A study often measures part of a large group and uses those results to learn about the whole group. Reliable inference begins with a clearly defined population and a sample selected by a method that avoids systematic favoritism.
Population, census, and sample
- Population
- The complete group the study aims to describe.
- Census
- Data collected from every member of the population.
- Sample
- A subset of the population from which data are actually collected.
- Define the population
State exactly which people or objects the conclusion is meant to describe.
- Select a random sample
Use a chance-based method that does not systematically favor part of the population.
- Measure the sample
Calculate a sample statistic, such as a proportion or mean.
- Estimate with caution
Use the sample result to estimate the population while acknowledging sampling variability.
Random selection and representativeness
| Method | What happens | Main concern |
|---|---|---|
| Simple random selection | A chance mechanism selects from a complete population list | Best supported here when properly carried out |
| Convenience sample | Researchers choose the easiest members to reach | May not represent harder-to-reach members |
| Voluntary response | People choose whether to participate | Strong opinions may be overrepresented |
| Single-location sample | Everyone is selected from one place or time | The location or time may differ from the population |
Identify a biased method
A city wants residents’ views about weekend bus service and surveys only riders waiting at the central station on Saturday morning.
- Population
All city residents whose views are meant to be described.
- Sample
Saturday-morning riders at one central station.
- Concern
The sample systematically favors current transit users at one time and place.
Estimate a population proportion or count
Here \(x\) is the number of sampled successes and \(n\) is sample size.
Multiply the sample proportion \(\hat{p}\) by population size \(N\). This is an estimate, not an exact census count.
Scale a random-sample result
In a random sample of \(240\) students, \(66\) prefer a later start time. The school has \(1{,}600\) students.
- Sample proportion
\(\hat{p}=66/240=0.275\).
- Scale
\(0.275(1600)=440\).
- Interpret
The result estimates, rather than counts exactly, how many students prefer the change.
Sample size and sampling variability
- Different random samples from the same population usually produce slightly different statistics; this is sampling variability.
- A larger random sample generally gives a more stable estimate than a smaller random sample selected by the same sound method.
- Reliability depends more directly on the random sample size than on the sample's percentage of a very large population at this source level.
- A sample of the same size and selection quality can have broadly similar reliability for two large populations, even when their total sizes differ greatly.
- No sample estimate becomes an exact population fact merely because the arithmetic is precise.
Random variability versus bias
Sampling variability
Chance differences among properly selected random samples; generally reduced by a larger random sample.
Selection bias
Systematic overrepresentation or underrepresentation caused by the method; not repaired by sample size alone.
Common sampling mistakes
- Confusing the population with the people actually surveyed.
- Calling a convenience sample random because the researcher did not choose specific names.
- Assuming a larger sample automatically eliminates bias.
- Scaling a sample count without first converting it to a proportion.
- Reporting a sample-based estimate as an exact population count.
- Generalizing to a broader population than the selection process represents.
- Assuming identical sample percentages guarantee identical reliability.
- Ignoring nonresponse or undercoverage when evaluating a method.
Check your understanding
A random sample of \(150\) customers finds that \(42\) use a feature. About how many of \(2{,}500\) customers would be estimated to use it?
- \(280\)
- \(600\)
- \(700\)
- \(1{,}050\)
Show answer and explanation
Answer: \(700\)
The sample proportion is \(42/150=0.28\). Scale it: \(0.28(2500)=700\).
What to remember
- Define the target population before judging a sample.
- Random selection supports representativeness; convenience and voluntary-response methods create concerns.
- A sample statistic estimates a population parameter.
- Use \(\hat{p}=x/n\) and estimated count \(=\hat{p}N\).
- Larger random samples reduce sampling variability but do not fix systematic bias.
- Generalize only to a population the selection method can reasonably represent.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.