Chapter 1: Introduction to Statistics Flashcards

Fundamentals of Statistics and Data Collection

• Definition of Statistics: The scientific discipline devoted to collecting, classifying, organizing, summarizing, analyzing, and interpreting data for the purpose of making informed decisions.

• Core Value: Statistics provides the foundation for evidence-based reasoning and critical decision-making across business, government, and the social, medical, and physical sciences.

• Data Definition: Any collection of numbers, characters, images, or items that convey information about entities.

  • Grammatical Note: Data is grammatically a plural noun (“data are…”). A single individual value is termed a datum.

  • Examples of Datasets:

  • Migration patterns and distribution of 118 bird species.

  • Worldwide gross earnings of the highest-grossing films as of 2020 (adjusted for inflation to 2020 dollars):

    1. Gone with the Wind (1939): $3,724,000,000\,USD$

    2. Avatar (2009): $3,273,000,000\,USD$

    3. Titanic (1997): $3,096,000,000\,USD$

    4. Star Wars (1977): $3,059,000,000\,USD$

    5. Avengers: Endgame (2019): $2,811,000,000\,USD$

    6. The Sound of Music (1965): $2,562,000,000\,USD$

    7. E.T. the Extra-Terrestrial (1982): $2,501,000,000\,USD$

    8. The Ten Commandments (1956): $2,368,000,000\,USD$

    9. Doctor Zhivago (1965): $2,244,000,000\,USD$

    10. Star Wars: The Force Awakens (2015): $2,213,000,000\,USD$

Descriptive vs. Inferential Statistics

• Descriptive Statistics:

  • Focuses on organizing, summarizing, and presenting data in an informative visual or numerical manner.

  • Standard graphical tools include Bar Charts, Pie Charts, Box Plots, Scatter Plots, and Heatmaps.

  • Example: A heat map displaying the total number of police reports across each 5-digit ZIP code in Austin, Texas.

• Inferential Statistics:

  • Involves methods for making decisions, predictions, estimations, or drawing conclusions about an entire population based on findings from a sample.

  • Example: Using a statistical model fitted to daily police report counts in Austin ($2003$ to $2022$) to project future daily report volumes for $2023$ through $2025$

Population, Sample, Parameters, and Statistics

• Population: The complete collection or set of all units (people, objects, events) of interest in a statistical study.

  • Finite Population: Possesses a fixed, countable number of members.

  • Infinite Population: Unlimited in extent, containing an uncountable number of theoretical members.

• Sample: A representative subset of units selected from the targeted population.

• Census: A data collection process that evaluates every single member of the target population.

• Parameter: A numerical value that summarizes a characteristic of an entire population.

  • Mnemonic: Population corresponds to Parameter.

• Statistic: A numerical value that summarizes data derived from a sample.

  • Mnemonic: Sample corresponds to Statistic.

• Unit / Subject / Individual: The single entity from which data are observed or measured in a study.

• Variable: A specific characteristic recorded for each unit in a dataset.

Classification of Variables and Levels of Measurement

• Qualitative (Categorical) Variables: Describe attributes, labels, or non-numerical properties.

  • Nominal Scale: Categorizes data into mutually exclusive groups with no natural rank or order.

  • Examples: Eye color (blue, brown, green, hazel), blood type, gender, nationalities, automobile colors, ZIP code, academic major field.

  • Ordinal Scale: Categorizes data into groups that possess a natural order or ranking, though exact mathematical differences between ranks are undefined.

  • Examples: Course letter grades ($A, B, C, D, F$), performance rating scales (poor, good, excellent), competition placement (1st, 2nd, 3rd place), socioeconomic status.

• Quantitative (Numerical) Variables: Represent numerical counts or physical measurements.

  • Discrete Level: Values obtained by counting items, resulting in distinct, separate numeric values.

  • Examples: Number of text messages sent per day, number of books on a shelf, number of spoken languages, number of siblings.

  • Continuous Level: Values obtained by measuring on a continuous numerical scale, capable of taking any value within a given interval.

  • Examples: Student height, body weight, age, temperature, wait time at a clinic.

