MA213 – Chapter 2: Methods for Describing Sets of Data

Where We’ve Been

• Reviewed foundational themes of statistics
  • Inferential Statistics: provide conclusions about a population based on sample data
  • Descriptive Statistics: organize, summarize, and present data
• Key elements of any statistical problem
  • $\text{Population}$ & $\text{Sample}$ definitions
  • Variable type identification
  • Data-collection methodology
• Two broad data types
  • Quantitative (numerical)
  • Qualitative (categorical)

Where We’re Going (Chapter Road-Map)

• Describe data visually
  • Graphs for quantitative variables
  • Graphs for qualitative variables
• Describe data numerically
  • Central tendency: Mean, Median, Mode
  • Variability: Variance, Standard Deviation
  • Relative standing, outliers, association
• Depict relationships between two quantitative variables
  • (e.g., scatterplots—later section)

2.1 Describing Qualitative (Categorical) Data

• Definition: qualitative values represent distinct classes—no intrinsic numeric meaning
  • Examples: eye color, gender, political party, geographic location
• Goal: organize raw category listings into useful summaries to support description & inference
• Two numerical summaries
  • Class Frequency: count of observations in each class
  • Class Relative Frequency: $\dfrac{\text{class frequency}}{\text{total}}$; multiply by $100$ for Class Percentage

Class Frequency (Illustrative Example)

• Transportation to school study
  • Classes: car, bus, bike, walk
  • Frequencies (sample): $12,7,6,11$ respectively

Adult Aphasia Study (Example 1)

• Raw listing of $22$ subjects → classes:
  • Anomic: $10$ patients
  • Broca’s: $5$ patients
  • Conduction: $7$ patients
• Relative measures
  • Anomic: $10/22 = 0.455 \; (45.5\%)$
  • Broca’s: $5/22 = 0.227 \; (22.7\%)$
  • Conduction: $7/22 = 0.318 \; (31.8\%)$
  • Totals check: $22/22 = 1.00 \; (100\%)$

Blood-Type Survey (Example 2; $n=24$)

• Tallied frequencies
  • $A = 5$, $B = 7$, $AB = 4$, $O = 8$
• Relative frequencies / percentages
  • $A: 5/24 \approx 0.208 \; (20.8\%)$
  • $B: 7/24 \approx 0.292 \; (29.2\%)$
  • $AB: 4/24 \approx 0.167 \; (16.7\%)$
  • $O: 8/24 \approx 0.333 \; (33.3\%)$

Graphical Displays for Qualitative Data

• Pie Chart
  • Slice angle: $\dfrac{\text{class freq}}{n}\times360^\circ$
  • Slice %: $\dfrac{\text{class freq}}{n}\times100$
  • Example (snack preferences): $\text{Potato chips}$ slice $= 37.3\%$ ⇒ $134^\circ$, etc.
• Bar Graph
  • Height = frequency, relative frequency, or % per class
  • Bars separated (qualitative axis has no inherent order)
  • Coffee reasons example
  • Taste $27.6\%$, Awake $30.3\%$, Coffeehouse $2.6\%$, Other $3.9\%$, Never drink $35.5\%$
• Pareto Diagram
  • Bars ordered left→right by descending height; superimposed line shows cumulative counts/percentages
  • Highlights principle that a few categories often account for majority

Practice: Pie Chart Construction (Blood Types)

• Angle/percentage computations reproduced above
• Resulting distribution
  • $O = 33.3\%$, $B = 29.2\%$, $A = 20.8\%$, $AB = 16.7\%$

2.2 Graphical Methods for Quantitative Data

• Quantitative values are measured on a true numeric scale (age, height, body temperature …)
• Primary small-to-large-data tools
  • Dot plots
  • Stem-and-leaf plots
  • Histograms

Dot Plots

• Each numeric observation plotted as a dot above a horizontal axis
• Construction steps
  1. Determine min/max → choose axis scale
  2. Draw baseline
  3. Plot a separate stacked dot for each occurrence (duplicates vertically stack)
• Pros/Cons: great for small datasets; preserves every value; quickly shows clusters & outliers

