Population vs. Sample - Algebra-1

1. Fundamental Concepts

Population: The entire set of individuals or all possible observations that are the focus of a study. It represents the overall scope researchers aim to understand or analyze.
Sample: A subset of representative individuals or observations selected from the population, used to infer or estimate the characteristics of the population.

Characteristics of a Population:
- Includes all individuals within the research scope, with a clear and complete range.
- Directly studying the population is often costly, difficult, or even impractical due to its potential size (e.g., the national population).
- Has well-defined boundaries, such as "all middle school students in a city" or "all smartphones of a brand."
Characteristics of a Sample:
- Is a subset of the population and must be representative (i.e., able to reflect the population’s characteristics); otherwise, research results may be biased.
- Typically smaller than the population. Analyzing samples allows indirect inference about the population, reducing research costs and difficulty.
- The method of sample selection (e.g., random sampling, stratified sampling) affects its representativeness.
Relationship Between the Two: Samples are derived from the population. The purpose of studying samples is to infer the characteristics of the population; the validity of a sample depends on its representativeness of the population.

1:
- Population: All 30 students in a class.
- Sample: 5 students randomly selected from the class.
2:
- Population: All registered shared bikes in Beijing (potentially millions).
- Sample: 200 shared bikes randomly selected from different districts in Beijing (used to inspect vehicle conditions or usage rates).
3:
- Population: All plastic waste in oceans worldwide (extremely broad and difficult to fully count).
- Sample: Plastic waste found in 100 seawater samples collected through scientific sampling in major ocean regions (Pacific, Atlantic, Indian Ocean, etc.) (used to estimate global marine plastic pollution levels).

Distinguishing Between Population and Sample:
- First, clarify the "overall scope" of the study (i.e., the population). For example, in "studying the eyesight of Chinese teenagers," the population is "all teenagers in China."
- A sample is a portion selected from the population. Ask, "Is it part of the population?" For example, "eyesight data of 100 students from a middle school" is a sample.
Methods to Ensure Sample Representativeness:
- Random Sampling: Each individual in the population has an equal chance of being selected (e.g., drawing lots, random number tables) to avoid subjective bias.
- Stratified Sampling: Divide the population into subgroups based on characteristics (e.g., age, gender), then sample from each subgroup (e.g., stratifying by urban/rural areas when studying national income).
Application Scenarios:
- When the population is small and data is easily accessible, study the population directly (e.g., grades of a class).
- When the population is large and full-scale study is difficult, infer the population through samples (e.g., sampling surveys in national censuses).
Analysis Logic: Estimate population parameters (e.g., population mean) using sample statistics (e.g., sample mean) and evaluate result reliability by considering sampling error.