• Levels of Measurement (Scales):

  • Nominal: Data divided into non-ordered categories.

  • Ordinal: Data categorized with a natural ordering scheme.

  • Interval: Ordered numerical data with constant unit differences, but lacking a true physical zero point. Ratios are not meaningful.

  • Examples: Temperature in $^\circ\text{C}$ or $^\circ\text{F}$, IQ score, credit score, SAT/ACT score, high school GPA.

  • Ratio: Ordered numerical data with equal intervals AND a true absolute zero point representing total absence of the measured quantity. Ratios between measurements are meaningful.

  • Examples: Height, weight, time, salary, age, plant yield.

• Converting Continuous Variables to Ordinal Variables:

  • Continuous quantitative variables like age, temperature, or income can be transformed into ordinal categorical variables by grouping precise measurements into ordered intervals or categories (e.g., Age grouped into $0-10$, $11-20$, $21-30$ years; or Income grouped into lower, middle, and upper income tiers).

Sampling Methods and Techniques

• Reasons for Using Samples:

  • Inaccessibility of the complete population (e.g., measuring the length of all fish in the Atlantic Ocean).

  • Resource limitations (excessive time, high costs, and heavy labor constraints).

  • Destructive testing methods (testing matchstick ignition quality, evaluating automobile crash safety, determining battery lifespan).

• Sampling Mechanics:

  • With Replacement: Selected members are returned to the population pool and may be chosen more than once.

  • Without Replacement: Selected members are permanently removed from the sampling pool and cannot be chosen again.

• Probability (Random) Sampling Methods:

  • Simple Random Sampling (SRS): Every possible sample combination of size $n$ has an equal chance of selection. Chosen directly from a complete population master list.

  • Stratified Random Sampling: The population is divided into non-overlapping, homogeneous subgroups called strata (sharing similar traits). Simple random sampling is then performed within every stratum.

  • Cluster Sampling: The population is divided into heterogeneous subgroups called clusters (often geographic, like city blocks or school districts). A simple random sample of clusters is selected, and a complete census is conducted on every member within those chosen clusters.

  • Multistage Cluster Sampling: Similar to cluster sampling, but instead of interviewing all members in the selected clusters, a secondary simple random sample of individuals is drawn from within each selected cluster.

  • Systematic Random Sampling: Population members are listed sequentially, a random starting point is selected, and every $k\text{th}$ member is selected for inclusion.

• Non-Probability Sampling Methods:

  • Convenience Sampling: Selecting individuals who are most easily accessible. Highly vulnerable to selection bias; generally unrepresentative and unsuitable for population inference.

• Structural Comparison: Stratified vs. Cluster vs. Multistage Sampling:

Sampling and Non-Sampling Errors

• Sampling Error:

  • Definition: The numerical discrepancy between a sample statistic and the true population parameter being estimated.

  • Causes: Natural population variability and random selection fluctuations.

  • Mitigation: Increasing total sample size ($n$) and using stratified random sampling.

• Non-Sampling Errors:

  • Definition: Errors caused during data collection, processing, or survey design, unrelated to sample selection.

  • Types and Causes:

  • Measurement and Processing Errors: Equipment calibration errors, inaccurate recording, data entry mistakes, and computational errors.

  • Response Bias: False or misleading answers given due to question wording, interviewer tone, question ordering, or social desirability.

  • Non-Response Bias: Systematically missing data when selected sample members refuse to participate or drop out.

  • Voluntary Response Bias: Unrepresentative results created when a sample relies on self-selected volunteers who possess strong or extreme opinions.

Observational vs. Experimental Studies

• Observational Studies:

  • Characteristics: Researchers observe and record measurements on subjects without applying treatments or manipulating variables.

  • Scope: Identifies associations or correlations between explanatory and response variables.

  • Key Constraint: Association does not imply causation. Unmeasured confounding factors prevent causal conclusions.

• Experimental Studies:

  • Characteristics: Researchers deliberately apply treatments to experimental units under controlled conditions and evaluate outcomes.

  • Scope: Designed to establish direct cause-and-effect relationships (causality).

