Section 2.1 Categorical Variables

SECTION 2.1 CATEGORICAL VARIABLES

One Categorical Variable

• Summary Statistics:

  • Frequency Table

  • Proportion

• Visualization:

  • Bar Chart

  • Pie Chart

Two Categorical Variables

• Summary Statistics:

  • Two-Way Table

  • Difference in Proportions

• Visualization:

  • Segmented Bar Chart

  • Side-by-Side Bar Chart

Observational Data and Generalization

• Causality:

  • Causality cannot be concluded from observational data collected from surveys.

• Generalization:

  • Sample may not represent the larger population due to lack of randomness.

Summarizing Data

• Need methods to summarize and visualize data for better understanding.

• Type of Summary Statistics and Visualization Methods:

  • Depends on type of variable (categorical or quantitative).

• Descriptive Statistics:

  • Also known as exploratory data analysis, targets summarizing and visualizing variables and their relationships.

Summarizing Data for One Categorical Variable

• Example: Cell Phone Ownership Survey (2012)

  • Categories: Android, iPhone, Blackberry, Non-smartphone, No cell phone.

Frequency Table

• Displays survey results in counts per category:

  • Android: 458

  • iPhone: 437

  • Blackberry: 141

  • Non-smartphone: 924

  • No cell phone: 293

  • Total: 2253

Proportion Calculation

• Proportion found using formula:

  • Formula: p = (number in category) / (total sample size)

• Notation:

  • Proportion for a Sample: (p-hat) p̂

  • Proportion for a Population: (p)

Example Calculation of Proportion

• Non Smartphone Owners:

  • p̂ = 293 / 2253 = 0.13

  • Equivalent Percentage: 13%

Relative Frequency Table

• Shows proportion of cases in each category:

  • Android: 0.203

  • iPhone: 0.194

  • Blackberry: 0.063

  • Non Smartphone: 0.410

  • No Cell Phone: 0.130

• Total of Table: Sums to 1 or 100%.

Examples of Proportions with Notation

• Problem a: For community college statistics, proportion of students preferring UCLA:

  • Total: 45, Preferred: 19

  • Answer: (p = \frac{19}{45} \approx 0.42)

• Problem b: For apples, with 2 out of 11 being rotten:

  • Proportion: (\hat{p} = \frac{2}{11} \approx 0.18)

Relative Frequency Construction

• Water Taste Preference Example:

  • Categories: Tap, Aquafina, Fiji, Sam's Choice

  • Total surveyed: 100

  • Relative Frequencies calculated as follows:

    • Tap: 0.10

    • Aquafina: 0.25

    • Fiji: 0.41

    • Sam's Choice: 0.24

Visualizing Data for One Categorical Variable

• Barplot:

  • Bar height corresponds to counts of cases in each category.

• Pie Chart:

  • Relative area of each slice represents proportions per category.

Summarizing Data for Two Categorical Variables

• Student Relationship Example: Surveyed university students on gender and relationship status.

Two-Way Table

• For relationship and gender:

  Female Male Total
  In a Relationship	32	10	42
  It’s Complicated	12	7	19
  Single	63	45	108
  Total	107	62	169

Proportion Calculations

• Proportions can be derived for categories using totals from the two-way table:

  • Sample Proportions:

    • Students in a relationship: [ \frac{42}{169} \approx 0.248 ]

    • Female students: [ \frac{107}{169} \approx 0.634 ]

    • Students in relationship that are female: [ \frac{32}{42} \approx 0.762 ]

Caution on Proportion Interpretation

• A distinction between proportions of categories is critical:

  • The proportion of females in relationships differs from that of relationships being female.

Difference in Proportions Concept

• Difference between proportions can show relationships between two categorical variables.

Example Calculations for Differences

• For singles by gender:

  • Female proportion: ( \frac{63}{107} \approx 0.589 )

  • Male proportion: ( \frac{45}{62} \approx 0.726 )

  • Difference: (0.589 - 0.726 = -0.137)

Example 5: Comparing Proportions of Female Students

• Calculate:

  • ( \hat{p}_1 = \frac{32}{42} \approx 0.762 ) (in relationship)

  • ( \hat{p}_2 = \frac{63}{108} \approx 0.583 ) (single students)

  • Difference: (0.762 - 0.583 = 0.179)

Visualizing Two Categorical Variables

• Side by Side Bar Chart:

  • Provides comparative visuals for two categories

• Segmented Bar Chart:

  • Displays stacked, relative proportions.

Example 6: Therapy and Sleep Improvement Study

• Two-Way Table Creation:

  • Groups: Therapy vs. No Treatment

• Proportion Calculations:

  • Therapy Group Improvements: ( \hat{p} ' = \frac{14}{20} = 0.7 )

  • No Treatment Improvements: ( \hat{p} ( = \frac{3}{20} = 0.15 )

  • Difference: ( \hat{p} ' - \hat{p} ( = 0.7 - 0.15 = 0.55)