Populations and Samples
Statistics through experiments · 5 of 16 · CC BY-SA 4.0
You cannot measure everyone. A carefully chosen handful can still tell you something.
A small town wants to know the average height of its 200 adult residents. Measuring everyone would settle it, but suppose you only have time for ten people.
The town is our population: the whole group we want to describe. The ten people we measure form a sample. Choose a sample below, then use its mean as your best guess before revealing the town.
A fictional town of 200 adults
Sample without replacement: nobody is measured twice in a single sample.
Each dot is a resident. Filled dots mark the town-wide sample.
No sample yet. What do you think the town’s average height will be?
One number to find, another to calculate
The town has a definite average height before we measure anyone. That population mean is a parameter. A parameter describes the population, even when we do not know its value.
The mean of the people we actually sampled is a statistic. A statistic is calculated from observed data. When we use it to guess the parameter, it acts as an estimate.
Reveal the town, then draw a few more samples of the same size. The statistic moves. The parameter stays put. That is not a calculation going wrong; different people have different heights.
A moment to think
What does a larger handful buy us?
Compare several samples of five with several samples of 100. Larger random samples generally give steadier estimates. An individual small sample can still land closer to the truth than an individual large one.
The experiment samples without replacement: once someone is selected, they cannot appear again in that sample. At 200, you have measured the whole town. This is a census, and sampling uncertainty disappears for this population.
Even a census can have measurement errors or missing answers. Here the measurements are exact and everyone responds, so we can concentrate on the uncertainty caused by choosing a sample.
A relationship to remember
The estimate points toward a target
Population (μ) → random sample → estimate (x̄)
μ, “mu,” names the population mean. x̄, “x-bar,” is the sample mean you calculated. Drawing another sample can change x̄ without changing μ.
The arrow matters: an estimate can tell us about the population only when the way we obtained the sample supports that inference.
Return to the experiment ↑A moment to think
Words you met
| Term | Meaning |
|---|---|
| Population | The complete group your question concerns. |
| Sample | The cases selected from that population. |
| Parameter | A numerical property of the population. |
| Statistic | A quantity calculated from sample data. |
| Census | Measuring every member of the population. |
Neighbors
- For more on study design, see Introduction to Data. For estimates, see Foundations for Inference.
- For the counting behind choosing a sample without replacement, see Combinatorics in the probability textbook. Combinations count the different groups we could select.
Written by June Kim. The conversational approach was inspired by Danielle Navarro’s Learning Statistics with R (CC BY-SA 4.0). The prose, experiments, and questions here were created for this book. This chapter’s text and illustrations are also shared under CC BY-SA 4.0.