STAT 110: Chapter 1 - Where Do Data Come From?

Scenario Context: A pharmaceutical company seeking FDA approval for a new drug that carries potential rare migraine side effects must estimate how many users experience this side effect.
Methodological Challenge: Testing every future user of the drug is impossible prior to approval.
Statistical Solution: The company conducts a well-designed experiment on a representative sample of subjects.
Core Principle of Data Integrity: Accurate data collection and analysis are essential prerequisites for generating trustworthy statistical results.
Fundamental Rule: Always ask, "Where do the data come from?" before trusting any statistic.

Definition of Statistics: Statistics is the science of data: how to collect, analyze, present, and interpret data, and how to make decisions using data.
General Conceptual Definition: Statistics is the science of understanding data and making decisions in the face of variability and uncertainty.
The Statistical Investigative Process: Statistical inquiry is an investigative process that involves three core steps:
- Step 1 (Data Collection): Collect information, often called data, from a sample.
- Step 2 (Data Analysis): Analyze the information by computing statistics, making plots, and looking for patterns.
- Step 3 (Statistical Inference): Make conclusions by using a sample to infer characteristics of a population.

Why Data is Needed:
- Decision-Making: Accurate and precise information is necessary to make informed decisions in real-world situations.
- Practical Scenario: Selecting a section for a statistics course based on professor recommendations or evaluating feedback on platforms like "Rate My Professor" requires assessing data trust and validity.
- Core Analytical Question: Evaluating what conclusions can be drawn about a situation using data collected from a sample.
Two Main Types of Statistics:
- Descriptive Statistics: Consists of methods for organizing, displaying, and describing data by using tables, graphs, and summary measures.
- Inferential Statistics: Consists of methods that use sample results to help make decisions or predictions about a target population.

Key Terms Across Data Science, Analytics, Machine Learning, and Database Management:
- Individuals: The objects or units described by a set of data. Individuals may be people, animals, plants, or things.
- Variable: Any one characteristic of an individual. A variable can take different values for different individuals.
- Data: The specific measurements recorded for a variable across individuals.
- Categorical Variable: Places an individual into one or several groups or categories.
- Quantitative Variable (Numeric Variable): Takes numerical values for which arithmetic operations such as adding and averaging make sense.
- Response Variable: Measures the major outcome or result of a study.
- Statistical Inference: Involves making conclusions or comments about a population based on data collected from a sample.

Student Dataset Example (Table 1.1: Student Data - End of the Semester):
- Advani, Sura: Major = COMM, Points = $397$ , Grade = B
- Barton, David: Major = HIST, Points = $323$ , Grade = C
- Brown, Annette: Major = LIT, Points = $446$ , Grade = A
- Chiu, Sun: Major = PSYC, Points = $405$ , Grade = B
- Cortez, Maria: Major = PSYC, Points = $461$ , Grade = A
Dataset Analysis Questions & Answers:
- Individuals: The students listed in the dataset (Advani, Sura; Barton, David; Brown, Annette; Chiu, Sun; Cortez, Maria).
- Number of Variables: Exactly $4$ variables are represented as columns.
- Names of Variables: Name, Major, Points, and Grade.
- Categorical Variables: Name, Major, and Grade.
- Quantitative Variables: Points (numerical points value where adding and averaging are arithmetic operations that make sense).

Thinking Ahead on Study Design: A knowledge of different study designs for gathering data helps explain how contradictory results can happen in scientific research studies and helps determine which studies deserve trust.
Observational Study:
- Definition: A study that observes individuals and measures variables of interest but does not intervene to influence the responses.
- Purpose: To describe some group or situation.
- Limitation: It is not possible to establish cause and effect definitively with observational studies.
Experiment:
- Definition: A study in which the researcher intentionally applies treatments to subjects and then measures a response variable to determine how the treatment affects the response.
- Purpose: To study whether the treatment causes a change in the response (establishing causality).
Study Identification Examples:
- Example 1: Researchers survey $1{,}000$ adults to record their current exercise habits and blood pressure levels at one point in time.
- Classification: Observational study.
- Example 2: Scientists randomly assign patients to receive either a new drug or a placebo and track their recovery over $6\, \text{months}$ .
- Classification: Experimental study (Experiment).

Population (Target Population): In a statistical study, the population consists of all elements, individuals, items, or objects whose characteristics are being studied.
Sample: The part or subset of the population containing the individuals that are actually observed.
- Inference Purpose: Data are collected from a sample in order to draw inferences about the entire population.
- Research Note: Scientific research studies are usually carried out on a sample of subjects rather than on whole populations.
Population and Sample Identification Example:
- Scenario: A university wants to know the average study time of all $20{,}000$ students. It collects data from $400$ students who volunteered.
- Population: All $20{,}000$ students.
- Sample: The $400$ students chosen / who volunteered.

Sample Survey:
- Definition: A study in which data are collected from a selected part, or sample, of the entire population.
- Scope: Does not include everyone, only a representative group.
- Advantages & Disadvantages: Faster and cheaper to conduct, but may have some error because it does not cover the whole population.
Census:
- Definition: A special survey that attempts to collect data from every single member of the population.
- Advantages & Disadvantages: More accurate because it covers everyone, but takes more time and money to carry out.
Historical and Legal Context of the United States Census:
- Constitutional Authority: The United States Constitution empowers Congress to carry out a census for the American people.
- Historical Frequency: Started in $1790$ and has occurred every $10\, \text{years}$ (decennially) since then.
- Broader Implications: Results of the decennial census have broad societal impacts, including deciding how many representatives each state will have in Congress and apportioning federal funds for underrepresented states or groups.