Population (Statistics)
Statistics & ProbabilityIn statistics, a population is the complete set of all individuals or items that a study aims to describe.
Definition
A population is the entire group of people, animals, or things that you want to learn about in a study. It includes every single member of that group.
Example
If a researcher wants to know the average height of all 7th graders in the United States, the population is every 7th grader in the country.
Key Insight
Populations can be huge (all people on Earth) or small (the $30$ students in your class). What makes it a population is that it is the complete group you care about.
Definition
A statistical population is the entire collection of individuals or measurements about which conclusions are desired. Populations are described by parameters. Because studying a full population is often impractical, researchers study samples instead.
Example
A company wants to know the average satisfaction rating of all $50{,}000$ of its customers (the population). Surveying all $50{,}000$ is too costly, so a sample is taken.
Key Insight
Parameters describe the population and are usually unknown. The goal of inferential statistics is to estimate those parameters from sample statistics.
Definition
A population is the set of all possible realizations of a random variable $X$ with distribution $F(x; \theta)$. The population is characterized by unknown parameters $\theta$ that belong to a parameter space. Inference involves estimating $\theta$ or testing hypotheses about it.
Example
If heights are normally distributed across a population with mean $\mu$ and variance $\sigma^2$, both $\mu$ and $\sigma^2$ are population parameters. A sample of size $n$ yields estimates $\bar{x}$ and $s^2$ that converge to these parameters as $n$ grows (by the law of large numbers and consistency of sample variance).
Key Insight
The distinction between population distribution and sampling distribution is fundamental: the sampling distribution of $\bar{x}$ has mean $\mu$ and standard error $\sigma/\sqrt{n}$, shrinking as sample size grows.