Statistics: The Art and Science of Learning From Data

Populations, Samples, Parameters, and Statistics

Statistics is the discipline concerned with collecting, analyzing, interpreting, and presenting data. Understanding the distinction between populations and samples is fundamental.

• Population: The entire group of individuals or objects of interest.

• Sample: A subset of the population, selected for study.

• Parameter: A numerical summary describing a characteristic of the population (e.g., population mean).

• Statistic: A numerical summary describing a characteristic of the sample (e.g., sample mean).

Exploring Data With Graphs and Numerical Summaries

Types of Variables

Variables are classified based on the nature of their values.

• Categorical: Values represent categories or groups (e.g., gender, color).

• Quantitative: Values are numerical and arithmetic operations make sense (e.g., height, weight).

  • Discrete: Countable values (e.g., number of siblings).

  • Continuous: Infinite possible values within a range (e.g., temperature).

Graphs for Quantitative Variables

• Dotplots, Stem-and-Leaf Plots, Histograms: Visualize distributions of quantitative data.

• Shapes of Distributions:

  • Symmetric

  • Left skewed

  • Right skewed

  • Bimodal

  • Uniform

• SOCS: Shape, Center, Spread, Outliers—framework for describing distributions.

Measures of Center, Spread, and Position

• Measures of Center: Mean, median, mode.

• Measures of Spread: Standard deviation, variance, range.

• Empirical Rule: For symmetric, bell-shaped distributions:

  • 68% within 1 standard deviation

  • 95% within 2 standard deviations

  • 99.7% within 3 standard deviations

• Effects of Outliers and Skewness: Outliers and skewness can distort measures of center and spread.

Measures of Position

• Five Number Summary: Minimum, Q1, Median, Q3, Maximum.

• Interquartile Range (IQR):

• Box-and-Whisker Plot: Visualizes the five number summary; side-by-side plots compare groups.

Exploring Relationships Between Two Variables: Regression

Scatterplots and Correlation

Scatterplots display the relationship between two quantitative variables.

• x: Explanatory variable

• y: Response variable

• DOTS: Direction, Trend, Strength, Outliers—used to describe scatterplots.

• Correlation Coefficient (r): Measures strength and direction of linear relationship; ; unitless.

Regression Line

• Equation:

• a: y-intercept (value of y when x = 0)

• b: Slope (change in predicted y for a one-unit increase in x)

• Calculation:

• Regression line passes through

• Residual: (difference between observed and predicted value)

• Coefficient of Determination (): Percentage of variation in y explained by x.

Cautions in Regression and Categorical Data

Cautions in Regression

• Extrapolation: Predicting outside the range of observed data can be unreliable.

• Influential Outliers: Points far from the trend can affect regression results.

• Correlation ≠ Causation: A strong association does not imply causality; lurking variables may exist.

Categorical Data Analysis

• Measurement: Frequency (count), proportion, percentage.

• Graphs: Bar chart, pie chart.

• Contingency Tables: Summarize two categorical variables.

• Calculations:

  • Conditional proportions

  • Difference of proportions

  • Relative risk

• Simpson’s Paradox: Association reverses when a third variable is considered.

Gathering Data

Types of Studies

• Observational Study: No treatment imposed; researcher observes.

• Experimental Study: Treatment imposed; allows for causal inference.

• Individuals: Subjects of study (people, animals, objects).

Sampling Methods

• Good Sampling:

  • Simple Random Sample (SRS)

  • Cluster Sample

  • Stratified Sample

  • Multistage Sample

• Bad Sampling:

  • Convenience Sample

  • Volunteer Sample

• Sample Surveys: Questionnaire, telephone interview, personal interview.

• Margin of Error:

• Sources of Bias: Undercoverage, nonresponse bias, response bias, wording of questions.

Experimental Design

• Response Variable: Variable measured in experiment.

• Experimental Units: Subjects receiving treatments.

• Treatments: Conditions applied to units.

• Placebo: Dummy treatment.

• Control Variability: Avoid lurking/confounding variables; use control group, blinding.

• Blinding:

  • Single blind: Subject unaware of treatment.

  • Double blind: Both subject and researcher unaware.

• Randomization: Random assignment to treatments.

• Replication: Number of units per treatment.

• Multifactor Experiments: Factors (categorical explanatory variables), levels, treatments (combinations).

• Matched Pairs, Blocking: Methods to control for confounding.

Types of Studies

• Cross-sectional: Snapshot in time.

• Case-control: Retrospective (looking back).

• Prospective: Forward-looking.

• IRB: Institutional Review Board—ensures ethical conduct.

Probability in Our Daily Lives

Basic Probability Concepts

• Sample Space: Set of all possible outcomes.

• Event: Subset of sample space.

• Probability: Between 0 and 1; sum of all outcome probabilities equals 1.

Probability Formulas

• Complement:

• Union:

• Disjoint Events: , so

• At Least One:

Independence and Conditional Probability

• Independence: Knowledge of one event does not affect the other.

• Independent Events:

• Conditional Probability:

• Independent Events:

• Sensitivity: ; Specificity:

• Tree Diagrams: Visual tool for organizing probabilities.

Random Variables and Probability Distributions

Continuous Probability Distributions

• Random Variable: Variable whose value is a numerical outcome of a random process.

• Continuous Random Variable: Infinite possible outcomes.

• Probability: Area under density curve; total area = 1.

• Normal Distribution: Bell-shaped, symmetric; parameters (mean), (standard deviation).

• Standard Normal: ,

• z-score:

• Standard Normal Table: Gives area to the left of z.

• Finding x from area: Use z-table and solve for x.

Discrete Probability Distributions

• Discrete Random Variable: Finite list of outcomes.

• Discrete Probability Distribution: Probabilities between 0 and 1; sum to 1.

• Mean (Expected Value):

• Binomial Distribution: Parameters n (number of trials), p (probability of success).

• BINS/NIBS: Binary, Independent, Fixed Number of Trials, Same Probability of Success.

• Mean:

• Standard Deviation:

• Probability of x successes: , where for

Summary Table: Types of Sampling Methods

Summary Table: Measures of Center and Spread