Introductory Statistics - Chapter 1: Sampling and Data

Definitions of Statistics, Probability, and Key Terms

• Statistics Definition: Statistics is the science dealing with the collection, analysis, interpretation, and presentation of data.

  • Descriptive Statistics: The practice of organizing and summarizing data. Data can be summarized using visual graphs or numerical summaries, such as calculating an average.

  • Inferential Statistics: Formal mathematical methods used to draw conclusions from data after studying probability and probability distributions. Inferential statistics uses probability to measure confidence in the correctness of drawn conclusions.

• Probability: A mathematical tool used to study randomness, dealing directly with the likelihood or chance of an event occurring. Probability calculations provide the foundation for statistical inference.

• Key Statistical Terms:

  • Population: A comprehensive collection of persons, things, or objects under study.

  • Sample: A portion or subset selected from the larger population to be studied in order to gather information about the population.

  • Parameter: A numerical characteristic that represents a property of an entire population.

  • Statistic: A numerical characteristic calculated from sample data that estimates a population parameter.

  • Representative Sample: A sample that accurately reflects the key characteristics of the population from which it was drawn.

  • Variable: A characteristic of interest measured for each entity in a population, conventionally denoted by capital letters such as $X$ or $Y$.

    • Numerical Variable: Takes on numerical values with equal units, such as weight in pounds or time in hours.

    • Categorical Variable: Places an entity into a specific category, such as political affiliation, eye color, gender, or ethnicity.

  • Data: The actual observed values recorded for a variable, which may be numbers or text.

    • Datum: A single individual value within a dataset.

• Sleep Duration Example & Analysis:

  • When recording average night sleep duration rounded to the nearest half-hour ($0.5\,\text{hours}$), data points cluster around specific central values.

  • Replicating the sleep study across different student populations (e.g., an English class versus a Physics class) would likely yield different distributions due to variances in workload, scheduling, or student demographics.

• Study Application: Cumulative GPA Analysis:

  • Context: A study conducted at a local college analyzes the cumulative GPA of graduating students.

  • Population: All students who graduated from the college last year.

  • Sample: A group of students who graduated from the college last year, randomly selected.

  • Parameter: The average cumulative GPA of students who graduated from the college last year.

  • Statistic: The average cumulative GPA of students in the sample who graduated from the college last year.

  • Variable: The cumulative GPA of one individual student who graduated from the college last year.

  • Data: The set of individual observed cumulative GPAs, such as $3.65$, $2.80$, $1.50$, and $3.90$.

  • Note: The phrase "all students who attended the college last year" describes a broader population than the target population of graduates.

• Study Application: Extracurricular Activities Survey:

  • Context: Researching the mean number of extracurricular activities in which high school students participate, based on a random survey of $100$ high school students.

  • Population: All high school students.

  • Sample: The $100$ high school students randomly surveyed.

  • Parameter: The mean number of extracurricular activities for all high school students.

  • Statistic: The mean number of extracurricular activities calculated from the $100$ surveyed students.

  • Variable: The number of extracurricular activities in which a single high school student participates.

  • Data: The observed individual counts of activities, such as $2$, $5$, and $7$.

Data, Sampling, and Variation in Data and Sampling

• Types of Data:

  • Qualitative Data: Results from categorizing or describing attributes of a population. Examples include hair color, blood type, ethnic group, and car models.

  • Quantitative Data: Values that are always numerical, generated by counting or measuring population attributes.

    • Quantitative Discrete Data: Data resulting from counting, taking on distinct, specific numerical values (often phrases beginning with "the number of"). Examples include counting phone calls per day ($0$, $1$, $2$, $3$).

    • Quantitative Continuous Data: Data resulting from precise physical measurement, capable of taking any real value within an interval. Examples include athlete heights ($183\,\text{cm}$, $172\,\text{cm}$, $191\,\text{cm}$).

• Data Type Classification Examples:

  • Gym machine counts ($12$, $15$, $10$, $22$, $20$): Quantitative discrete.

  • Lawn area measurements ($144\,\text{sq. feet}$, $160\,\text{sq. feet}$, $190\,\text{sq. feet}$, $180\,\text{sq. feet}$, $210\,\text{sq. feet}$): Quantitative continuous.

  • House exterior colors (white, yellow, white, red, white): Qualitative.

  • Number of pairs of shoes owned: Quantitative discrete.

