In statistics, a population is the entire group you want to learn about, while a sample is the smaller group you actually measure. A parameter is a number that describes a population, such as the true mean height of all students in a school. A statistic is a number computed from a sample, such as the mean height of 50 selected students.
This distinction matters because most real populations are too large, costly, or impossible to measure completely.
We use sampling to collect data from part of the population and then use sample statistics to estimate population parameters. For example, the sample mean x̄ can estimate the population mean μ, and the sample proportion p̂ can estimate the population proportion p. Different samples usually give different statistics, so estimates have uncertainty.
Good sampling methods reduce bias and make statistics more reliable for drawing conclusions about the population.
Key Facts
- Parameter: a numerical value that describes a population, such as μ, σ, or p.
- Statistic: a numerical value computed from a sample, such as x̄, s, or p̂.
- Population mean: μ = sum of all population values / N.
- Sample mean: x̄ = sum of sample values / n.
- Sample proportion: p̂ = number of successes in sample / n.
- Statistics estimate parameters, so x̄ estimates μ and p̂ estimates p.
Vocabulary
- Population
- The complete group of individuals, objects, or measurements that a study is about.
- Sample
- A smaller subset selected from a population to collect data from.
- Parameter
- A fixed numerical value that describes a characteristic of an entire population.
- Statistic
- A numerical value calculated from sample data.
- Sampling variability
- The natural change in a statistic from one random sample to another.
Common Mistakes to Avoid
- Calling x̄ a parameter is wrong because x̄ is calculated from a sample, so it is a statistic.
- Calling μ a statistic is wrong because μ describes the entire population, even if its value is unknown.
- Assuming one sample statistic equals the exact parameter is wrong because samples vary and estimates usually contain sampling error.
- Using a biased sample is wrong because a statistic from an unrepresentative sample may give a poor estimate of the population parameter.
Practice Questions
- 1 A school has 1,200 students, and a random sample of 60 students has an average height of 167 cm. Identify the population, sample, parameter being estimated, and statistic calculated.
- 2 In a sample of 200 voters, 118 support a proposal. Calculate the sample proportion p̂ and state which population parameter it estimates.
- 3 A researcher surveys only people leaving a gym to estimate the average weekly exercise time for an entire city. Explain why the sample may be biased and how that affects the statistic as an estimate of the parameter.