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Week 3 · Lecture outline

Week 3 — Lecture Outline · Numerical Summaries: Center & Spread

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objectives covered: Objective 2 — Summarize and display univariate data.
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 week's big question "Can one number honestly stand in for a hundred numbers — and which number should it be?"
By the end of the week, students can… (1) compute and interpret the mean, median, and mode, and say which one a dataset deserves; (2) explain resistance — why one wild value drags the mean but barely touches the median; (3) compute the range, variance, and standard deviation of a small dataset by hand and read SD as the typical distance from the mean; (4) build a five-number summary, find the IQR, apply the 1.5×IQR rule, and read a boxplot; (5) compute a z-score and use it to compare values from different scales.
Key vocabulary mean (x̄, μ), median, mode, resistant measure, skew and the mean's pull, range, deviation, variance (s², σ²), standard deviation (s, σ), the n−1 divisor (named, not derived), five-number summary, quartiles (Q1, Q3), IQR, 1.5×IQR rule, outlier (formal), boxplot, z-score
Materials slides (Deck 3), the Week 3 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. "Your maps app says your commute averages 26 minutes. A listing site says apartments in your area rent for an average of $1,650. A syllabus says the class average on a quiz was 82. You made three decisions this week off numbers like these. What exactly is an 'average' — and when is it lying to you?"

  • Last week we turned piles of numbers into pictures (histograms, shapes). A picture is honest but bulky — you can't text someone a histogram and expect a decision.
  • This week we compress a whole dataset into two or three numbers: one for center ("where's the middle of this?") and one for spread ("how much do the values wander?"). Compression loses information — the skill is knowing what got lost.

The promise (write it on the board): "By the end of this week you can summarize any list of numbers with the right center, the right spread — and you can spot the exact moment an 'average' stops telling the truth."

Why it matters line (memory hook): "Center without spread is half a story. The average tells you where the middle is; the spread tells you whether to believe it."

Segment 2 — Three Centers: Mean, Median, Mode (22 min)

Plain language first.
- The mean is the everyday "average": add everything up, divide by how many. It's the balance point — where the data would balance on a seesaw.
- The median is the middle value once the data are sorted — half the values below, half above. (Even count? Average the middle two.) Sorting first is not optional.
- The mode is the most frequent value — and it's the only center that works for categorical data (the modal answer, the most common species).

Memory hook (put it on a slide):

Mean = balance point. Median = middle person. Mode = most common.

Notation, after the idea (callback to Week 1's letters): the sample mean is written x̄ ("x-bar"); the population mean is μ ("mu"). Same split as p̂ and p: samples get Latin letters, populations get Greek. The statistic x̄ estimates the parameter μ.

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

Five one-way commute times, in minutes: 15, 20, 20, 25, 30.
- Mean: 15 + 20 + 20 + 25 + 30 = 110; 110 ÷ 5 = 22 minutes.
- Median: already sorted; the middle (3rd) value is 20 minutes.
- Mode: 20 appears twice, everything else once → 20 minutes.
Three defensible "centers," two different numbers — and that gap is about to become the whole story.

Quick check to plant: "Why do the mean (22) and median (20) disagree at all?" (The 30 pulls the mean up a little; the median only counts positions.) Hold that thought — Segment 3 blows it wide open.

Segment 3 — Resistance: the Day the Mean Lied (20 min)

The setup (worked example, part 1). A rideshare driver logs five airport runs: 12, 18, 20, 22, 28 minutes.
- Sum = 12 + 18 + 20 + 22 + 28 = 100 → mean = 100 ÷ 5 = 20 minutes. Median (sorted, middle value) = 20 minutes. Lovely agreement.

The twist (part 2). Same route, but one day a crash closes the highway and the 28-minute run becomes 88 minutes: 12, 18, 20, 22, 88.
- New sum = 12 + 18 + 20 + 22 + 88 = 160 → new mean = 160 ÷ 5 = 32 minutes.
- New median: sorted it's still 12, 18, 20, 22, 88 → median = 20 minutes. Unmoved.
- One bad day moved the mean 12 minutes. It moved the median zero.

