Week 8 · Lecture outline

Week 8 — Lecture Outline · The Normal Distribution + Midterm Review Kickoff

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objectives covered: Objective 5 — Use the normal model to compute and interpret proportions and percentiles (the normal-distribution portion; sampling distributions follow in Week 10).
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	"One bell-shaped curve keeps showing up — in heights, lifetimes, and fill volumes. How do we use it to turn 'how far from typical?' into an exact percentage?"
By the end of the week, students can…	(1) read a density curve (total area 1; area = proportion); (2) apply the empirical rule (68–95–99.7) to any normal model; (3) compute and interpret a z-score, and use z to compare values from different distributions; (4) do forward (value → percent) and inverse (percent → value) normal calculations with a small z-table and with technology; (5) check whether a normal model fits before trusting any of it.
Key vocabulary	density curve, normal distribution/model, N(μ, σ), empirical rule (68–95–99.7), standardize, z-score, standard normal, percentile, left-tail area, symmetric, inverse normal, assessing normality
Materials	slides (Deck 8), the Week 8 chapter (with the friendly z-table), the week's readings + video links, a spreadsheet (Google Sheets or Excel), a Desmos-class graphing/statistics tool, 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 8 doubles as the midterm review kickoff.

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

Hook (callback to Week 7). Put last week's binomial histogram back on the screen: 100 free throws at p = 0.25 — mean np = 25, SD √18.75 ≈ 4.33. Ask: "What shape did we keep seeing as n grew?" A bell. "That bell was not an accident. This week we meet the curve the bars were trying to become — the most famous curve in statistics."

Heights bunch around a middle value and thin out symmetrically. So do machine-filled volumes, battery lifetimes, and measurement errors. One curve describes them all — you only need two numbers to draw it.
"This week is the hinge of the course: it caps the descriptive-and-probability half (the midterm covers Weeks 1–8 next week), and it is the engine of everything in the inference half."

The promise (write it on the board): "By the end of this week you can take any 'how unusual is this value?' question — a battery that died early, a bottle that's overfilled, a very tall friend — and turn it into an exact percentage, in both directions: value → percent and percent → value."

Why it matters line (memory hook): "The normal curve turns a ruler into a percentage."

Segment 2 — Density Curves: Area Is the Whole Story (18 min)

Plain language first. A histogram of a large dataset is bumpy; a density curve is the smooth idealized outline we model it with. Two rules define every density curve:

The total area under the curve is exactly 1 (that's 100% of the data).
The area between two values is the proportion of the data in that interval.

That's it. Height on its own is not a proportion — area is. For a continuous variable, the probability of landing exactly on one value is 0; only intervals carry area.

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

A game app assigns each player a bonus timer by picking a random value between 0 and 4, every value equally likely. The density curve is a flat rectangle from 0 to 4.
- For the total area to be 1, the height must be 1 ÷ 4 = 0.25.
- P(timer < 1) = area from 0 to 1 = width × height = 1 × 0.25 = 0.25 → 25%.
Not every density curve is a bell — this one is a rectangle, and the area rules still run everything.

Now the star of the week: the normal distribution — the bell-shaped, symmetric density curve. Write the model notation: N(μ, σ) — pronounce it "normal with mean mu and standard deviation sigma."

μ (the mean) says where the center is. σ (the SD, from Week 3) says how spread out it is.
Symmetric ⇒ mean = median ⇒ half the area (0.50) lies below μ.
Two numbers draw the entire curve. That economy is why the normal model is everywhere.

Misconception + cure:
- ❌ "The curve's height at my value is the percent of data there."
✅ Cure: area, not height. Shade an interval every single time you set up a problem — the shaded region is the answer.

Segment 3 — The Empirical Rule: 68–95–99.7 (22 min)

Plain language first. For any normal model — and only approximately-normal shapes — three landmark facts hold:

About 68% of values fall within 1 SD of the mean.
About 95% fall within 2 SDs.
About 99.7% fall within 3 SDs.

