Unit 1: Exploring One-Variable Data

Categorical Variables

• Definition: Take on values that are category names or group labels.
• Representation:
  • Frequency tables or relative frequency tables.
  • Graphical displays: Bar graphs, dot plots, pie charts.
• Example 1.1: Survey of 2,000 parents about desired school year length:
  • 180 days: 1,100 parents (55%).
  • 160 days: 300 parents (15%).
  • 200 days: 500 parents (25%).
  • No Opinion: 100 parents (5%).

Quantitative Variables

• Definition: Take on numerical values for measured or counted quantities.
• Categories:
  • Discrete: Finite or countable values with gaps.
  • Continuous: Infinite values with no gaps.
• Graphical Representation: Dotplots, histograms, stemplots, cumulative plots, boxplots.
• Example 1.2: AP classes taken by 2,200 seniors:
  • 0 classes: 400 seniors (18%).
  • 1 class: 500 seniors (23%).
  • 2 classes: 900 seniors (41%).
  • 3 classes: 300 seniors (14%).
  • 4 classes: 100 seniors (5%).

Describing Quantitative Data Distribution

• Center: Separates values roughly in half.
• Spread: Range of values.
• Patterns:
  • Clusters: Natural subgroups.
  • Gaps: Areas with no values.
• Shape:
  • Unimodal: Single peak.
  • Bimodal: Two peaks.
  • Symmetric: Mirror images.
  • Skewed: Asymmetric distribution.
  • Bell-shaped: Symmetric with center mound.
  • Uniform: Even distribution.
• Example 1.3: Hodgkins lymphoma age distribution shows bimodal pattern.

Summary Statistics for Quantitative Variables

• Descriptive Statistics: Summarization of data.
• Inferential Statistics: Inferences from data.
• Average: Representative value (mean or median).
  • Median: Middle value of ordered set.
  • Mean: Sum of values divided by count.
• Variability:
  • Range: Difference between max and min.
  • Interquartile Range (IQR): Middle 50% range.
  • Variance: Average of squared differences from mean.
  • Standard Deviation: Square root of variance.
• Example 1.4 & 1.5: Home run distances and salaries illustrating mean and variability.

Graphical Representations

• Skewness: Mean vs. median indicates skewness.
• Example 1.7: Faculty salaries suggesting right skew.

Comparing Distributions

• Methods: Back-to-back stemplots, side-by-side histograms, parallel boxplots, cumulative plots.
• Example Comparisons:
  • Example 1.9: NBA team wins via stemplot.
  • Example 1.10: Student sleep hours via histograms.
  • Example 1.11: Stock price trends via boxplots.
  • Example 1.12: Population age comparisons via cumulative frequency plots.

The Normal Distribution

• Characteristics: Symmetric, bell-shaped.
• Mean = Median: Centered distribution.
• Empirical Rule: 68-95-99.7% rule for normal distributions.
• Example 1.13: Taxicab mileage under normal distribution.