Use sample mean to estimate population mean.
Use a random sample mean as the best estimate for the population mean.
Core Idea
The mean of a random sample is the best single-number estimate for the mean of the entire population it was drawn from.
Understanding
When you can't measure everyone, you measure a subset and use that average as your estimate. If a researcher randomly selects 150 students from a university and finds their average study time is 14.2 hours per week, 14.2 hours is the estimate for the average across all students at that university.
This works because random sampling gives every member of the population an equal chance of being included. Without randomness — say, surveying only students in the library — the sample is biased and the mean won't reliably reflect the population.
The sample mean is only useful as an estimate when the sample is randomly selected from the population of interest. The SAT often tests whether you notice a non-random or mismatched sample.
Step by Step
- Confirm the sample was randomly selected from the population.
- Identify the sample mean from the given data.
- State that this sample mean is the estimate for the population mean.
- Check that the population described in the conclusion matches the population the sample was actually drawn from.
Misconceptions
- Thinking the sample mean equals the population mean exactly — it's an estimate, not an exact value.
- Ignoring how the sample was collected — a convenience sample (e.g., volunteers) doesn't reliably estimate the population mean.
- Generalizing beyond the population sampled — a sample of high school juniors can't estimate the mean for all high school students.
Worked Example
A researcher randomly selected 200 households in a city and found that the mean monthly electricity bill was $127. Which of the following is the best estimate of the mean monthly electricity bill for all households in the city?
Select an answer to see the explanation