Types of Observational Studies

• Retrospective (Case-Control) Study:

  • Analyzes historical records or pre-existing data (medical charts, databases) over a past timeframe.

  • Example: Investigation of California's 2010 pertussis (whooping cough) outbreak comparing medical records of $682$ infected children to $2,016$ uninfected controls to assess vaccination history.

• Cross-Sectional Study:

  • Measures explanatory factors and health outcomes simultaneously at a single point in time.

  • Primary Function: Estimating disease or attribute prevalence within a population.

  • Example: Isolating Propionibacterium acnes from $80$ acne patients during a clinic visit to test antibiotic resistance rates.

• Prospective (Longitudinal / Cohort) Study:

  • Tracks a defined cohort sharing common demographic factors forward in time over an extended period.

  • Example: The Nurses' Health Study, initiated in $1976$ with over $100,000$ registered female nurses tracked via biennial questionnaires, which demonstrated that physical walking in one's $50\text{s}$ and $60\text{s}$ is associated with improved memory performance at age $70+$.

Principles and Types of Experimental Design

• Key Terminology:

  • Response Variable: The measured outcome of interest (dependent variable).

  • Explanatory Variable: The factor manipulated by researchers (independent variable).

  • Treatment: A specific experimental condition applied to subjects.

  • Experimental Unit / Subject: The entity receiving a treatment.

  • Confounding Variable: A variable correlated with both explanatory and response variables, obscuring the true relationship.

  • Lurking Variable: An unmeasured background variable influencing the observed response.

  • Placebo: An inactive dummy treatment (saline, sugar pill) given to a control group.

  • Placebo Effect: Measurable health improvement caused by psychological expectation.

  • Single-Blind Experiment: Experimental units do not know which treatment group they are in.

  • Double-Blind Experiment: Neither the experimental units nor the assessing researchers know treatment assignments.

• Four Principles of Experimental Design:

  1. Control: Comparing experimental treatment groups against a baseline control group.

  2. Randomization: Randomly assigning subjects to treatment groups to balance unmeasured background factors.

  3. Replication: Using sufficiently large sample sizes and repeating experiments to confirm findings.

  4. Blocking: Grouping subjects into homogeneous blocks based on known background factors (e.g., gender, age group) before random treatment allocation.

• Main Experimental Frameworks:

  • Completely Randomized Design: Subjects are assigned directly to treatment groups purely through random selection.

  • Randomized Block Design: Subjects are categorized into homogeneous blocks based on a characteristic (e.g., gender), then randomly assigned to treatments within each block.

  • Matched Pairs Design: Subjects are matched in pairs (e.g., identical twins, or pre- and post-test assessments on the same subject), and treatments are randomly assigned within each pair.

• Inferential Scope Matrix:

Practice Problems and Comprehensive Solutions

• Exercise 1: Sunflower Height Analysis

  • Scenario: 1) Mean height of a random sample of East Texas sunflowers was $140\,\text{cm}$. 2) Mean height of a random sample of West Texas sunflowers was $134\,\text{cm}$. 3) Therefore, East Texas sunflowers tend to be taller than West Texas sunflowers.

  • Solutions:

  • Statement 1: Descriptive statistics (summarizes sample measurement).

  • Statement 2: Descriptive statistics (summarizes sample measurement).

  • Statement 3: Inferential statistics (generalizes beyond sample data to draw a conclusion about the broader population).

• Exercise 2: Exam Scores, Internet Access, and Salon Customers

  • Scenario: Classify as descriptive or inferential statistics:

  1. The average STAT 302 final exam score last semester was $89$.

  2. Households with children under 18 are more likely to have Internet access ($77\%$) compared to those without ($68\%$).

  3. A home salon operator served $25$ customers last week.

  • Solutions:

  1. Descriptive statistics.

  2. Inferential statistics.

  3. Descriptive statistics.

• Exercise 3: Suicidal Ideation Among Adolescents

  • Study Details: Survey by Dr. Matthew Nock (Harvard University) analyzing $6,482$ U.S. adolescents aged 13–18. Found that over $12\%$ experience suicidal ideation.