  • Vehicle type driven: Qualitative.

  • Vacation destination choice: Qualitative.

  • Distance from home to grocery store: Quantitative continuous.

  • Number of classes taken per school year: Quantitative discrete.

  • Class tuition amounts: Quantitative continuous.

  • Calculator model used: Qualitative.

  • Movie ratings (e.g., G, PG, PG-13, R or star ratings): Qualitative / Ordinal.

  • Political party affiliation: Qualitative.

  • Weights of sumo wrestlers: Quantitative continuous.

  • Poker winnings in dollars: Quantitative continuous.

  • Number of correct quiz answers: Quantitative discrete.

  • Attitude ratings toward government: Qualitative.

  • IQ scores: Quantitative discrete (when recorded as integer scores).

  • Student academic classifications (Freshman, Sophomore, Junior, Senior) in a pie chart: Qualitative.

  • Standardized exam scores grouped into class boundaries ($50$ to less than $60$, $60$ to less than $70$, etc.) in a histogram: Quantitative continuous.

• Graphical Displays for Qualitative Data:

  • Pie Charts: Circular charts divided into wedges, where each wedge is proportional in size to the percentage of individuals in that category.

  • Bar Graphs: Displays where bar lengths (vertical or horizontal) are proportional to the count or percentage of individuals in each category.

  • Pareto Charts: Specialized bar graphs where category bars are sorted in descending order from largest to smallest size.

• Random Sampling Methods:

  • Simple Random Sampling: Every sample of a given size has an equal probability of selection. Example: Computer-generated random numbers selecting student names from an alphabetical list of $31$ students.

  • Stratified Sampling: The population is partitioned into non-overlapping subgroups called strata, and a proportional simple random sample is taken from each stratum. Examples: Sampling by academic department; selecting $25$ students from each class level (Freshman, Sophomore, Junior, Senior).

  • Cluster Sampling: The population is partitioned into clusters; a subset of clusters is randomly selected, and every member of the chosen clusters is surveyed. Examples: Randomly selecting four academic departments and surveying all students in them; selecting two class years randomly out of four and surveying all students in those years.

  • Systematic Sampling: A starting point is chosen randomly, and every $n\text{th}$ individual is selected from an ordered list. Examples: Surveying every $10\text{th}$ person at a checkpoint; selecting every $50\text{th}$ student from an alphabetical list starting from a randomly generated index.

• Non-Random Sampling and Replacement Rules:

  • Convenience Sampling: Non-random sampling utilizing readily available results or subjects. Examples: Interviewing classmates in an algebra class; surveying the first $100$ students walking past a library. Highly prone to bias.

  • Sampling with Replacement: Selected individuals are returned to the population prior to subsequent selections, allowing an individual to be chosen multiple times.

  • Sampling without Replacement: Selected individuals are permanently removed from the selection pool for subsequent draws. When the sample size is small relative to the population, sampling without replacement mathematically approximates sampling with replacement.

• Sampling Errors, Nonsampling Errors, and Bias:

  • Sampling Errors: Errors caused by the actual process of sampling, such as having a sample size that is too small.

  • Nonsampling Errors: Errors caused by factors unrelated to the sampling process, such as defective counting equipment or biased question phrasing.

  • Sampling Bias: Created when data collection systematically favors certain population members over others, rendering conclusions invalid.

• Sampling Method Identification Exercises:

  • Selecting $6$ players (ages $8\text{--}10$), $7$ players (ages $11\text{--}12$), and $3$ players (ages $13\text{--}14$): Stratified sampling.

  • Interviewing all HR personnel across five high-tech companies: Cluster sampling.

  • Interviewing $50$ female and $50$ male high school teachers: Stratified sampling.

  • Interviewing every $3\text{rd}$ cancer patient from an hospital roster: Systematic sampling.

  • Using computer-generated random numbers to select $50$ students: Simple random sampling.

  • Interviewing algebra classmates regarding average jean ownership: Convenience sampling.

• Representativeness Analysis of Specific Samples:

  • High school GPA study using university honor students: Non-representative and heavily biased upward.

  • Most popular cereal under age 10 surveyed outside a supermarket for 3 hours: Biased by store location, time of day, and demographic access.

  • U.S. adult average income surveyed via state-based cluster sampling of U.S. congressmen: Severely biased; congressmen earnings do not reflect general population income.

