Constructing Graphical and Tabular Displays of Data

Variable Classification

• Categorical Variable: Consists of names or labels identifying groups of individuals.

  • Examples: Academic major, ZIP code of a home in Arkansas, and variables where numbers represent categories (e.g., $0$ for man, $1$ for woman).

• Numerical Variable: Consists of measurable quantities.

  • Examples: Age of a hip-hop artist (e.g., Travis Scott at $28$ years or Kendrick Lamar at $32$ years) and maximum speed of a car (e.g., Toyota Camry at $114 \text{ mph}$ or Porsche 911 at $201 \text{ mph}$).

Frequency and Distributions

• Frequency: The number of observations in a specific category.

• Frequency Table: Lists all categories and their respective frequencies.

• Relative Frequency: The proportion of total observations belonging to a category, calculated as:

$\text{Relative Frequency} = \frac{\text{Frequency of Category}}{\text{Total Number of Observations}}$

• Sum of Relative Frequencies: For any categorical variable, the sum of relative frequencies across all categories always equals $1$.

• Example Calculations (Movie Genres):

  • Action: Frequency of $6$ out of $42$ observations leads to a relative frequency of $6/42 = 3/21 \approx 0.143$ ($14.3\%$).

  • Comedy: Largest relative frequency at $16/42 = 8/21 \approx 0.381$ ($38.1\%$).

Logical Grouping (AND and OR)

• AND: Refers to the intersection of criteria. For November 2020, dates that are in the third week AND are Thursdays includes only the date $19$.

• OR: Refers to the union of criteria. Dates in the third week OR Thursdays in November 2020 include: $5$, $12$, $15$, $16$, $17$, $18$, $19$, $20$, $21$, and $26$.

Graphical Data Interpretation

• Frequency Bar Graph: Displays categories on the horizontal axis and frequencies on the vertical axis with bars representing the count for each category.

• Multiple Bar Graph: Used to compare proportions between different groups (e.g., men vs. women).

• Political Survey Analysis (1960 adults):

  • Women identifying as Democrats: $0.37$ proportion.

  • Men identifying as Independents: $0.43$ (the greatest proportion for men).

  • Proportion vs. Count: A smaller proportion can represent a larger count if the total group size is larger. For example, a $0.38$ proportion of $1081$ women ($411$) is more individuals than a $0.43$ proportion of $879$ men ($378$).

• Sampling Error: Findings from a sample (e.g., $24\%$ of men surveyed are Republicans) cannot be generalized as an exact percentage for the entire population (e.g., all American men) due to potential fluctuations in data.