  • Identification:

  1. Population: All adolescents aged 13 to 18 residing in the United States.

  2. Sample: The $6,482$ surveyed adolescents aged 13 to 18.

  3. Variable: Suicidal ideation status (Categorical: Yes / No).

  4. Population Parameter: The proportion of all U.S. adolescents aged 13–18 who experience suicidal ideation.

  5. Sample Statistic: The observed proportion ($>12\%$) of adolescents in the sample who experienced suicidal ideation.

• Exercise 4: Handedness in College Students

  • Study Details: Approximately $11\%$ of Americans are left-handed. An instructor counts $3$ left-handed students out of $95$ enrolled in STAT 302 Section 510.

  • Identification:

  1. Population: All Americans.

  2. Sample: The $95$ students in STAT 302 Section 510.

  3. Variable: Hand dominance (Categorical: Right-handed / Left-handed).

  4. Population Parameter: The proportion of all Americans who are left-handed ($11\%$).

  5. Sample Statistic: The proportion of left-handed students in the section ($\frac{3}{95} \approx 3.16\%$).

• Exercise 5: Medication Information Survey

  • Study Details: Survey of $1,000$ U.S. adults finding that $75\%$ consult a physician/pharmacist for medication questions, while $8\%$ consult a medication website.

  • Identification & Conclusion:

  1. Population: All U.S. adults. Sample: The $1,000$ surveyed U.S. adults.

  2. Inferential Conclusion: The majority of U.S. adults consult healthcare professionals rather than websites for medication questions.

• Exercise 6: College Application Data Classification

  • Classification Table:

• Exercise 7: Classification of Common Variables

  • DMV wait time: Quantitative (Continuous, Ratio scale).

  • Rainforest mammal species: Qualitative (Nominal scale).

  • Family member birth year: Quantitative (Discrete, Interval scale).

  • Soccer jersey numbers: Qualitative (Nominal scale).

  • Bank PIN numbers: Qualitative (Nominal scale).

• Exercise 8: Contextual Sampling Scenarios

  • Scenario 1: FDA randomly selects $25$ hospitals and surveys every doctor in them -> Cluster Sampling.

  • Scenario 2: Sample of $10$ students randomly selected from each of 12 STAT sections -> Stratified Sampling.

  • Scenario 3: Politician interviews $30$ neighbors in his community -> Convenience Sampling.

  • Scenario 4: Dormitory survey picking a random room and selecting every $5\text{th}$ room sequentially -> Systematic Sampling.

  • Scenario 5: Computer program generates random student ID numbers from a full registry -> Simple Random Sampling (SRS).

• Exercise 9: Ann Landers Parent Survey Bias

  • Background: Advice columnist asked readers: "If you had it to do over again, would you have children?" Over $10,000$ responded, with $70\%$ answering "No". A subsequent national representative random sample by Newsday found $91\%$ answered "Yes".

  • Primary Bias Type: Non-Response / Voluntary Response Bias (Disgruntled parents actively responded, while satisfied parents largely ignored the open call).

• Exercise 10: Chia Seeds Study Analysis

  • Details: $38$ men and $38$ women randomly assigned to $25\,\text{g}$ chia seeds twice daily or matching placebo for $12\,\text{weeks}$. Double-blinded. Measured appetite and weight loss; found no significant differences.

  • Analysis:

  1. Study Type: Randomized Block Experiment (blocked by gender).

  2. Response Variables: Appetite score and weight loss.

  3. Explanatory Variable: Chia seed consumption.

  4. Treatments: $25\,\text{g}$ chia seeds vs. placebo.

  5. Blinding: Double-blind.

  6. Causal Statement Permissible: Yes (due to experimental control and randomized assignment).

• Exercise 11: Aspirin Heart Attack Study Analysis

  • Details: $400$ men aged 50–84 randomly assigned to daily aspirin or placebo for $3\,\text{years}$.

  • Analysis:

  1. Study Type: Randomized Experiment.

  2. Population: Men aged 50 to 84.

  3. Sample: The $400$ recruited men.

  4. Experimental Units: Individual participating men.

  5. Explanatory Variable: Daily aspirin intake.

  6. Response Variable: Occurrence of heart attack.

  7. Treatments: Aspirin pill vs. placebo pill.

• Exercise 12: Light, Noise, and Gender Experiment

  • Scenario: Study examining light levels and noise levels on student exam performance, maintaining equal representation of males and females in all groups.

