Descriptive Statistics – Summarising Quantitative Data

Context & Purpose

• Lecture continues the Descriptive Statistics block, focusing on summarising quantitative data.
• Goal: transform raw numerical data into tables and graphs that reveal structure and patterns.

Quantitative Variables Refresher

• Defined as variables that assume numerical values (e.g.
  audit-time in days).

Frequency-Distribution Table: Step-by-Step Construction

1. Decide on Number of Classes (Categories)

• Use Sturges’ Rule: $k = 1 + 3.3 \log n$
  • $n$ = sample size.
  • Always round up the result.
• Example data set: 30 observations.
  • $k = 1 + 3.3 \log 30 = 5.875 \rightarrow 6\;\text{classes}$

2. Compute Class Width $c$

• Formula: $c = \frac{\text{max} - \text{min}}{k}$
• Example: max $= 32$, min $= 11.5$
  • $c = \frac{32-11.5}{6}=3.417 \rightarrow 4$ (round up to next integer, even if only 0.0001 over).

3. Establish Class Boundaries

• If min value is an integer → start exactly at min.
• If min value is not an integer → start at next lower integer.
• Example:
  • Min = 11.5 → first lower boundary = $11$.
  • Upper boundary of each class = lower boundary $+ c$.
  • Six intervals produced (square bracket = inclusive, round bracket = exclusive):
  1. $[11,15)$
  2. $[15,19)$
  3. $[19,23)$
  4. $[23,27)$
  5. $[27,31)$
  6. $[31,35)$
  • Boundary logic: value $15$ belongs to 2nd class, $19$ to 3rd, etc.

4. Tally Observations

• Scan raw list once, mark a stroke (|||| then ) in the correct class.
• Example tallies lead to frequencies:
  • $f = [8,7,3,8,3,1]$ (sum = $30$, matches $n$).

5. Calculate Additional Columns

• Relative frequency: $rf = \frac{f}{n}$. Sum = 1.
• Cumulative frequency: $F<em>i = f</em>i + \sum<em>{j<i} f</em>j$.
  • Example: $F = [8,15,18,26,29,30]$.
• Relative cumulative frequency: $\frac{F}{n}$.
• Class midpoint ((xm)): $x</em>m = \frac{\text{lower} + \text{upper}}{2}$.
  • Example midpoints: $[13,17,21,25,29,33]$.
• Optional columns: percentage, proportion, etc.

Minimum required for a basic table: Class Intervals + Frequencies.

Graphical Methods for Quantitative Data

A. Histogram

• X-axis: class intervals (continuous scale).
• Y-axis: frequencies.
• Bars touch because data are continuous.
• Example bar heights: 8,7,3,8,3,1.
• Label axes: e.g. “Audit Time (days)” and “Number of Clients”.

Interpreting Histogram Shape

• Symmetric: frequencies cluster near centre, mirror-like tails.
• Uniform: all classes have ~equal frequency.
• Negatively skewed (skew-left): bulk of data on right, tail extends left.
• Positively skewed (skew-right): bulk on left, tail extends right.

B. Ogive (Cumulative Frequency Curve)

• Requires Class Boundaries + Cumulative Frequencies.
• Plot upper class boundary vs. cumulative frequency; join with straight segments.
• Always non-decreasing.
• Interpretation:
  • At 19 days, $F=15$ → 15 clients finished in <19 days.
  • Use vertical then horizontal tracing to answer “≤ value” questions (e.g.
    ≤27 days → 26 clients).

C. Frequency Polygon

• Uses midpoints vs. frequencies.
• Add two extra (arbitrary) midpoints so curve starts/ends on X-axis:
  • Start: first midpoint $-c$ ( $13-4=9$ ) with $f=0$.
  • End: last midpoint $+c$ ( $33+4=37$ ) with $f=0$.
• Plot points (9,0), (13,8), (17,7), (21,3), (25,8), (29,3), (33,1), (37,0) and connect with straight lines.
• Entire polygon touches the X-axis only at the two artificial endpoints.

Practical & Pedagogical Notes

• Choice of extra columns/graphs depends on message & audience.
• Frequency tables suffice for numerical summaries; graphs convey visual intuition.
• Check sums: $\sum f = n$ and $\sum rf = 1$.
• Rounding-up principle applies to both $k$ and $c$, ensuring full coverage without data loss.
• Always label graphs fully (title, axes, units).
• Contrast: histogram (touching bars, continuous) vs.
  bar chart (separate categories, bars separated).
• Cumulative frequency tools (table or ogive) allow percentile-type inquiries without raw data.

Course Logistics Mentioned

• Unit 2 concluded; Practice Assignment 2 open (work over next 7-10 days).
• Memo/solutions available in 5-7 days.
• Additional video provided for full worked example; slides alone are “sufficient”.
• Encouragement to “work diligently”.

Summary of Key Formulas & Symbols

• Sturges: $k = 1 + 3.3 \log n$
• Class width: $c = \dfrac{\max - \min}{k}$
• Relative frequency: $rf = \dfrac{f}{n}$
• Cumulative frequency: $F<em>i = \sum</em>{j\le i} f_j$
• Class midpoint: $x_m = \dfrac{\text{lower}+\text{upper}}{2}$
• Notation: square bracket [ = inclusive, round bracket ) = exclusive.

Ethical & Practical Implications

• Transparent summarisation prevents misinterpretation—e.g.
  incorrect class widths or mis-labelled histograms can bias audience perception.
• Proper rounding avoids hiding observations outside chosen boundaries.
• Graph choice impacts how variability & skewness are communicated to stakeholders (auditors, managers, regulators, etc.).