Categorical Data Representations, Contingency Tables, and Conditional Distributions

Categorical Data Representations and Misleading Graphs

• Primary graphical displays for categorical data include pie charts and bar charts.

• Visual Analysis Caution:

  • Graphs and charts can easily be visually misleading.

  • Data presented graphically must be critically analyzed to verify that visual representations accurately reflect underlying numeric values.

• Example of a visually misleading graph:

  • A graph displaying run time ranges where a blue slice visually appears to be the largest section (occupying nearly a quarter or $25\%$ of the total display area).

  • Numeric analysis reveals that the yellow slice is actually the largest section at $12.6\%$, or a slice labeled $10.21\%$ fails to match its visual proportions.

Frequency Distributions and Categorical Graphing Methods

• Bar Charts:

  • Structure: Displays category names or labels on one axis (typically the x-axis) and frequency or relative frequency on the opposing axis (typically the y-axis).

  • Orientation: Can be rendered vertically or horizontally.

  • Structural Separation Rule: In a categorical bar chart, individual bars MUST NOT touch each other. This distinct gap differentiates bar charts from quantitative histograms, where bars touch.

  • Category Inclusion: Categorical datasets frequently incorporate an "Other" category to account for uncollected or remaining data points so that the total relative frequency equals $100\%$.

  • Electricity Generation Example:

  • Categories include Coal, Hydroelectric, Nuclear, Natural Gas, Petroleum, and Other Renewable Sources.

  • Standard bar charts often require visual estimation of values relative to axis gridlines unless explicit numerical labels are provided.

• Pie Charts:

  • Structure: Displays proportions as relative slices of a complete circle ($100\%$).

  • Displays precise numerical percentages directly alongside each corresponding category slice (e.g., Coal at $46.1\%$, Petroleum at $1.2\%$).

Desktop Conferencing Market Case Study

• Dataset Context: A $2013$ report detailing market share for desktop conferencing applications.

• Provided Market Share Data:

  • Cisco Systems: $58\%$ share

  • Citrix: $13\%$ share

  • Microsoft: $11\%$ share

• Constructing the Relative Frequency Table:

  • Sum of provided market shares: $58\% + 13\% + 11\% = 82\%$

  • Calculated "Other" app category: $100\% - 82\% = 18\%$

  • Frequency Distribution Table:

  • Cisco: $58\%$ relative frequency ($0.58$

  • Citrix: $13\%$ relative frequency ($0.13$

  • Microsoft: $11\%$ relative frequency ($0.11$

  • Other: $18\%$ relative frequency ($0.18$

  • Total: $100\%$ ($1.00$

• Graph Construction Rules:

  • Bar Chart: Place market categories on the horizontal axis (Cisco, Citrix, Microsoft, Other). Set vertical scale up to at least $58\%$ (e.g., intervals of $10\%$ up to $60\%$). Ensure bars do not touch.

  • Pie Chart: Draw a circle with an approximate center point. Partition slices for Cisco ($58\%$), Other ($18\%$), Citrix ($13\%$), and Microsoft ($11\%$).

• Examination Protocol Note: Exams consist of multiple-choice questions where manual graph construction is not required, but identifying correct graphs and reading graph data accurately is required.

Contingency Tables for Two Categorical Variables

• Definition: A contingency table displays frequencies for two categorical variables simultaneously.

• Structure:

  • Rows represent categories of one categorical variable.

  • Columns represent categories of the second categorical variable.

  • Cells contain joint frequencies (counts of cases meeting specific combined conditions of both variables).

  • Margins display row totals and column totals.

• Proportions in Contingency Tables:

  • Joint Proportions (Relative Frequencies): Computed by dividing an individual cell frequency by the overall total sample size ($N$) of the entire table.

  • Conditional Proportions: Computed by restricting/conditioning analysis to a specific row or column. That specific row total or column total becomes the denominator.

• Rounding Conventions:

  • Proportions: Round to $4$ decimal places (e.g., $0.5625$).

  • Percentages: Round to $2$ decimal places (e.g., $56.25\%$).

Optometry Customer Dataset and Joint vs. Conditional Probabilities

• Optometry Dataset Summary:

  • Collected from an optometry shop with variables Gender (Male, Female) and Eye Condition (Nearsighted, Farsighted, Needs Bifocals).

  • Raw Contingency Table Data:

  • Nearsighted Row: Male = $6$, Female = $4$, Row Total = $10$

  • Farsighted Row: Male = $20$, Female = $16$, Row Total = $36$

  • Needs Bifocals Row: Male = $6$, Female = $12$, Row Total = $18$

  • Column Totals: Male Total = $32$, Female Total = $32$, Overall Sample Total ($N$) = $64$

• Step-by-Step Probability Calculations:

  • Question 1: What percent of all customers are farsighted?

  • Denominator / Condition: All customers ($N = 64$).

  • Numerator: Total farsighted customers ($36$).

  • Proportion: $\frac{36}{64} = 0.5625$

  • Percentage: $56.25\%$

  • Question 2: What percent of nearsighted customers are females?

  • Denominator / Condition: Nearsighted customers only ($10$).

  • Numerator: Female nearsighted customers ($4$).

  • Proportion: $\frac{4}{10} = 0.4000$

  • Percentage: $40.00\%$

  • Question 3: What percent of all customers are nearsighted AND female?

  • Denominator / Condition: All customers ($N = 64$).

  • Numerator: Nearsighted females overlapping cell ($4$).

  • Proportion: $\frac{4}{64} = 0.0625$

  • Percentage: $6.25\%$

  • Question 4: What percent of all customers are nearsighted OR female?

  • Denominator / Condition: All customers ($N = 64$).

  • Logical Operator Distinctions:

    • "AND" requires meeting both categories simultaneously.

    • "OR" includes meeting either category or both categories.

  • Double-Counting Warning: Adding total nearsighted ($10$) to total females ($32$) equals $42$, which double-counts the $4$ nearsighted females.

  • Calculation Method 1 (Inclusion-Exclusion): $\frac{10 + 32 - 4}{64} = \frac{38}{64}$

  • Calculation Method 2 (Sum of Non-Overlapping Cells): $\frac{6 + 4 + 16 + 12}{64} = \frac{38}{64}$

  • Proportion: $\frac{38}{64} = 0.59375 \approx 0.5938$

  • Percentage: $59.38\%$

Conditional Distributions by Category

• Examination Calculator Requirements:

  • Calculators are required for exams to accurately compute decimals, proportions, and statistical functions.

• Conditional Distribution of Eye Condition by Gender:

  • Conditioning by gender requires evaluating Male ($N = 32$) and Female ($N = 32$) columns independently as relative percentage distributions.

  • Male Conditional Distribution ($N = 32$):

  • Nearsighted: $\frac{6}{32} = 0.1875 = 18.75\%$

  • Farsighted: $\frac{20}{32} = 0.6250 = 62.50\%$

  • Needs Bifocals: $\frac{6}{32} = 0.1875 = 18.75\%$

  • Column Sum Verification: $18.75\% + 62.50\% + 18.75\% = 100.00\%$

  • Female Conditional Distribution ($N = 32$):

  • Nearsighted: $\frac{4}{32} = 0.1250 = 12.50\%$

  • Farsighted: $\frac{16}{32} = 0.5000 = 50.00\%$

  • Needs Bifocals: $\frac{12}{32} = 0.3750 = 37.50\%$

  • Column Sum Verification: $12.50\% + 50.00\% + 37.50\% = 100.00\%$