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Week 2 · Lecture outline

Week 2 — Lecture Outline · Summarizing Data with Tables & Graphs

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objectives covered: Objective 2 — Summarize and display univariate data (this week: the tables-and-graphs half; Week 3 adds the numerical summaries).
SLOs touched: A (reason quantitatively from data) · B (communicate results to a non-technical audience)
Meeting pattern: planned as 2 sessions × ~75 min = ~150 min. Segment minutes below total ~150; scale them to your own pattern.

Week at a Glance

The table below is the week's one-page map — question, abilities, vocabulary, materials, timing.
The week's big question "How do you turn a pile of numbers into a picture your eyes can read — and how do you spot the pictures built to fool you?"
By the end of the week, students can… (1) build a frequency table and a relative-frequency table for categorical or quantitative data; (2) choose and read the right display — bar chart, pie chart, histogram, dot plot, stem plot — for a given variable; (3) describe a distribution's shape (symmetric, right-/left-skewed, uniform, bimodal) and flag informal outliers; (4) name the design tricks that make a misleading graph and state the fix.
Key vocabulary frequency, relative frequency, frequency table, class (bin/bucket), bar chart, pie chart, histogram, dot plot, stem-and-leaf plot (stem plot), distribution, shape, symmetric, skewed right/left, uniform, bimodal, peak (mode of a graph), outlier (informal), misleading graph, truncated axis
Materials slides (Deck 2), the Week 2 chapter, the week's readings + video links, a spreadsheet (Google Sheets or Excel), the student's chatbot for the AI-critique moment and the tutorial
Timing note 8 segments, ~150 min total. Session 1 = Segments 1–4 (~75). Session 2 = Segments 5–8 (~75).

Segment 1 — Hook & the Promise (8 min) · Session 1 opens

Hook. Put 25 raw numbers on the screen — a jumbled list, no order. Give the class ten seconds: "What can you tell me about these numbers?" (Silence is the point.) Then reveal the same 25 numbers as a histogram. Instantly everyone can answer: where they cluster, how far they spread, which value sits off by itself.

  • "A pile of numbers is unreadable. A picture is instant. Today: how statisticians turn columns into pictures — honestly."
  • Last week you learned to ask who was measured and how they were picked. This week is the first thing we do with the answers.

The promise (write it on the board): "By the end of this week you can take any column of raw data and produce the right table and the right graph — and you can look at a graph in an ad or a news feed and tell whether it's informing you or playing you."

Why it matters line (memory hook): "Every dataset has a shape. Your job is to see it — and to notice when someone's hiding it."

Segment 2 — Frequency & Relative-Frequency Tables (22 min)

Plain language first.
- A frequency is just a count — how many data values fall in a category or a class.
- A relative frequency is that count's share of the whole: count ÷ total. It answers "what fraction?" instead of "how many?"
- A frequency table lists every category (or class) with its count; add a relative-frequency column and totals become comparable across datasets of different sizes. Relative frequencies always sum to 1 (percents sum to 100%) — that's your built-in error check.

Memory hook: "Frequency counts it. Relative frequency shares it."

One fully worked example (do every step out loud).

A student's workout playlist has 40 songs. Tallying by genre: pop 14, hip-hop 10, rock 8, country 5, other 3.
- Check the counts: 14 + 10 + 8 + 5 + 3 = 40 ✓
- Relative frequencies: 14 ÷ 40 = 0.35 · 10 ÷ 40 = 0.25 · 8 ÷ 40 = 0.20 · 5 ÷ 40 = 0.125 · 3 ÷ 40 = 0.075
- Check the shares: 0.35 + 0.25 + 0.20 + 0.125 + 0.075 = 1.000 ✓ — as percents: 35% + 25% + 20% + 12.5% + 7.5% = 100% ✓
- Read one aloud both ways: "14 songs are pop" (frequency) and "35% of the playlist is pop" (relative frequency).

Quantitative data need one extra move: numbers don't come pre-grouped, so we cut the number line into classes (also called bins or buckets) of equal width first — "20 to under 25 minutes, 25 to under 30, …" — then count into the classes. Convention: each class includes its left edge and excludes its right edge, so a value of exactly 25 goes in the 25–<30 class, and nothing is counted twice. (The full quantitative example runs in Segment 5.)

Land the key idea: the table is the bridge between raw data and every graph this week — get the counts right and the picture draws itself.

