Populations, Samples, and Representative Sampling

12.1: Populations, Samples, and Representative Sampling. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 63 parts Practice Flashcards

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Population and sample

Objective: Distinguish the target group from observed participants.

Prerequisite skills: percentages and sample vocabulary. A population is the full group the study seeks to describe; a sample is the subset actually measured. A statistic summarizes a sample, while a parameter describes a population. Clear definitions prevent unjustified extrapolation. Common error: calling the surveyed subset the entire target population. Restrictions and interpretation: A large sample does not repair a systematically biased selection method.

Formulas

sample ⊆ target population

Worked examples

Try itA district wants the mean commute time of all 4100 students. It surveys 120 students. What is the target population?Show answer

The question concerns every district student, so they form the population. The 120 respondents are the sample. Neither a subset defined by commute length nor a larger state group matches the target. Therefore the requested answer is All students in the district.

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Population and sample

Objective: Distinguish the target group from observed participants.

Formulas

sample ⊆ target population

Worked examples

Try itA district wants the mean commute time of all 4100 students. It surveys 120 students. What is the target population?Show answer

Representative selection

Objective: Choose a method that gives the target group a fair opportunity for inclusion.

A random sample selected from a complete population roster reduces selection bias. Convenience samples and voluntary online polls can systematically overrepresent particular groups. Representation depends on selection and nonresponse, not only size. Common error: assuming a large volunteer sample must represent everyone. Restrictions and interpretation: A large sample does not repair a systematically biased selection method.

Formulas

random sampling supports population generalization

Worked examples

Try itA college wants to estimate all students’ library use. Which plan best supports a representative sample of its 5200 students?Show answer

A random sample from the complete roster gives students with differing library habits an opportunity for selection. The other methods systematically favor library users or particular schedules. Therefore the requested answer is Randomly select student IDs from the complete enrollment roster.

Sample size and bias

Objective: Explain why selection method matters even for large samples.

Sampling variation arises from chance differences among random samples; selection bias is a systematic distortion in who is included. Increasing a biased sample’s size can make it more precise around the wrong target without removing the bias. Common error: claiming that a huge biased survey becomes unbiased. Restrictions and interpretation: A large sample does not repair a systematically biased selection method. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

size reduces random variation, not systematic selection bias

Worked examples

Try itA city collects 33000 responses from a cycling-club mailing list to estimate all residents’ cycling habits. What is the main concern?Show answer

The mailing list overrepresents cycling enthusiasts. Its large size does not fix the selection bias. Random assignment concerns treatments, whereas random sampling concerns who is included. Therefore the requested answer is Club members may differ systematically from all residents.

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