  • Public transit proportions evaluated by interviewing Central Park bench sitters in NYC: Biased toward park visitors and non-representative of general transit usage.

  • Average stay cost in Massachusetts hospitals surveyed via simple random sampling of 100 hospitals statewide: Representative.

• Critical Evaluation Factors for Statistical Studies:

  • Non-representative / Biased Samples: Yield invalid, inaccurate results.

  • Self-Selected Samples: Call-in or voluntary response surveys reflect overrepresented extreme views.

  • Sample Size Issues: Small samples reduce reliability, though unavoidable in constraints like automobile crash testing or rare medical trials.

  • Undue Influence: Framing questions or environment to distort subject responses.

  • Non-response or Refusal: High non-response shifts representative traits toward respondents with strong opinions.

  • Causality Fallacy: Confusing correlation with causation; association between variables does not imply one causes the other.

  • Self-Funded or Self-Interest Studies: Studies financed by parties with outcome interests require rigorous scrutiny regarding impartiality.

  • Misleading Data Presentation: Scaled graph manipulation, missing baseline context, or selective reporting.

  • Confounding: Overlapping effects of multiple factors preventing isolation of individual variable impact.

• Detailed Study Scenarios & Bias Evaluation:

  • ABC College Textbook Spending Study ($10,000$ upperclassmen population):

    • Sample 1: $10$ organic chemistry students spending $\{\$128, \$87, \$173, \$116, \$130, \$204, \$147, \$189, \$93, \$153\}$.

    • Sample 2: Every $5\text{th}$ senior in P.E. classes ($10$ total) spending $\{\$50, \$40, \$36, \$15, \$50, \$100, \$40, \$53, \$22, \$22\}$.

    • Evaluation: Neither sample represents the $10,000$ student population. Chemistry students take high-cost STEM courses, biasing spending upward; P.E. seniors take low-cost courses, biasing spending downward.

  • Radio Station Listener Preference Study ($20,000$ audience size):

    • Method: Convenience sample of $200$ attendees at a station concert event ($24$ preferred talk shows, $176$ preferred music).

    • Evaluation: Biased. Concert attendees inherently skew toward music preferences, underrepresenting home or commuting talk-show listeners.

Frequency, Frequency Tables, and Levels of Measurement

• Rounding Off Rules:

  • Carry final calculated statistics to one additional decimal place beyond the raw data.

  • Perform rounding only on the final answer; do not round intermediate steps.

• Four Levels of Measurement:

  1. Nominal Scale Level: Qualitative data consisting of categories, names, labels, colors, or yes/no responses. Data cannot be meaningfully ordered (e.g., placing pizza before sushi has no ordinal meaning).

  2. Ordinal Scale Level: Categorical data that can be ordered or ranked, but differences between ranks cannot be calculated or measured meaningfully (e.g., top five national parks ranked $1$ to $5$).

  3. Interval Scale Level: Ordered numerical data where differences between values are meaningful and measurable, but no absolute natural zero point exists ($0$ does not denote complete absence). Example: Celsius ($^\circ\text{C}$) and Fahrenheit ($^\circ\text{F}$) temperature scales ($40^\circ = 100^\circ - 60^\circ$, but $0^\circ$ does not represent absence of thermal energy; $-10^\circ\,\text{F}$ and $-15^\circ\,\text{C}$ exist).

  4. Ratio Scale Level: Ordered numerical data with measurable differences AND a true natural zero starting point ($0$ denotes complete absence of the property). Ratios between values are mathematically valid. Example: Exam scores ($0$ to $100$ points), where a score of $80$ is four times greater than a score of $20$.

• Measurement Scale Examples:

  • High school soccer player athletic ability (superior, average, above average): Ordinal.

  • Baking temperatures ($350^\circ$, $400^\circ$, $325^\circ$, $250^\circ$, $300^\circ$): Interval.

  • Colors of crayons in a 24-count box: Nominal.

  • Social Security numbers: Nominal.

  • Incomes measured in dollars: Ratio.

  • Satisfaction survey ($1 = \text{very satisfied}$, $2 = \text{somewhat satisfied}$, $3 = \text{not satisfied}$): Ordinal.

  • Preferred TV shows (comedy, drama, sci-fi, sports, news): Nominal.

  • Time of day on an analog watch: Interval.

  • Distance in miles to the closest grocery store: Ratio.