Segment 3 — Categorical Displays: Bar Charts & Pie Charts (20 min)

Plain language first.
- A bar chart draws one bar per category; the bar's length shows the count (or relative frequency). Bars are separated — the gaps say "these are separate labels, not a number line." You may order the bars however helps (alphabetical, biggest-first).
- A pie chart shows how one whole splits into parts: each slice's share of the circle is the category's share of the total. A pie only works when the categories are non-overlapping pieces of one whole — the slices must total 100%.

Same playlist, both displays (worked, on the board):

Bar chart: five bars — pop 14, hip-hop 10, rock 8, country 5, other 3 — counts on the vertical axis, starting at zero.
Pie chart: slices of 35%, 25%, 20%, 12.5%, 7.5% — total 100% ✓. Every song is in exactly one genre, so the "parts of one whole" requirement holds.

When the pie breaks (name it now, it returns in Segment 7):
- Categories that overlap (one person can belong to several clubs — club-membership counts don't form a whole).
- Slices that don't total 100% (a tell that something was double-counted or left out).
- Too many skinny slices — a bar chart stays readable; a 15-slice pie doesn't.

Rule to post: "Pie = parts of one whole. Anything else = bars." And bar charts start the count axis at zero — the bar's whole length is the data.

Segment 4 — Misconceptions + Quick Interaction (25 min) · Session 1 closes (~75)

Name the misconceptions out loud, then cure each:

  • ❌ "A histogram is just a bar chart with a different name."
    ✅ Cure: a bar chart shows categories (separated bars, reorder freely); a histogram shows a number line (touching bars, fixed order). "Bars apart = categories. Bars touching = a number line." You can sort a bar chart tallest-first; sorting a histogram would scramble the number line.
  • ❌ "Tall bars on the left means skewed left."
    ✅ Cure: skew is named for the tail — the long, thin side — not the peak. Say it twice: "The tail tells the tale." (Full treatment in Segment 6; plant it now.)
  • ❌ "Any list of numbers can be a pie chart."
    ✅ Cure: a pie needs parts of one whole summing to 100%. If the categories overlap, or the percents don't total 100, the pie is lying — use bars.
  • ❌ "An outlier is a mistake — delete it."
    ✅ Cure: an outlier is a flag, not a verdict. Investigate: a real-but-unusual value stays (and gets reported); a data-entry error gets fixed and documented. Silently deleting data is the one unforgivable move.

Interaction — Think-Pair-Share ("which display?", ~12 min):
Put 6 situations on a slide; students choose the display solo (30 sec), compare with a neighbor (1 min), then vote. Suggested items: the blood types of 500 donors · how one household's electricity bill splits across appliances · the weights of 400 pumpkins at a county fair · the number of houseplants owned by the 12 people on a team · the daily rainfall amounts in a city across 90 days · the share of a pizza shop's sales from each of its five menu categories.
(Answers: bar chart · pie chart · histogram · dot plot · histogram · pie chart.)
Debrief the two that split the room: houseplants (a count is quantitative — a dot plot puts the 12 values on a number line; a bar-per-person is a labeled lineup, not a distribution) and blood types (there is no number line for A/B/AB/O — a histogram is impossible, bars or a pie only).

Segment 5 — Quantitative Displays: Histograms, Dot Plots & Stem Plots (25 min) · Session 2 opens

Hook back in: "Last session: categories. Today: actual numbers — where the word shape starts meaning something."

Dot plot — the display you can draw in 30 seconds.
- One dot per data value, stacked above a number line. Nothing is lost: every individual value stays visible. Best for small datasets.

Worked example (draw it live).

Ask 15 people how many books they read last year: 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 14.
Stack dots: one dot at 2, two at 3, three at 4, four at 5, two at 6, one each at 7 and 8 — and one dot alone at 14, past a visible gap. (Check: 1+2+3+4+2+1+1+1 = 15 ✓.)
Read it: values cluster from 2 to 8, the tallest stack is at 5, and 14 sits apart — our first informal outlier (more in Segment 6).

Histogram — the workhorse for bigger data.
- Cut the number line into equal-width classes, count each class (that's Segment 2's table), draw one touching bar per class. Touching is meaningful: the bars share edges because the classes share edges.

Worked example (table → histogram, all arithmetic shown).