Name the idea: a measure is resistant if extreme values (outliers) barely budge it. The median is resistant; the mean is not — every value tugs on the balance point, and a far-away value tugs hardest.

Skew, connected to Week 2's shapes: in a right-skewed distribution (long tail toward high values — incomes, home prices, rents), the tail's extreme values drag the mean above the median. Left-skew drags it below. Memory hook: "The mean chases the tail." That's why rent and income reports lead with the median — one penthouse shouldn't redefine "typical."

Which center to report (the judgment call):
- Roughly symmetric, no wild values → mean (it uses every value's actual size).
- Skewed or outlier-prone → median (report the mean too if you like — the gap between them is itself a diagnostic).
- Categorical → mode is the only option.

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

Name the misconceptions out loud, then cure each:

  • ❌ "The average IS the typical value."
    ✅ Cure: only when the shape cooperates. In skewed data most values sit below (or above) the mean — a few extremes finance the difference. Always ask for the shape (Week 2) before trusting the center. The crash-day mean of 32 describes a commute that basically never happens.
  • ❌ "An outlier moves every summary, so it doesn't matter which one I use."
    ✅ Cure: the whole point of resistance — 12/18/20/22/88 moved the mean 12 minutes and the median not at all. Which summary you pick decides how much one weird value gets to vote.
  • ❌ "The mode is useless — or there can be only one."
    ✅ Cure: the mode is the only center for categorical data ("the most common apartment type is a one-bedroom"), and ties are legal (bimodal — often a sign two groups got mixed, a Week 2 shape idea).
  • ❌ "If a value looks extreme, just delete it."
    ✅ Cure: never silently. First ask: typo or truth? A 9,999-gram penguin is a data-entry error; an 88-minute commute really happened. Investigate, report with-and-without, and use a rule (coming in Segment 6) — not a feeling — to flag it.

Interaction — Think-Pair-Share (rapid-fire, ~12 min):
Put 6 quick calls on a slide; students decide solo (30 sec), compare with a neighbor (1 min), class votes. For each: does the MEAN or the MEDIAN better summarize a typical value?
1. Household incomes in a city (long right tail) → median
2. Backpack weights of 30 students, roughly symmetric → mean
3. Home sale prices with three luxury estates in the data → median
4. Points scored by one player across 20 games, no wild games → mean
5. Repair-shop turnaround times, most jobs quick, a few disasters → median
6. "What's the most common jacket size we should order?" → neither — mode
Debrief #6 hard: it's a categorical/what-to-order question — center isn't always mean-or-median.

Segment 5 — Spread: from Range to Standard Deviation (25 min) · Session 2 opens

Hook back in: "Two grocery stores. Your last five weekly bills at Store A: 90, 95, 100, 105, 110 dollars. At Store B: 70, 85, 100, 115, 130. Both average exactly $100. Are they the same store experience?" (No — B is a rollercoaster.) Center without spread is half a story.

Step 1 — the range (and its flaw). Range = max − min. Store A: 110 − 90 = 20. Store B: 130 − 70 = 60. Quick, intuitive — but it uses only two values, so one outlier owns it completely (not resistant, and blind to everything in between).

Step 2 — deviations, the raw material. A deviation is (value − mean): how far, and which direction, each value sits from the balance point. Deviations always sum to zero (that's what "balance point" means), so we can't just average them — we square first.

Step 3 — the standard deviation recipe (fully worked; go slowly).

Five quiz scores (out of 10): 6, 6, 8, 10, 10.
1. Mean: 6 + 6 + 8 + 10 + 10 = 40; 40 ÷ 5 = 8.
2. Deviations: 6−8 = −2 · 6−8 = −2 · 8−8 = 0 · 10−8 = 2 · 10−8 = 2. (Check: they sum to 0. ✓)
3. Square them: 4, 4, 0, 4, 4.
4. Add: 4 + 4 + 0 + 4 + 4 = 16.
5. Divide by n − 1 = 4: 16 ÷ 4 = 4. This is the sample variance, s² = 4 (units: points²).
6. Square root: √4 = 2. The sample standard deviation is s = 2 points.
Say it in words: a typical quiz score sits about 2 points from the mean of 8.

