Introducing Statistics: Questions, Samples, and Context

Statistics begins with a question about a group. The useful question is seldom “What numbers can I calculate?” It is “What do these numbers help us learn?” A collection of measurements becomes a statistical study when we connect the data we can collect to the people or things we want to understand.

Population, Sample, and Context

Suppose Urbanopolis wants to improve the trip to school. The district has 2,400 students, but interviewing every student would require time, staff, and repeated attempts to reach absent students. A population consists of all the items or individuals of interest: here, all 2,400 students currently enrolled in the district. We use $N$ for population size. A sample is the subset from which data are actually obtained. If 120 selected students report their usual travel time, the sample size is $n=120$.

A datum is one piece of information about one item or individual. “Student 037 usually travels for 18 minutes” supplies one travel-time datum. The 120 reported travel times together form a data set. Students who were not selected still belong to the population; having no recorded travel time does not remove them from the group we want to understand.

In a statistical study, we collect data from a sample to answer an investigative question about a larger population. Sampling is useful when the population is too large to measure completely or when contacting every item is too difficult. Sampling is not simply taking fewer observations and forgetting the rest of the group. The connection from sample to population must remain visible, and the way the sample was selected affects how convincing that connection will be.

Study component	Urbanopolis example	What it tells us
Population	All 2,400 currently enrolled students	The group the district wants to understand
Sample	The 120 students from whom travel times are obtained	The group represented in this data set
Datum	One student’s usual travel time, in minutes	One piece of recorded information
Data set	All 120 recorded travel times	The observations available for analysis
Result	66 of the 120 report more than 20 minutes	A description of this particular sample

The last row gives a sample percentage of 55%. In context means connecting that calculation to its real-world meaning: 55% of the sampled students reported a usual school trip longer than 20 minutes. It does not mean 55% of students everywhere have a long trip. A formula, an individual observation, and a conclusion each need an identifiable counterpart in the study. Keep the population, time period, variable, and units alongside the numbers.

Worked Example: A Different Population

A cafeteria manager samples 40 lunches served on Tuesday and records the amount of uneaten food in grams. If the manager’s question concerns all 600 lunches served on that Tuesday, then the population is those 600 lunches, not all students in the school. A lunch is the item selected, 40 is the sample size, and one measured leftover mass is a datum. If the sample mean is 72 grams, the contextual statement is “The 40 sampled Tuesday lunches had an average of 72 grams of uneaten food.” Whether this describes other days requires a different study connection.

A Question Before the Results

An investigative question states a defined purpose for a specific study. “Among currently enrolled Urbanopolis students, what proportion usually takes more than 20 minutes to reach school?” specifies a group, a measurable response, and a quantity to estimate. We can ask each selected student about the same kind of trip, record minutes, classify trips relative to 20 minutes, and calculate a sample proportion.

Compare that with “Is school travel bad?” The purpose is unclear: bad could mean slow, expensive, dangerous, or unpleasant. A useful revision names what we will measure and why. A question must also be feasible. A plan to ask students about their usual trip this week is collectable; a plan to learn every student’s exact travel time on every school day ten years ago may require records that do not exist. Before collecting data, explain what information is needed and how that information can answer the question.

The question should not be changed because the results happen to be disappointing. If a team originally studies trips longer than 20 minutes and later switches to 35 minutes because that threshold creates a more striking headline, the new purpose was selected after seeing the data. An unexpected pattern can motivate a clearly labeled new investigation. It cannot retroactively become the original question.

Worked Example: Planning a Library Study

The library wants to decide whether to extend Saturday hours. A feasible question is “For residents who used the Urbanopolis library in the past month, what proportion would use it after 5 p.m. on Saturday?” The intended quantity and population are clear, and a survey can collect relevant responses. This question concerns stated intentions, so the eventual conclusion should describe intentions, not claim to know exactly how many people will visit. Keeping the purpose fixed also helps the team choose its sampling plan and analyze every response rather than only favorable ones.

Independent Practice

Use the information in each question to make your own calculation, representation, or argument. Explain what your result means in its setting.

Original generated practice for the Fall 2026 course.

Who belongs to the study?

A museum has 8,000 annual members. It selects 160 members and obtains a preferred-opening-time response from every selected member.

Part a

Identify the population, sample, N, and n. Explain why a member not selected still belongs to the population.

Your reasoning

Worked answer and rubric

One record or many?

For each of 160 museum members, a staff member records an opening-time preference. One entry says “10 a.m.”.

Part a

Identify a datum and the data set. State whose information the datum describes.

Your reasoning

Worked answer and rubric

Why collect a sample?

A wildlife team wants information about 40,000 trees spread over a mountain region. Measuring every tree would exceed the season’s fieldwork budget.

Part a

Explain why a sample-based statistical study is useful. Identify the larger group the study is meant to describe.

Your reasoning

Worked answer and rubric

Keep the question honest

A team asks whether more than half of library users prefer Saturday opening after 5 p.m. After seeing that only 38% do, it changes its headline question to whether at least one-third prefer those hours.

Part a

Explain the problem with treating the changed question as the original purpose. Describe a transparent way to investigate the new question.

Your reasoning

Worked answer and rubric

Plan the garden decision

Urbanopolis has 900 community-garden plot holders. The council can survey 90 of them. It wants to decide whether to install storage lockers next spring. Existing records identify current plot holders but do not record their equipment-storage needs.

Part a

Construct one specific investigative question that would help the locker decision and can be answered yes or no by each current plot holder. Define the intended-use condition and time period clearly. Name the population, the response you would collect, and the quantity you would analyze.

Your reasoning

Part b

Identify N, n, one possible datum, and the complete data set for your planned study. Explain which people are the sample.

Your reasoning

Part c

Explain why surveying the 90 can form a statistical study about the 900. Give two practical reasons the council might not contact all 900.

Your reasoning

Part d

Suppose 54 of the 90 sampled holders answer yes to your binary question in part (a). Calculate the sample percentage and state exactly what it means using your own intended-use condition and time period. Name a population claim the calculation alone does not establish.

Your reasoning

Part e

Before collecting responses, outline how the council could collect and analyze data for the exact binary question you constructed. Keep your chosen yes/no condition and time period consistent. Explain what to do if an unexpected pattern suggests a different question.

Your reasoning

Worked answer and rubric