25 finishing times (minutes) from a neighborhood 5K race:
21, 23, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 41, 43.
Classes of width 5, counted out loud:
- 20–<25 → 3 (21, 23, 24) · relative frequency 3 ÷ 25 = 0.12
- 25–<30 → 8 · 8 ÷ 25 = 0.32
- 30–<35 → 7 · 7 ÷ 25 = 0.28
- 35–<40 → 5 · 5 ÷ 25 = 0.20
- 40–<45 → 2 (41, 43) · 2 ÷ 25 = 0.08
Checks: 3 + 8 + 7 + 5 + 2 = 25 ✓ and 0.12 + 0.32 + 0.28 + 0.20 + 0.08 = 1.00 ✓
The histogram: five touching bars with heights 3, 8, 7, 5, 2. One glance now says what 25 raw numbers couldn't: a single peak in the upper 20s and a thinning tail toward the slow side.

Stem plot — the histogram that keeps its digits.

Same 25 times. Stem = tens digit, leaf = ones digit, leaves sorted:
2 | 1 3 4 5 6 6 7 8 8 9 9
3 | 0 1 1 2 3 4 4 5 6 7 8 9
4 | 1 3
(Leaf counts 11 + 12 + 2 = 25 ✓.) Turn it sideways and it's a histogram — but every actual value is still readable. With only three stems it's coarse; note (one line) that split stems — each stem listed twice, leaves 0–4 then 5–9 — would reproduce our five-class picture exactly.

Choosing among the three: dot plot for small sets when every value matters; histogram for anything big; stem plot when you want the picture and the raw digits.

Segment 6 — Shape: Describing What You See (18 min)

Plain language first. Once data are on a number line, the graph has a shape, and the shape is information:

  • Symmetric — left and right halves are rough mirror images. (Bell-shaped is the famous special case: the weights of same-variety apples in an orchard bin pile up evenly around a middle.)
  • Skewed right — the long, thin tail points right (toward large values). Classic: household incomes — most bunch low-to-middle, a few enormous values stretch the tail.
  • Skewed left — the tail points left (toward small values). Classic: retirement ages — most cluster in the 60s, a trickle of early retirees stretches down into the 40s.
  • Uniform — flat; every value about equally common. Classic: the last digit of phone numbers.
  • Bimodal — two distinct peaks, often two groups in one dataset. Classic: a restaurant's customer counts by hour — a lunch peak and a dinner peak.

The direction trap (cure it with the worked data): our 5K histogram (3, 8, 7, 5, 2) peaks early and trails off toward the high times — the tail points right, so it is skewed right, even though the tall bars sit left. Memory hook, said twice: "The tail tells the tale."

Informal outliers. An outlier is a value that sits far from the bulk of the data, usually past a visible gap — like the 14-book reader in the dot plot (nearest neighbor 8, a six-book gap). This week the eye test is enough; Week 3 adds an arithmetic rule (the 1.5×IQR fence).
- Investigate, don't delete: the 14 might be a genuine bookworm (keep it, mention it) or a typo for 4 (fix it, document the fix).

Describe-a-distribution checklist (post it): shape, center (roughly where the middle sits), spread (how far values run), and surprises (outliers, gaps, second peaks). This week we point at these by eye; Week 3 turns center and spread into numbers.

Segment 7 — Misleading Graphs: The Same Data, Told Slant (20 min)

Frame it: "Last week's villain was a biased sample — the data were broken before anyone drew a chart. This week's villain draws a chart designed to make honest data lie."

One fully worked example (compute the lie).

An ad compares two internet providers' average download speeds: Provider K: 95 Mbps. Provider L: 100 Mbps. The ad's bar chart starts its vertical axis at 90 instead of 0.
- Bar heights as drawn: K rises 95 − 90 = 5 units; L rises 100 − 90 = 10 units.
- Visual impression: 10 ÷ 5 = 2.0 — L's bar is twice K's. "Twice the speed!"
- The truth: 100 ÷ 95 ≈ 1.05 — about 5% faster.
- One axis choice inflated 5% into 100%. Redraw from zero and the two bars look almost identical — which is the honest picture.

The four classic tricks (name them; students will find all four in the wild):
1. Truncated axis — bars start above zero, so small gaps look giant (the example above).
2. Area tricks — scaling a 2-D picture: double the value but double both dimensions, and the image grows 2 × 2 = 4×. Pictogram ads love this.
3. Missing labels — no axis numbers, no total, no sample size; a graph you cannot check is a decoration, not evidence.
4. Cherry-picked classes — bucket widths chosen to bury a dip or manufacture a peak. (Same data, different buckets, different story — students will do this to the penguins in Data Lab 2, on purpose.)

The nuance (be honest about it): for a bar chart, the axis must start at zero — the bar's length is the data. A line chart tracking small-but-real changes may legitimately zoom — with the axis loudly labeled. Rule to post: "Bars start at zero. If you zoom, label it loudly." (This tension — when is zooming honest? — is exactly Discussion 2.)