  • Calendar dates ($1066$, $1492$, $1644$, $1947$, $1944$): Interval.

  • Heights of adult women aged 21–65: Ratio.

  • Letter grades (A, B, C, D, F): Ordinal.

• Frequency Definitions:

  • Frequency ($f$): The number of times a specific value occurs within a dataset.

  • Relative Frequency ($RF$): The proportion or fraction of times a value occurs relative to total outcomes $n$: $RF = \frac{f}{n}$.

  • Cumulative Relative Frequency ($CRF$): The accumulation of consecutive relative frequencies up to the current row.

• Frequency Table Construction 1: Student Work Hours ($n = 20$ students):

  • Raw Data: $5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3$

  • Data Value $2$: Frequency $f = 3$, Relative Frequency $RF = \frac{3}{20} = 0.15$, Cumulative Relative Frequency $CRF = 0.15$

  • Data Value $3$: Frequency $f = 5$, Relative Frequency $RF = \frac{5}{20} = 0.25$, Cumulative Relative Frequency $CRF = 0.40$

  • Data Value $4$: Frequency $f = 3$, Relative Frequency $RF = \frac{3}{20} = 0.15$, Cumulative Relative Frequency $CRF = 0.55$

  • Data Value $5$: Frequency $f = 6$, Relative Frequency $RF = \frac{6}{20} = 0.30$, Cumulative Relative Frequency $CRF = 0.85$

  • Data Value $6$: Frequency $f = 2$, Relative Frequency $RF = \frac{2}{20} = 0.10$, Cumulative Relative Frequency $CRF = 0.95$

  • Data Value $7$: Frequency $f = 1$, Relative Frequency $RF = \frac{1}{20} = 0.05$, Cumulative Relative Frequency $CRF = 1.00$

• Frequency Table Construction 2: Soccer Player Heights ($n = 100$ male semiprofessional players):

  • Interval $59.50\text{--}61.95$: Frequency $f = 5$, Relative Frequency $RF = 0.05$, Cumulative Relative Frequency $CRF = 0.05$

  • Interval $61.95\text{--}63.95$: Frequency $f = 3$, Relative Frequency $RF = 0.03$, Cumulative Relative Frequency $CRF = 0.08$

  • Interval $63.95\text{--}65.95$: Frequency $f = 15$, Relative Frequency $RF = 0.15$, Cumulative Relative Frequency $CRF = 0.23$

  • Interval $65.95\text{--}67.95$: Frequency $f = 40$, Relative Frequency $RF = 0.40$, Cumulative Relative Frequency $CRF = 0.63$

  • Interval $67.95\text{--}69.95$: Frequency $f = 17$, Relative Frequency $RF = 0.17$, Cumulative Relative Frequency $CRF = 0.80$

  • Interval $69.95\text{--}71.95$: Frequency $f = 12$, Relative Frequency $RF = 0.12$, Cumulative Relative Frequency $CRF = 0.92$

  • Interval $71.95\text{--}73.95$: Frequency $f = 7$, Relative Frequency $RF = 0.07$, Cumulative Relative Frequency $CRF = 0.99$

  • Interval $73.95\text{--}75.95$: Frequency $f = 1$, Relative Frequency $RF = 0.01$, Cumulative Relative Frequency $CRF = 1.00$

  • Percentage Analysis:

    • Percentage of heights less than $65.95\,\text{inches}$: $23\%$ (corresponds directly to $CRF = 0.23$ at upper boundary $65.95$).

    • Percentage of heights falling between $61.95$ and $65.95\,\text{inches}$: $3\% + 15\% = 18\%$ (or $0.23 - 0.05 = 0.18$).

Experimental Design and Ethics

• Experimental Design Concepts:

  • Explanatory Variable: The independent variable manipulated by researchers to determine its effect on another variable.

  • Response Variable: The dependent variable that measures the outcome of interest caused by changes in the explanatory variable.

  • Treatments: Specific values or conditions of the explanatory variable assigned to experimental subjects.

  • Experimental Unit: A single individual object or entity measured within the experiment.

• Observational Studies vs. Experiments:

  • Observational Study: Researchers measure variables without manipulating treatments or directly controlling environments. Observational studies cannot prove causality due to lurking variables.

  • Lurking Variables: Unmeasured or uncontrolled additional variables that obscure cause-and-effect relationships.