The vocabulary, locked down:
- Variance (s²) = the average squared deviation (with the n−1 divisor). Its units are squared — points², dollars² — which is why nobody speaks variance out loud.
- Standard deviation (s) = √variance, back in the data's own units. SD = the typical distance from the mean. SD is never negative, and SD = 0 means every value is identical.
- Why n − 1? One line at this level: a sample's deviations are measured from x̄ (which came from the same sample), so they run slightly small; dividing by n − 1 corrects the books. Your calculator and spreadsheet do this automatically for sample data. Population versions: σ and σ² (Greek again — populations get Greek letters).
- Like the mean it's built on, the SD is not resistant — the crash-day 88 would inflate it wildly.

Back to the stores: technology gives Store A s ≈ 7.9 and Store B s ≈ 23.7 (same mean, triple the wobble) — "typical distance from $100" is the sentence a non-statistician can actually use.

Segment 6 — Five Numbers, One Picture: Quartiles, IQR & the Outlier Rule (22 min)

Plain language first. The median splits the data in half. Do it twice more: Q1 is the median of the lower half (25% below it), Q3 is the median of the upper half (75% below it). The five-number summary — min, Q1, median, Q3, max — is a five-word biography of any dataset, and the boxplot is its picture (Week 2's plots get a sibling).

One fully worked example (every step).

Monthly rents for eight studio apartments, already sorted (dollars):
950, 1000, 1100, 1150, 1250, 1300, 1400, 2400.
- Median: even count → average the 4th and 5th: (1150 + 1250) ÷ 2 = 1200.
- Q1: median of the lower half (950, 1000, 1100, 1150) = (1000 + 1100) ÷ 2 = 1050.
- Q3: median of the upper half (1250, 1300, 1400, 2400) = (1300 + 1400) ÷ 2 = 1350.
- Five-number summary: 950 · 1050 · 1200 · 1350 · 2400.
- IQR = Q3 − Q1 = 1350 − 1050 = 300 — the width of the middle 50%, and it's resistant (the 2400 never touched it).
- The 1.5×IQR rule (the formal outlier fence): 1.5 × 300 = 450. Lower fence = 1050 − 450 = 600; upper fence = 1350 + 450 = 1800. The $2,400 rent is above 1800 → flagged as an outlier (some landlord is dreaming).
- Mean here = 10,550 ÷ 8 = 1318.75 vs. median 1200 — the luxury listing drags the mean; the median shrugs. (Callback to Segment 3.)

Boxplot read-through: box from Q1 to Q3 (the middle 50%), line at the median (not the mean!), whiskers to the most extreme values inside the fences, dots beyond the fences = flagged outliers. A boxplot is the five-number summary you can see — and skew shows as a lopsided box/whisker.

The pairing rule (put it on a slide):

Mean travels with SD. Median travels with IQR. Report the resistant pair for skewed/outlier-prone data, the classic pair for well-behaved data.

Segment 7 — z-Scores: How Many SDs from Home? (16 min)

Plain language first. A raw number means little without context: is a 44-minute commute bad? Depends on your typical commute and your typical wobble. The z-score converts any value into universal units: how many standard deviations from the mean.

z = (value − mean) ÷ SD

  • z = 0 → exactly average. Positive → above the mean; negative → below. |z| ≈ 2 or more → starting to be genuinely unusual (a fuller rule arrives with the normal model, later in the course).
  • A z-score is not a percentage — z = 2 means "2 SDs above the mean," not "top 2%."

One fully worked example (comparing across different scales).

You scored 85 on a statistics exam where the class mean was 75 with SD 5. You scored 80 on a chemistry exam with class mean 70 and SD 10. Which performance is more impressive?
- Statistics: z = (85 − 75) ÷ 5 = 10 ÷ 5 = 2.0 → two SDs above the class.
- Chemistry: z = (80 − 70) ÷ 10 = 10 ÷ 10 = 1.0 → one SD above.
Both are 10 raw points above average — but statistics wins: in a tighter distribution, the same 10 points is a rarer feat. z-scores make different scales comparable.

