Graphs for Categorical Data – Pie Charts & Bar Charts

Module Context

• Continuing from earlier modules where measures of central tendency (mean), variability (range, standard deviation) and histograms (Module 1) were introduced.

• Current focus: Pictorial/graphical summaries of behaviour—especially for categorical data.

• Graph types covered in this mini-lecture:
  • Pie charts
  • Bar charts
  (Mentioned but not detailed here: histograms, box-and-whisker plots for continuous data.)

Pie Charts

• Purpose: Visualise categorical data by partitioning a circle into slices.

• Mechanism:
  • Area of each slice ∝ category frequency or percentage.
  • Useful legend or colour coding needed.

• Example 1 — Favourite Season Survey
  • Sample size $N = 150$ (fictitious).
  • Categories: spring, summer, winter, autumn.
  • Observation: largest slice = autumn ⇒ majority preference.

• Advantages
  • Intuitive, quick at-a-glance comparison of parts to whole.
  • Good when categories ≤ ≈ 10 and each occupies a sizeable portion.

• Pitfalls & Limitations
  • "Too many slices" problem – visual clutter when categories exceed ≈10.
  • Hard to discriminate tiny percentages.
  • Labels often overlap; legends become unreadable.

• Example 2 — Causes of Death in Shakespeare’s Plays (many categories)
  • Stabbing > 50 % readily visible.
  • Remainder categories indistinguishable.

• Example 3 — US 2007 Federal Budget Allocation (Wikipedia)
  • Large categories readable; small-budget items effectively invisible.
  • Demonstrates difficulty with small percentages.

Bar Charts

• Also visualise categorical data; employ bars whose height encodes a number.

• Horizontal (x-) axis = categories; Vertical (y-) axis = numerical value.

• Flexibility of y-axis:

  1. Raw counts (frequencies).

  2. Summary statistics (mean, median, proportion, etc.).

Bar-Chart Example 1 — Favourite Season (Counts)

• Same $N = 150$ dataset shown as bars.

• Height of each bar = count of votes.

• Simple comparison: autumn > summer > winter > spring.

Bar-Chart Example 2 — Athlete Sprint Times (Means)

• Hypothetical study: 20 athletes per sport.

• Categories: bowling, athletics, swimming, hockey, football.

• y-axis = mean 100 m race time (seconds).
  • Bowling ≈ 25 s, Athletics ≈ 10 s, etc.

• Demonstrates summary-statistic usage rather than counts.

Error Bars & Variability

• Bar charts can incorporate variability indicators:
  • e.g., ±1 SD error bars.
  • Graph conveys both central tendency (mean) and dispersion.

Multiple Variables & Interactions (Grouped / Clustered Bars)

• Extension: Compare categories across a second variable (country, condition, gender…).

• Example 3 — Athlete Type × Country (A vs B)
  • Each athlete type has two bars (country A & country B).
  • Interpretation paths:
  – Main effects: bowling slower than hockey in both countries.
  – Interaction:
  • Bowling: country A slower than B.
  • Swimming: country A faster than B.
  → Effect of athlete type depends on country.

• Bar charts are especially intuitive for spotting such interactions.

Bar-Chart Pitfalls

1. Category Overload
   • Re-plotting Shakespeare deaths as bars results in tall “stabbing” bar beside numerous near-zero bars—pattern still unreadable.

2. Axis Manipulation / Misleading Scales
   • Ethical & interpretive risk: truncating the y-axis exaggerates perceived differences.
   • Case Study — Fox News Infographic
   – Data: welfare recipients $\approx 108{,}000{,}000$ vs full-time employees $\approx 101{,}000{,}000$.
   – Original graph y-axis starts at $100{,}000{,}000$ → makes gap look huge.
   – Re-plot starting at $0$ shows modest difference.
   • Lesson: Always inspect/label axes; zero baseline is recommended for bar charts unless strong justification.

Practical & Ethical Implications

• Graph choice matters for clarity, transparency, and integrity.

• Over-complicated pie/bar charts hinder comprehension and invite misinterpretation.

• Axis manipulation can intentionally or inadvertently bias viewers—ethical responsibility to present data honestly.

• When data have many categories or wide dynamic ranges, consider alternatives (e.g., dot plots, treemaps, lollipop charts, log scales, or collapsing categories).

Summary Cheat-Sheet

• Use pie charts when:
  • ≤ 10 categories, fairly even or large slice sizes, need part-to-whole insight.

• Avoid pie charts when: many tiny segments or need precise comparison.

• Use bar charts when:
  • Need to compare magnitudes across categories, display summary statistics, add error bars, or visualise interactions between variables.

• Watch-outs:
  • Too many categories → clutter.
  • Truncated or inconsistent axes → misleading impressions.
  • Provide clear labels, scales, legends.

• Both graphs supplement numerical summaries (mean, SD, etc.) and can powerfully communicate behavioural data when designed correctly.