Stem-and-Leaf Plots

• Split each number into stem (leading digit(s)) & leaf (trailing digit)
  • $34 \to 3 | 4$; $356 \to 35 | 6$
• Example 6: $20$-day cardiogram counts
  • Raw data: ${25,31,20,32,13,14,43,02,57,23,36,32,33,32,44,32,52,44,51,45}$
  • Organized display (stem | leaves):
  • $0 | 2$
  • $1 | 3 \; 4$
  • $2 | 0 \; 3 \; 5$
  • $3 | 1 \; 2 \; 2 \; 2 \; 2 \; 3 \; 6$
  • $4 | 3 \; 4 \; 4 \; 5$
  • $5 | 1 \; 2 \; 7$
• Strength: retains raw values and shows order; useful for moderate (<100ish) sample sizes

Histograms

• Group quantitative values into class intervals (bins) on horizontal axis
• Vertical axis = frequency or relative frequency within each bin; bars touch (continuity)
• Example 7: $20$ test scores $[45\dots 90]$
  • Possible 5-class grouping (given solution)
  • $45{-}53:3$, $54{-}62:4$, $63{-}71:4$, $72{-}80:5$, $81{-}90:4$
  • Visual histogram depicts distribution shape across score intervals

Choosing Number of Classes

• General guideline: $5 \text{–} 20$ classes (depends on $n$)
• As $n$ increases ⇒ narrower class width beneficial
• Classes must be
  • Mutually exclusive
  • Exhaustive (cover entire range)
  • Continuous (no gaps even if frequency $=0$)
  • Equal width
• Practical width calculation: \text{Class width} = \dfrac{\text{max} - \text{min}}{\text{desired # classes}} then round up to a convenient value

Bin-Number Illustration (MPG Data)

• Too few bins (2) ⇒ oversmoothing; too many (100) ⇒ noise; select middle ground (e.g., 10–20) for balance

Grouped Frequency Distribution Example (50-State Record Highs)

• Data range $\approx 100 \text{–} 134$
• Using $7$ classes
  • Determine class width $= \dfrac{134-100}{7} \approx 4.9$ ⇒ round to $5$
  • Create intervals (e.g., $100{-}104,105{-}109,\dots$) & tally frequencies
  • Histogram plotted accordingly (details in slides)
  • Analysis targets distribution’s center (~$115{-}120$°F), spread (~$100{-}134$°F), possible skewness

Interpreting Histograms & Exploring Distributions

• Identify shape
  • Symmetric bell-shaped
  • Uniform (flat)
  • J-shaped / Reverse J
  • Left-skewed or Right-skewed
  • Bimodal, U-shaped, etc.
• Center: midpoint where half data lie on each side (visual estimate) – later formalized by mean/median
• Spread: range or anticipated variability (min⇢max)
• Outliers: isolated bars/dots far from main cluster; flag for error check or substantive investigation

Common Shapes (visual palette)

• Bell-shaped (Gaussian)
• Uniform (rectangular)
• J / Reverse-J
• Left- or Right-skewed (tail direction)
• Bimodal (two peaks) or U-shaped

Comparative Summary of Graph Types

• Dot Plot
  • Precise values shown, simple, small $n$
• Stem-and-Leaf
  • Values shown & ordered; still compact; moderate $n$
• Histogram
  • Conceals individual values but better for large $n$; reveals overall pattern more clearly

Conceptual & Practical Implications

• Choosing the correct summary/graph depends on
  • Data type (qualitative vs quantitative)
  • Sample size
  • Audience need: raw detail vs overall pattern
• Ethical responsibility
  • Avoid misleading through inappropriate class widths or selective category ordering
  • Clearly label axes & include units
• Real-world relevance
  • Data visualization underpins decision-making in business, healthcare, public policy
  • Recognizing skew/outliers prevents faulty “average” interpretations (e.g., median salary vs mean)

Key Formulas (LaTeX)

• Relative frequency: $\text{RF}<em>i = \dfrac{f</em>i}{n}$
• Class percentage: $\%<em>i = \text{RF}</em>i \times 100$
• Pie-slice angle: $\theta<em>i = \text{RF}</em>i \times 360^{\circ}$
• Histogram class width (rounded): $w = \left\lceil\dfrac{\text{max}-\text{min}}{k}\right\rceil$ where $k$ = desired # classes

Study Tips & Connections

• Master categorical vs numerical distinction first; drives all later analytic choices
• Re-draw given examples by hand to reinforce procedure memory (dot plot, stem-leaf, histogram)
• Link shapes to potential real processes (e.g., right-skewed income, symmetric test errors)
• Practice converting raw tallies to relative frequencies & percentages—vital for reports
• Preview upcoming numeric measures: mean & standard deviation integrate with the visual tools learned here