must you identify?