Descriptive Statistics – Summarising Qualitative Data

Raw Data & Why We Summarise

• Raw data = non-processed, just-collected observations

  • May appear as questionnaires, lists, tables, spreadsheets, etc.

  • Example given: table of 20 student responses showing province of origin alongside other variables.

• Disadvantages of raw data

  • Contains “too much information” → cognitively heavy and time-consuming to read.

  • Lacks visual impact → the underlying “story” is hidden.

  • Scaling problem: 100 or 1 000 observations exacerbate both issues.

• Practical implication: always convert raw qualitative data into concise tables or graphs before analysis or presentation.

Key Definitions (Revision)

• Qualitative variable: records non-numeric information (e.g., province, gender, colour).

• Descriptive statistics: methods for summarising, organising, presenting data.
  Focus of the lecture: descriptive techniques for qualitative variables.

Frequency Table – The First Summarising Step

• Structure (minimum two columns)

  1. Categories (e.g., Eastern Cape, Free State, Gauteng, …).

  2. Frequencies ($f$) = counts for each category.

• Building procedure

  1. List all observed categories (order can be alphabetical or by size).

  2. Perform tallies (////) for each raw observation.

  3. Add tallies → obtain $f$ for every row.

  4. Check: $\sum f = n$ (sample size).
     • Example: $7+4+\dots = 20$ students.

• Why important?

  • Gives immediate insight into “how often” each outcome appears.

  • Underlies every subsequent graph (pie, bar, etc.).

Extra Informative Columns

1. Relative frequency ($rf$)
   Formula: $rf=\frac{f}{n}$
   • Properties: $0\le rf\le1$ and $\sum rf =1$.

2. Percentage (%)
   $\text{Percentage}=rf\times100$
   • Properties: $\sum\text{Percentages}=100$.

3. Angle size (°) – required for pie charts
   $\text{Angle}=rf\times360$
   • Properties: $\sum\text{Angles}=360$.

• Flexibility: include only the columns that match the story you wish to convey.
  • Minimalists: categories + f.
  • Presenters: add % and angle for automatic charting.

Graphical Summaries for One Qualitative Variable

Pie Chart

• Preparation: compute angle sizes as above.

• Construction rules

  • Slice angle $\propto rf$.

  • Use colour/legend to identify categories.

  • Label or annotate slices OR place legend beside chart.

  • Always give a descriptive title (e.g., “Pie Chart of Province of Origin”).

• Interpretation: area of slice visualises proportion (immediate percentage feel).

Bar Chart (Simple / Unstacked)

• Axes

  • $x$-axis = categories.

  • $y$-axis = frequency or relative frequency or percentage.

• Drawing rules

  • Bars of equal width, do not touch (space distinguishes categories).

  • Scale $y$-axis to highest frequency (e.g., 0 → 7 for 7 entries).

  • Label both axes and supply graph title.

• Pros: easy comparison of heights; quick spotting of most/least common categories.

Two Qualitative Variables – Contingency Table

• Definition: frequency table that records joint occurrences of two (categorical) variables.

  • Example: Province × Gender (Male/Female).

• Layout

  • Rows = categories of variable 1, columns = categories of variable 2 (or vice-versa).

  • Cell gives count $f_{ij}$.

Graphical Options for Two-Way Tables

1. Stacked Bar Chart

   • $x$-axis = primary categories (e.g., province).

   • Each bar’s height = overall frequency for that province.

   • Bar is subdivided (stacked) by second variable (gender) using different colours.

   • Legend identifies segments (e.g., red = Male, blue = Female).

   • Alternative orientation: swap axes (make gender primary, stack provinces).

2. Multiple (Clustered) Bar Chart

   • Bars for each sub-category drawn side-by-side within the same primary category.

   • Facilitates direct visual comparison between sub-categories at each category level.

• Choice criteria

  • Stacked: highlights composition of totals.

  • Clustered: highlights direct comparison of sub-groups.

Best-Practice Checklist for Categorical Graphs

• Verify $\sum f = n$, $\sum rf =1$, \sum\text{%}=100, $\sum\text{Angle}=360$.

• Use meaningful, non-abbreviated category labels or provide a clear legend.

• Keep colours distinct & colour-blind-friendly; avoid misleading 3-D effects.

• Scales must start at 0 for bar charts (avoids exaggeration of differences).

• Titles should state what and for whom/when (e.g., “Bar Chart of Gender Distribution among First-Year Students, 2023”).

• Mention sample size $n$ either in caption or footnote.

Real-World Relevance & Ethical Notes

• Appropriate summaries prevent information overload for audiences (managers, policy makers, the public).

• Proper labelling avoids misinterpretation; mis-labelled axes or missing totals can lead to unethical miscommunication of results.

• In surveys with sensitive categories (e.g., gender identity, ethnicity), anonymised frequency tables protect respondent privacy.

Connections to Previous & Upcoming Lectures

• Builds directly on earlier definitions of qualitative variable and descriptive statistics (Revision ✓).

• Sets the foundation for upcoming lecture: Summarising quantitative data (histograms, stem-and-leaf, etc.).

• Practice Assignment 1 should now be complete; expect Assignment 2 after the next session → keep pace to avoid falling behind.

Quick Formula Recap (All in One Place)

• $\text{Relative Frequency}=\frac{f}{n}$

• $\sum f = n$

• $\sum \text{Relative Frequencies}=1$

• $\text{Percentage}=\text{Relative Frequency}\times100$

• $\sum \text{Percentages}=100$

• $\text{Angle Size}=\text{Relative Frequency}\times360$

• $\sum \text{Angle Sizes}=360$

End of qualitative-data summarising techniques — be ready to apply these to homework and projects.