Comprehensive Study Notes on Statistical Variables, Sampling Methods, and Bias

Introduction to Categorical and Quantitative Variables

• Categorical Variables:

  • Definition: A categorical variable places an individual into one of several groups or categories based on an attribute, characteristic, or label.

  • Examples: Gender/gender identity, hair color, religion, ethnicity, academic major, movie genre, car model, and physical attributes.

  • Special Categorical Examples:

    • ZIP Code: Even though it consists of numbers, a ZIP code is categorical because it describes a physical location where an individual lives (an attribute). Taking an average ZIP code makes no logical sense.

    • Area Code: Similar to a ZIP code, an area code uses numerical digits to identify a geographical region, making it a categorical attribute rather than a quantitative measurement.

• Quantitative Variables:

  • Definition: A quantitative variable takes numerical values for which arithmetic operations (such as calculating averages or sums) make sense.

  • Examples: Height, weight, income, distance, runtime/duration, box office revenue, and fuel economy.

Populations, Samples, and Censuses

• Population:

  • Definition: The entire group of individuals about which information is desired.

  • Examples: All citizens in the United States eligible to vote, all students enrolled at Georgia State University (GSU), or all artifacts recovered from an archaeological site.

• Census:

  • Definition: A study that collects data from every single individual in the entire population.

  • Practicality & Constraints: Conducting a true census is rarely feasible for large populations due to time, cost, and non-response limitations. For example, the United States attempts to conduct a national census every $10\,\text{years}$ targeting citizens aged $18$ and older, but it fails to be a perfect census due to non-replacements and unreachable individuals.

• Sample:

  • Definition: A subset of individuals drawn from the population from which data is actually collected.

  • Purpose: Sample statistics are calculated to estimate unknown population parameters.

Observational Studies vs. Experiments

• Observational Study:

  • Definition: A study in which variables of interest are observed and measured on individuals, but no attempt is made to influence or modify the responses.

  • Purpose: Used to describe characteristics, groups, or existing situations.

  • Examples: Observing wildlife interactions on a safari, reviewing patient medical records at a hospital, or noting hair colors in a room.

  • Critical Limitation: Observational studies cannot establish cause-and-effect relationships due to confounding variables.

• Experiment:

  • Definition: A study in which treatments are deliberately imposed on individuals to observe and measure the resulting responses.

  • Examples: Testing a new blood pressure medication by giving half a sample the standard drug and the other half a new formula; applying different fertilizer brands to plants to measure growth height; administering varying dosages of aspirin to evaluate heart attack risk.

  • Causation: Properly designed experiments are the only valid statistical method to establish direct cause-and-effect relationships.

Study Review Questions and Platform Instructions

• Platform System Rules & Test Review Tips:

  • Checkpoint questions directly populate the test review materials, which in turn form the pool of questions for exams.

  • Checkpoint questions allow unlimited attempts. However, clicking the Solution tab locks the question permanently and awards $0\,\text{points}$. Students must always select "Keep Working" or "Try Again" to preserve attempt eligibility.

• Question 1: An individual in a dataset must be a human being.

  • Answer: False. Individuals can be objects, animals, geographic areas, or entities (e.g., pottery shards, cars, movies).

• Question 2: Which of the following is an example of a categorical variable?

  • Answer: Genre (e.g., thriller, romantic comedy). Runtime and revenue are quantitative.

• Question 3: What does sample size refer to?

  • Answer: The number of individuals selected from a population for a study.

• Question 4: What is the difference between a population and a sample?

  • Answer: A population includes all members of a group, whereas a sample includes only some members.

• Question 5 (Car Dealer Dataset):

  • Data Variables Classification:

    • Buyers: Individuals

    • ZIP Code: Categorical

    • Sex: Categorical

    • Distance from Dealer: Quantitative

    • Car Model: Categorical

    • Fuel Economy: Quantitative

    • Price: Quantitative

• Question 6 (Archaeological Dig):

  • Scenario: Students catalog pottery shards, stone tools, and artifacts. The director randomly checks $2\,\%$ of the artifacts.

  • Population: All artifacts collected from the dig.

  • Sample: The $2\,\%$ subset of artifacts chosen for verification.

• Question 7 (General Motors Smoking Study):

  • Scenario: General Motors sponsored a smoking cessation study where $439\,\text{volunteers}$ received up to $750\,\text{dollars}$ for quitting smoking for a year, while $439\,\text{volunteers}$ were simply encouraged to quit. After one year, the incentive group was $3\,\text{times}$ more likely to quit.

