Chapter 1: Introduction to Data & Study Design

What is Statistics?

Statistics is the science of collecting, organizing, analyzing, and interpreting data to draw conclusions. It helps us make informed decisions based on evidence rather than guesses.

• Collecting data: Gathering information from populations or samples.

• Organizing data: Structuring data for analysis.

• Analyzing data: Using statistical methods to summarize and interpret data.

• Interpreting data: Drawing conclusions and making predictions.

Population vs Sample

Understanding the difference between population and sample is fundamental in statistics.

• Population: The entire group of interest (e.g., all college students).

• Sample: A subset of the population selected for study (e.g., 100 surveyed students).

• Census: Collecting data from every member of the population.

Variables

A variable is a characteristic measured on individuals or objects. Variables are classified as qualitative (categorical) or quantitative (numerical).

• Categorical (Qualitative) Variables: Describe qualities or categories (e.g., eye color, gender).

• Numerical (Quantitative) Variables: Describe quantities obtained through counting or measuring (e.g., height, weight).

Subtypes of Quantitative Variables

• Discrete Variables: Counted values (e.g., number of pets).

• Continuous Variables: Measured values (e.g., height, weight).

Descriptive vs Inferential Statistics

• Descriptive Statistics: Summarize and organize data (mean, median, graphs).

• Inferential Statistics: Use sample data to make conclusions about a population.

Types of Studies

• Observational Study: Researchers observe subjects without assigning treatments. Can show association but not causation.

• Experiment: Researchers assign treatments to subjects. Can establish causation.

Sampling Methods

• Simple Random Sample: Every member has an equal chance of selection.

• Stratified Sample: Divide population into groups (strata), then randomly sample from each.

• Cluster Sample: Divide population into clusters, randomly select clusters, and sample everyone in selected clusters.

• Systematic Sample: Select every nth individual.

• Convenience Sample: Select individuals easiest to reach; often biased.

Chapter 2: Picturing Variation with Graphs

Describing Distributions

Graphical methods help visualize data and identify patterns. Key features to describe:

• Shape: Symmetric, skewed right, skewed left.

• Center: Typical value (mean or median).

• Variability (Spread): How spread out the data are.

• Outliers: Unusual observations.

Graphs for Numerical Data

• Dotplot: Best for small datasets.

• Stemplot: Shows original values.

• Histogram: Best for larger datasets; bars touch, shows frequency.

Graphs for Categorical Data

• Bar Chart: Most common for categorical data; bars separated.

• Pie Chart: Shows percentages as slices of a circle.

• Pareto Chart: Bar chart arranged largest to smallest.

Distribution Shapes

• Symmetric: Mean ≈ Median.

• Skewed Right: Tail points right; Mean > Median.

• Skewed Left: Tail points left; Mean < Median.

Modes

• Unimodal: One peak.

• Bimodal: Two peaks.

• Multimodal: Three or more peaks.

Chapter 3: Numerical Summaries of Center and Variation

Measures of Center

• Mean (x̄): Arithmetic average.

• Median: Middle value when data are ordered.

Mean vs Median

Measures of Spread

• Range:

• Variance (s²): Square of standard deviation.

• Standard Deviation (s): Measures typical distance from the mean.

• Interquartile Range (IQR):

Quartiles and Five-Number Summary

• Q1: 25th percentile

• Q2: Median (50th percentile)

• Q3: 75th percentile

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

Boxplots

Boxplots display the five-number summary and show center, spread, skewness, and outliers.

Outlier Detection

• Lower Fence:

• Upper Fence:

• Values outside the fences are considered outliers.

Empirical Rule (68-95-99.7 Rule)

• For bell-shaped distributions:

• Within 1 SD: 68% of data

• Within 2 SD: 95% of data

• Within 3 SD: 99.7% of data

Z-Score

• Formula:

• Interpretation: Positive z = above mean; Negative z = below mean

Chapter 4: Regression Analysis – Exploring Associations Between Variables

Scatterplots

Scatterplots are used to study relationships between two numerical variables. Always describe trend, strength, shape, and outliers.

• Positive Trend: As x increases, y increases.

• Negative Trend: As x increases, y decreases.

• No Association: No pattern.

Correlation (r)

• Measures strength and direction of a linear relationship.

• Range:

• Interpretation: +1 = perfect positive, 0 = no linear relationship, -1 = perfect negative

Regression Equation

• Formula:

• Where is the predicted value, is the intercept, is the slope.

Slope Interpretation

• Positive slope: For every 1-unit increase in x, y increases by slope units.

• Negative slope: For every 1-unit increase in x, y decreases by slope units.

Residual

• Formula:

• Interpretation: Positive residual = actual > predicted; Negative residual = actual < predicted

Coefficient of Determination (R²)

• Formula:

• Interpretation: Percent of variation explained by the model

Extrapolation

• Making predictions outside the observed data range is risky and should be avoided.

Summary Table: Variable Types

Exam Preparation Tips

• Master the differences between population vs sample, categorical vs numerical, discrete vs continuous, descriptive vs inferential, and observational vs experimental studies.

• Be able to describe distributions using shape, center, spread, and outliers.

• Know how to calculate and interpret mean, median, standard deviation, IQR, z-scores, correlation, regression equations, and R².

• Understand the risks of extrapolation and the difference between association and causation.

Additional info: The included image visually reinforces the classification of variables, directly supporting the explanation of variable types and their subcategories.