Misconception + cure:
- ❌ "A negative z-score means something went wrong."
✅ Cure: negative just means below the mean — for commute times or grocery bills, negative z is the good direction. Sign is direction, size is rarity.

Tease: "Right now a z-score is a ruler. In the normal-distribution week it becomes a probability machine — same z, big upgrade."

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

Technology workflow — the whole week in five functions (exact steps):
1. Put a dataset in column A (say A2:A21).
2. Center: =AVERAGE(A2:A21) · =MEDIAN(A2:A21) · =MODE(A2:A21).
3. Spread: =STDEV(A2:A21) for the sample SD (that's the one this course means) · =VAR(A2:A21) for s² · =MAX(A2:A21)-MIN(A2:A21) for the range.
4. Quartiles: =QUARTILE(A2:A21,1) and =QUARTILE(A2:A21,3); subtract for the IQR.
5. Same names in Google Sheets and Excel. Two footnotes worth saying out loud: =STDEVP() is the population divisor (n) — using it on sample data is this week's classic tech error; and software computes quartiles by interpolation, so =QUARTILE can differ slightly from our split-the-halves hand method. Neither is "wrong" — say which method you used.

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

Paste this to your chatbot: "Find the sample standard deviation of 6, 6, 8, 10, 10. Show every step."
Then check it against today's worked example (s = 2, from Σ(x−x̄)² = 16 and 16 ÷ 4 = 4). Chatbots frequently divide by n = 5 instead of n − 1 (giving √3.2 ≈ 1.79 — the population SD) or fumble the squaring arithmetic. If its answer isn't 2, make it show its divisor. The tool drafts, you judge — and this exact trap is waiting in the data lab.

Callback + tease:
- Callback: "Week 1 taught you to ask who was measured; Week 2 gave you the shape; this week compressed the shape into honest numbers. Notice how the ideas stack: you can't pick mean-vs-median without first seeing skew."
- Tease next week: "So far every dataset has been one column. Next week: two variables at once — does studying more actually track with scoring more? Scatterplots, correlation, and the two-variable versions of 'careful now.'"

Hand-off (the week's work):
- Chapter 3 (the primary reading) if they haven't read it — then Lecture Tutorial 3 (AI tutor, share-link submission) — centers, spread, five-number summaries, z-scores.
- Data Lab 3 (penguins again: center & spread by species, plus a sabotage experiment) · Quiz 3 (end of week) · Discussion 3 (the average that misled you) · Assignment 3 (AI-coached).

Instructor FAQ — Common Stumbles

Student says / does Quick cure
Finds the median of an unsorted list. Sort first, every time. The median is a position — positions only exist in sorted data. Same for quartiles.
"The deviations sum to zero, so the spread is zero?" That zero is the balance point doing its job — it's why we square the deviations before averaging. Spread lives in the squares.
Reports the variance (s² = 4) as "the spread is 4 points." Variance is in squared units (points²). Take the square root and report s = 2 points — the typical distance from the mean.
Uses =STDEVP() (or divides by n) on sample data. That's the population divisor. This course's data are samples: divide by n − 1, i.e., =STDEV(). On 6, 6, 8, 10, 10 the wrong divisor gives ≈1.79 instead of 2 — a visible tell.
"My =QUARTILE value doesn't match my hand answer." Software interpolates; our hand method is median-of-the-halves. Small differences are expected — name your method, don't panic.
Calls any big-ish value an outlier. Feelings don't flag outliers — the 1.5×IQR fences do. Compute Q1 − 1.5·IQR and Q3 + 1.5·IQR and check. Then investigate (typo vs. truth), never silently delete.
"The line in the middle of a boxplot is the mean." It's the median. Boxplots are built entirely from the resistant five-number summary; the mean isn't in the picture.
"z = 2 means top 2%." z = 2 means 2 SDs above the mean — a distance, not a percentile. Turning z into percents needs the normal model (later in the course).

Scope flag

This outline stays within Objective 2. The one-line why n−1 rationale and the =QUARTILE-vs-hand-method footnote are added context (not strictly required by the objective) — kept because they prevent the two most common technology-generated confusions; cut them for a leaner session. z-scores appear here as relative standing only — all probability/area uses wait for the normal-distribution week.