Variables Part Two: Organizing & Displaying Numeric Data

Categorical vs. Numeric Data

• Categorical (qualitative)
  • Order of categories does not matter; they can be alphabetized or arranged arbitrarily.
• Numeric (quantitative)
  • Natural order from smallest → largest.
  • Must be organized respecting that order.
  • Primary tools introduced:
  • Frequency & relative-frequency distributions.
  • Graphs: dot plots, stem-and-leaf plots, histograms.

Frequency & Relative-Frequency Distributions

• Purpose: tabulate how often each value (or interval) occurs.
• Two grouping styles:
  1. Single-value grouping — each class is one specific value.
  2. Cut-point grouping — each class is an interval
  • Lower cut point (class lower limit) is included.
  • Upper cut point is excluded (becomes lower limit of next class).
• Relative frequency formula: $\text{rel. freq.}=\frac{f}{n}$ where $f$ is class frequency, $n$ is total observations.

Single-Value Example: Ages of 10 Statistics Students

• Raw ages: 18, 18, 18, 19, 19, 20, 17, 18, 19, 20.
• Youngest = 17; oldest = 20 ⇒ list all integers 17–20 even if absent.
• Tally → Frequency table
  • 17 (1), 18 (4), 19 (3), 20 (2)  (sum = 10).
• Relative frequencies: 0.10, 0.40, 0.30, 0.20.
• Always include: title, variable label ("Age, yrs"), frequency & relative-frequency columns, and grand total.

Cut-Point Grouping

Rules

• Recommended number of classes $k$: between 5 and 20.
• All classes same width $w$ (prefer whole number when possible).
• Compute width
  $w=\frac{\text{max}-\text{min}}{k}$ then round up.
• Each observation belongs to exactly one class.

Egg-Weight Example (20 eggs, grams)

• Min = 54.4 g; Max = 62.1 g.
• Range $R=62.1-54.4=7.7$ (small).
• Choose $k=5$ ⇒ $w=\frac{7.7}{5}=1.54\to2$.
• Start at nearest convenient whole number: 54.
• Classes (lower inclusive, upper exclusive):
  • 54–<56, 56–<58, 58–<60, 60–<62, 62–<64.
• After tally: frequencies 2, 6, 6, 4, 2 (total 20) → relative frequencies .10, .30, .30, .20, .10.
• Midpoint of a class: $\text{mid}=\frac{\text{lower}+\text{upper}}{2}$.

Retirement-Home Ages (20 people, 81–90)

• Range $R=90-81=9$ ⇒ pick $k=5$ ⇒ $w=\lceil1.8\rceil=2$.
• Start 81 → classes 81–<83, 83–<85, … 89–<91.
• Fill tally → complete frequency & relative-frequency columns, add title.

College-Coach Ages (100 coaches, 35–80)

• Range $R=80-35=45$; choose $k=8$ for larger spread.
• $w=45/8=5.625\to6$.
• Classes starting 35: 35–<41, 41–<47, …, 77–<83 (8 total).
• After tally one could compute frequencies, rel. frequencies, add descriptive title.

Graphical Displays

Dot Plot

• Small/medium data sets; each dot represents one observation positioned above its value on a number line.
• Example: Resting heart rates of 15 ASU students (52–93 bpm).
  • Axes: horizontal = heart rate (beats per minute), vertical often omitted; number of stacked dots = frequency.
  • Must include title & unit; speaker illustrated missing labels as a teaching point.

Stem-and-Leaf Plot

• Shows raw data while giving distribution shape.
• Split each value into:
  • Stem = all but final digit.
  • Leaf = final (right-most) digit.
• Draw vertical line; stems on left, leaves on right (ascending order).
• Two formats:
  1. One-line-per-stem — every stem appears once.
  • Resting heart rates example produced rows for 5|, 6|, 7|, 8|, 9|.
  1. Two-lines-per-stem — first line holds leaves 0–4, second line 5–9.
  • GPA example (2.0–4.0): stems 2, 3, 4 duplicated; first row for 0–4 leaves, second for 5–9 leaves.
  • Demonstrated sorting, handling duplicates, and possible >4.0 honors cases.
• Power-walking 10 k times (60–89 min) used two-lines-per-stem; blank stems retained (no skipping like number line).
• Advantages: preserves individual observations, quick to construct by hand.
• Disadvantages: cluttered with very large datasets (≥ hundreds); use histogram instead.

Histogram

• Bar-like graph for numeric data; bars touch (continuous scale).
• Y-axis: frequency or relative frequency; X-axis: class intervals.
• Two versions:
  • Frequency histogram (counts).
  • Relative-frequency histogram (proportions).
• Egg-weight example displayed both; bars spanned 54–<56,…,62–<64 g.
• Choosing $k$ too small (e.g., 2 classes → $w=8$) hides structure; too large (e.g., 20 classes) shows noisy spikes. Aim for balance guided by range and the 5–20 rule.

Distribution & Shape Terminology

• Distribution: table, graph, or formula indicating possible values of a variable and their frequencies.

Modality (Number of Peaks)

• Unimodal: one peak (e.g., normal bell curve).
• Bimodal: two peaks.
• Multimodal: three or more peaks (no special “trimodal” term; all 3+ fall here).
  • Instructor analogy: unicycle (1 wheel) = unimodal.

Symmetry & Skewness

• Symmetric distribution: can be split into mirror halves.
• Right-skewed: long tail extends to larger values (right side “pulls” out).
• Left-skewed: long tail toward smaller values (left side elongated).
  • Visual analogy: kid grabbing one side of the bell curve and running.

Practical & Pedagogical Notes

• Always include:
  • Descriptive title.
  • Variable name & measurement units on axes or table headings.
• Whole numbers preferred for class boundaries ("the world doesn’t like decimals"), but scientists may tolerate precise values.
• When deciding $k$:
  • Large range (≈1000) → lean toward upper limit (≈20).
  • Small range (≈10) → lean toward lower limit (≈5).
  • Trial-and-error acceptable until display "looks best".
• Empty classes should still appear in tables/plots (analogous to not skipping numbers on a number line).
• Dot plots & stem-and-leaf ideal for quick insight, homework, or small n; histograms preferred for large datasets or presentations.

Key Formula Recap

• Range: $R = \text{max} - \text{min}$.
• Class width (before rounding): $w=\frac{R}{k}$.
• Relative frequency: $\text{rel. freq.}=\frac{f}{n}$.
• Class midpoint: $\text{mid}=\frac{\text{lower cut point}+\text{upper cut point}}{2}$.

Ethical & Real-World Connections

• Examples contextualized: nutritional supplements in poultry industry, retirement-home demographics, power walking & joint health, academic GPAs.
• Instructor encourages adult learners ("went back in mid-30s – you can too") to reduce intimidation and foster inclusive education.