  • Variable Classification: $2$ explanatory variables (light level, noise level), $1$ blocking variable (gender), and $1$ response variable (exam performance).

Case Studies in Statistical Analysis

• Case Study 1: Cognitive Behavior Therapy for Chronic Fatigue Syndrome (Deale et al., 1997)

  • Participant Pool: $142$ patients referred by primary care physicians. $60$ enrolled ($82$ excluded or declined).

  • Design: Randomized controlled trial with $30$ patients assigned to Cognitive Behavioral Therapy (CBT: activity management education) and $30$ to Relaxation Control (muscle relaxation/visualization).

  • Follow-up Results at $6\,\text{months}$ ($7$ dropouts total: $3$ in treatment, $4$ in control):

• Outcome Proportions:

  • CBT Treatment: $\frac{19}{27} \approx 70.37\%$

  • Relaxation Control: $\frac{5}{26} \approx 19.23\%$

• Statistical Interpretation: The absolute difference of $51.14\%$ is substantial. Causal attribution to CBT is justified because random assignment balances background confounding factors (e.g., depression level, gender).

• Generalizability: Restricted to clinic-referred volunteers matching specific entry criteria.

  • Case Study 2: Light Wavelength Impact on Elodea Photosynthesis

• Participant Pool: $40$ young Elodea aquatic plants sampled from the Great Lakes region.

• Experimental Treatments: Exposed to Blue light, Red light, Green light, or No light ($10$ plants per group).

• Mean Oxygen Bubble Production ($5\,\text{minute}$ period):

  • Blue Light: $14.6\,\text{bubbles}$

  • Red Light: $12.2\,\text{bubbles}$

  • Green Light: $7.1\,\text{bubbles}$

  • No Light (Control): $0.2\,\text{bubbles}$

• Causal Inference: Allowed if plants were randomly assigned to treatment groups.

• Generalizability: Confined strictly to Elodea plants within the Great Lakes region.

Appendix: Statistical Software and Technology Instructions

• Microsoft Excel (Windows) Add-In Activation:

  1. Click File > Options.

  2. Select Add-ins from the left navigation panel.

  3. Set the Manage drop-down menu to Excel Add-ins and click Go….

  4. Check the Analysis ToolPak box and click OK.

  5. Access via the Data tab ribbon under Data Analysis.

• Microsoft Excel (macOS) Add-In Activation:

  1. Click Tools on the top menu bar and select Excel Add-ins….

  2. Check Analysis ToolPak and click OK.

• Generating Random Numbers in Excel:

  • Function =RANDBETWEEN(bottom, top): Returns a random integer between lower and upper bounds (with replacement).

  • Function =RAND(): Returns a random decimal value between $0$ and $1$.

  • Analysis ToolPak Sampling Tool: Data > Data Analysis > Sampling > Select Input Range > Set Method to Random > Enter Number of Samples > Select Output Range.

• TI-83 / TI-84 Graphing Calculator Instructions:

  1. Press MATH key.

  2. Scroll right to highlight the PRB (Probability) menu.

  3. Select item 5:randInt(.

  4. Syntax: randInt(lower_bound, upper_bound, n).

  5. Example: randInt(1, 6, 3) outputs a set of three random integers between 1 and 6, such as {4, 3, 5}.

• R and RStudio Programming Commands:

  • Generate Simple Random Sample:

  x <- sample(1:90, size = 10, replace = FALSE)
  print(x)
  &nbsp;&nbsp;&nbsp;&nbsp;```
  * Setting Working Directory in RStudio:
  * Click **Session** > **Set Working Directory** > **To Source File Location**.
  * Importing Data Files into R:
  

  r

  Read Text File

  data_txt <- read.table("file.txt", header = TRUE, sep = "\t")

  Read CSV File

  data_csv <- read.csv("file.csv")

  Read Excel File

  install.packages("readxl") library(readxl) data_excel <- read_excel("file.xlsx")     ```