Introduction to Statistics: Populations, Samples, and Variables

Statistics: A collection of tools for collecting and analyzing data in large quantities to infer information about a population from a representative sample.
Data: Information gathered from experiments and surveys.

Subjects: The specific entities or individuals measured in a study.
Population: The set of ALL subjects of interest (e.g., all $5,000,000$ South Carolina residents).
Parameter: A numerical summary or result calculated from a population.
Sample: A subset of the population for which data is actually collected (e.g., a study of $20,000$ residents).
Statistic: A numerical summary or result calculated from a sample.

Descriptive Statistics: Summarizing and describing patterns in observed data.
Inferential Statistics: Making predictions or decisions about a larger population based on sample statistics.

State Flag Study:
- Population: All state residents.
- Parameter: The governor’s claim that $15\%$ of all residents favor a change.
- Sample: A random sample of $1,000$ residents.
- Statistic: The $20\%$ of the sample found to be in favor of the change.

Simple Random Sample: Every item in the population has the same probability of being selected (e.g., randomly picking $100$ students from a USC freshman list).
Systematic Sample: Selecting a random starting point and sampling every $k^{th}$ item (e.g., starting with the twentieth freshman at Russell House and sampling every fifth thereafter).
Stratified Sample: Grouping data by a common characteristic and randomly sampling some from every group.
Cluster Sample: Grouping data by naturally occurring units (clusters), randomly selecting certain groups, and sampling all members within those groups.
Convenience Sample: Sampling items that are easy or inexpensive to reach, such as selecting the first $100$ individuals seen on the street.

Measurement Error: Bad data resulting from inaccurate information (e.g., subjects lying about drug use) or recorded typos (e.g., entering $0.02$ instead of a value in the tens of thousands).
Non-response Error: Occurs when a subject cannot be contacted (e.g., after $5$ calls) or fails to provide data.
Sampling Error: Natural variation in results that occurs because different samples contain different subjects.
Coverage Error: Occurs when certain groups are excluded from the selection process, failing to cover the entire population.

Variable: A characteristic or property of an individual that varies among occurrences.
Data: The set of values associated with one or more variables.
Categorical (Qualitative): Values representing distinct categories (e.g., hair color or gender).
Quantitative: Values representing a count or measured quantity.
- Discrete: Values representing a count (key phrase: "the number of").
- Continuous: Values measured on a continuum (e.g., time, age, or salary).