  • Random Assignment: Assigning subjects to treatment conditions randomly to distribute lurking variables equally across all groups.

• Control Mechanisms and Blinding:

  • Control Group: A baseline group receiving a dummy or placebo treatment to isolate active treatment effects from experimental influence.

  • Placebo: A neutral treatment (e.g., sugar pill) with no active ingredient, controlling for the power of suggestion.

  • Single-Blind Experiment: Participants do not know whether they are assigned to active treatments or placebos.

  • Double-Blind Experiment: Both the participants AND the researchers interacting directly with them are unaware of treatment assignments.

• Experimental Study Analysis 1: Aspirin & Heart Attack Study:

  • Context: $400$ men aged $50$ to $84$ are randomly split into two groups (aspirin daily vs. placebo daily for 3 years) to test heart attack prevention.

  • Population: Men between the ages of $50$ and $84$

  • Sample: The $400$ recruited men participating in the study

  • Experimental Units: Each individual man participating in the trial

  • Explanatory Variable: Regular daily intake of aspirin

  • Response Variable: Heart attack occurrence status

  • Treatments: Daily aspirin pill vs. daily placebo pill

• Experimental Study Analysis 2: Smell, Taste & Maze Performance:

  • Context: Participants complete mazes three times wearing floral-scented masks and three times wearing unscented masks, assigned at random to mask order sequence. Time and scent perception ratings are recorded.

  • Explanatory Variable: Mask scent type (floral-scented vs. unscented)

  • Response Variable: Maze completion time and subject impression rating of scent (positive, negative, or neutral)

  • Treatments: Floral-scented mask trial condition vs. unscented mask trial condition

  • Lurking Variables: Maze practice effect (learning curve across trials), mental fatigue, individual baseline spatial ability

  • Blinding Feasibility: Full participant blinding is impossible if the floral scent is noticeable, but double-blind data recording and timing analysis can be maintained.

• Experimental Study Analysis 3: Birth Order and Personality:

  • Cannot be conducted as a randomized experiment because researchers cannot randomly assign birth order to individuals.

  • Main Problem: Observational nature introduces uncontrolled confounding factors (family size, socioeconomic status, parental age differences) that prevent causal conclusions.

• Experimental Study Design 4: Texting While Driving Simulation:

  • Explanatory Variable: Driving condition (distracted by texting vs. undistracted driving)

  • Response Variable: Driver reaction brake time (seconds elapsed when leading vehicle brakes)

  • Treatments: Driving while actively texting vs. driving without phone interaction

  • Participant Selection: Must consider age, driving experience, reaction speed, and phone usage habits

  • Design Considerations: Randomly assigning participants to a repeated-measures design (where each driver tests under both conditions in randomized order) controls for individual driving skill variability better than splitting into separate groups.

  • Lurking Variables: Vehicle simulator familiarity, phone interface familiarity, fatigue, ambient lighting

  • Blinding Application: Drivers cannot be blinded to texting, but timing evaluators can be blinded to driver treatment conditions during video measurement analysis.

• Ethical Standards in Statistical Research:

  • Minimize risks to participants and ensure risks are reasonable.

  • Obtain formal informed consent from human participants.

  • Protect participant privacy and data confidentiality strictly.

• Unethical Behaviors & Corrective Actions in Neighborhood Surveying:

  • Selecting a comfortable street block: Introduces convenience selection bias. Correction: Randomly select blocks across the entire target community.

  • Skipping unreached homes without returning: Introduces non-response bias. Correction: Record unreached addresses and make repeated call-backs at varied times.

  • Fabricating responses for missed homes: Constitutes fraudulent data fabrication. Correction: Exclude uncollected responses entirely and document the non-response rate.

• Unethical Behaviors & Corrective Actions in Commercial Juice Testing:

  • Study commissioned by apple juice vendor: Financial conflict of interest. Correction: Utilize independent third-party research administration.

  • Restricted choice options (only apple and cranberry): Selection bias. Correction: Include comprehensive choices or open-ended preference options.

  • Unblinded taste testing: Allows brand recognition bias. Correction: Implement blind taste testing.

  • Misleading advertising claims: Advertising $25\%$ Brand X vs $33\%$ Brand Y as "Most teens like Brand X as much as or more than Brand Y" distorts data where $42\%$ had no preference. Correction: Accurately report full comparative statistical distributions.