  • Type of Study: Experiment (treatments were deliberately imposed).

  • Categorical Variables: Smoking status after one year (quit vs. didn't quit) and Treatment offer (financial incentive vs. encouragement).

  • Quantitative Variables: None (receiving money is recorded as a categorical offer type, not a varying numerical quantity per individual).

  • Population: All General Motors employees who smoke.

  • Sample: The $878\,\text{employees}$ ($439 + 439$) who volunteered.

• Question 8 (Newspaper Survey):

  • Scenario: A publisher inserts surveys into $1{,}000\,\text{copies}$ of a weekly paper; $189\,\text{surveys}$ are returned.

  • Population: All readers of the local weekly newspaper.

  • Sample: The $189\,\text{readers}$ who actually completed and returned the survey.

• Question 9 (Movie Analysis - Avengers: Endgame Dataset):

  • Dataset Details: $12\,\text{popular movies}$ evaluated across several metrics.

  • Individuals: The $12\,\text{movies}$.

  • Categorical Variables: Release Year (attribute/label), Genre, Rating (PG, PG-13, R, NC-17).

  • Quantitative Variables: Duration/Runtime (measured in time), Box Office Revenue.

  • Excluded Attributes: Number of moviegoers (not measured in the provided dataset).

Sampling Methods and Sources of Bias

• Sampling Requirements:

  • A sample must accurately represent the target population to make valid statistical inferences.

• Bias:

  • Definition: Systematic errors in the design of a study that systematically favor certain outcomes, leading to consistent overestimation or underestimation of the population parameter.

• Biased Sampling Methods:

  • Voluntary Response Sampling:

    • Occurs when individuals choose themselves to participate in a sample (e.g., receipt QR code surveys at Chili's or Applebee's, dorm wall flyers, online Yelp reviews).

    • Flaw: Overrepresents individuals with strong, extreme opinions (either very positive or very negative). Results cannot be generalized to the broader population.

  • Convenience Sampling:

    • Occurs when researchers select individuals who are easiest to reach or readily available.

    • Example: Standing outside a dining hall to ask exiting students about food quality, or surveying students getting off a school bus regarding parking lot expansion.

    • Flaw: Fails to produce a representative sample of the overall population.

Types of Statistical Bias

• Undercoverage:

  • Definition: Occurs when certain groups in a population are intentionally or accidentally excluded from the sampling frame.

  • Examples: Conducting a landline-only telephone survey (excludes individuals without landlines); estimating general student metrics using only collegiate athletic teams.

  • Real-World Case Study (2016 Presidential Polling): National polls predicted a victory for Hillary Clinton over Donald Trump primarily due to undercoverage. Pollsters oversampled urban regions (New York City, Atlanta, Boston, San Francisco, Miami) and systematically undercovered rural demographics.

• Nonresponse:

  • Definition: Occurs when a selected individual cannot be reached or refuses to participate in the study.

  • Example: Mail-in surveys where a large percentage of recipients discard the questionnaire.

• Response Bias:

  • Definition: Occurs when there is a systematic pattern of inaccurate, untruthful, or influenced responses.

  • Causes: Authoritative interviewers, social desirability concerns, misleading or leading question wording.

  • Real-World Example (Ohio School Drug Study, 2002): To evaluate a new drug prevention program, a school resource officer walked down hallways directly asking students, "Do you do drugs?" All students answered "No," leading the school to falsely report a $100\,\%$ effective prevention program due to severe response bias.

  • Leading Question Example: Framing a survey as "Do you support decreasing tuition at Georgia State even if $75\,\text{children}$ die because of it?" artificially alters responses.

Simple Random Sampling and Variability

• Simple Random Sample (SRS):

  • Definition: A sampling design of size $n$ chosen in such a way that every group of $n$ individuals in the population has an equal chance of being selected as the sample.

  • Mechanism Example: Assigning every individual in a population of $250\,\text{people}$ a unique integer from $1$ to $250$, placing those numbers into a random generator or hat, and selecting $30\,\text{numbers}$.

  • Benefit: SRS is the gold standard of sampling because it eliminates selection bias.

• Sampling Variability:

  • Definition: The natural variation of sample statistics observed when taking multiple random samples of the same size from the exact same population.

  • Core Concept: Different random samples will yield different sample statistics. The foundational goal of inferential statistics is using a single random sample's result to draw accurate, valid inferences about population parameters despite inherent sampling variability.