Callback: Week 1's Literary Digest poll was a sampling failure — garbage in. A truncated axis is a display failure — garbage drawn. A statistician guards both doors.

Segment 8 — Technology Workflow + AI-Critique, Callback & Hand-off (12 min) · Session 2 closes (~75)

Technology workflow — from column to picture (exact steps, demo live):
1. Counts without tallying: with categories in column B (say B2:B41), =COUNTIF(B2:B41,"pop") returns the frequency; copy for each genre. For a quantitative class like 25–<30: =COUNTIFS(A2:A26,">=25",A2:A26,"<30").
2. Relative frequency: in the next column, =C2/40 (count ÷ total), then check the column sums to 1 with =SUM(...).
3. Histogram in Google Sheets: select the data column → Insert ▸ Chart → under Chart type choose Histogram → Customize ▸ Histogram ▸ Bucket size to try different class widths. In Excel: select the column → Insert ▸ Statistic Chart ▸ Histogram; right-click the axis → Format Axis to set bin width.
4. Watch the shape move: re-bucket the same column at width 250, 500, 1000 — the shape story should survive reasonable bucket choices; if a "pattern" appears only at one cherry-picked width, distrust it. (A Desmos-class stats tool draws dot plots and histograms from a pasted list, if you prefer it for quick pictures.)

AI-critique moment (students verify, not consume):

Paste this to your chatbot: "A histogram has classes 20–<25, 25–<30, 30–<35, 35–<40, 40–<45 with frequencies 3, 8, 7, 5, 2. Which way is this distribution skewed, and why?"
Then check its answer against the tail rule. Chatbots regularly flip the direction ("the bars are tall on the left, so it's left-skewed") or call the histogram a "bar chart." The correct read: peak in the 25–<30 class, tail stretching toward the high values → skewed right. The tool drafts, you judge — same habit as Week 1, now aimed at graphs. (The tutorial and this week's lab both drill it.)

Callback + tease:
- Callback: "Week 1: trust the number only if the sample was picked honestly. Week 2: trust the picture only if it's drawn honestly. Same skepticism, new door."
- Tease next week: "Your eyes said centered near the upper 20s, spread out, skewed right. Next week we put numbers on those words — mean, median, standard deviation — and meet data's most famous tug-of-war: mean vs. median."

Hand-off (the week's work):
- Chapter 2 (the primary reading) if they haven't read it yet — then Lecture Tutorial 2 (AI tutor, share-link submission) — tables, displays, shape, misleading graphs.
- Data Lab 2 (the penguins return: first pictures of real data — and a deliberately lying axis) · Quiz 2 (end of week) · Discussion 2 (the zoomed-in axis) · Assignment 2 (AI-coached).

Instructor FAQ — Common Stumbles

The table below pairs the predictable Week 2 stumbles with their quick cures.

Student says / does Quick cure
"The tall bars are on the left, so it's left-skewed." Skew is named for the tail, not the peak. Trace the long thin side; that direction is the skew. The tail tells the tale.
Draws a histogram with gaps between bars (or reorders its bars). Gaps mean categories. A histogram's bars touch because the classes share edges on a number line, and their order is fixed. Bars apart = categories; bars touching = a number line.
Puts data with overlapping categories into a pie chart. Pie = parts of one whole, summing to 100%. Overlapping categories (people can pick several) don't form a whole — use a bar chart.
"Where does a value of exactly 25 go — class 20–<25 or 25–<30?" Left edge in, right edge out: 25 belongs to 25–<30. The convention exists so nothing is counted twice.
Relative frequencies sum to 0.99 or 1.01 and the student "fixes" a count. Rounding, not error — say so in a note. Only a sum far from 1 (like 1.10) signals a real mistake (double-counting or a missed case).
"There's an outlier — should I delete it?" Investigate first: real value → keep and report; entry error → fix and document. Never silently delete.
"How many classes should my histogram have?" Rule of thumb: 5–15, equal widths; try a few bucket sizes in your tool. Too few hides the shape, too many shatters it.
"So every graph that doesn't start at zero is a lie?" Bars must start at zero (length = value). A line chart may zoom to show small real changes — honestly, with the axis loudly labeled.

Scope flag

This outline stays within Objective 2's tables-and-graphs half. The split-stem aside (Segment 5), the "zoom honestly" line-chart nuance (Segment 7), and the bucket-size experiment (Segment 8) are added context beyond the strict objective — kept because they cure the week's stickiest misconceptions; cut them for a leaner session. Time-series graphs are deliberately out of scope.