These three numbers are the only ones students memorize this week; everything else comes from a table or technology.

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

In one large health survey, adult women's heights are approximately normal with mean 65 inches and SD 2.5 inches — the model N(65, 2.5).
- Within 1 SD: 65 − 2.5 = 62.5 to 65 + 2.5 = 67.5 → about 68% of women.
- Within 2 SDs: 65 − 5 = 60 to 65 + 5 = 70 → about 95%.
- Within 3 SDs: 65 − 7.5 = 57.5 to 65 + 7.5 = 72.5 → about 99.7%.
Then read a tail off the symmetry: outside 2 SDs lies 5%, split evenly — so about 2.5% of women are taller than 70 inches (and 2.5% shorter than 60).

Halving trick (say it explicitly): between the mean and 1 SD above sits 34% (half of 68). Between 1 and 2 SDs above sits 13.5% (half of 95 − 68 = 27). Sketch the curve and label the slices — students who draw it stop guessing.

Misconception + cure (the week's #1):
- ❌ "About 95% of ANY dataset lies within 2 SDs of the mean."
✅ Cure: the empirical rule is a property of the normal curve, not of data in general. A strongly skewed dataset can badly break it. IF bell-shaped, THEN 68–95–99.7 — the rule has a password, and the password is the shape. (Segment 7 shows how to check.)

Segment 4 — z-Scores: One Ruler for Everything + Think-Pair-Share (27 min) · Session 1 closes (~75)

Plain language first. The empirical rule only speaks in whole SDs. For everything else, we first ask the universal question: how many SDs from the mean is my value? That count is the z-score (Week 3 introduced it as relative standing — today it becomes the key to the whole curve):

z = (x − μ) ⁄ σ — "value minus mean, divided by SD."

z = 2 → two SDs above the mean. z = −1.5 → one-and-a-half SDs below. z = 0 → exactly average.
Standardizing converts any normal model to the standard normal — N(0, 1) — which is why one table serves every problem.

Worked example 1 (do every step):

How unusual is a 70-inch woman under N(65, 2.5)?
z = (70 − 65) ÷ 2.5 = 5 ÷ 2.5 = 2.0. Two SDs above the mean — by the empirical rule, taller than about 97.5% of women. (Next segment sharpens that to 97.72% with the table.)

Worked example 2 — comparing across different rulers (the signature use):

Who is relatively taller: a 70-inch woman (women ~ N(65, 2.5)) or a 74-inch man (men ~ N(70, 4))?
- Her z = (70 − 65) ÷ 2.5 = 2.0. His z = (74 − 70) ÷ 4 = 1.0.
- She is 2 SDs above her distribution's mean; he is only 1 above his. She is relatively taller, even though he is 4 inches taller in raw inches. z-scores let you compare apples to oranges by putting both on the SD ruler.

Misconceptions + cures:
- ❌ "A negative z-score means something went wrong (or the value is bad)."
✅ Cure: the sign is just direction — below the mean. Half of all z-scores are negative. For tire wear or wait times, a negative z can be great news.
- ❌ "z = 2 means twice as tall (or twice the mean)."
✅ Cure: z counts SD steps, not multiples. Say it in words every time: "two standard deviations above the mean."
- ❌ "Divide by the variance."
✅ Cure: z uses σ, the SD — same units as the data — never σ². If your z has units, you divided by the wrong thing.

Interaction — Think-Pair-Share (rapid-fire, ~10 min). Six quick items on a slide; solo 30 sec, pair 1 min, fingers vote. All use N(65, 2.5) unless stated:
1. z for a 67.5-inch woman → +1
2. z for a 60-inch woman → −2
3. Which is more unusual: z = −2 or z = +1.5? → z = −2 (farther from 0 — distance, not sign)
4. About what percent of women are between 62.5 and 67.5 inches? → 68%
5. A woman at the 15.87th percentile has what z? → −1 (15.87% below ↔ 1 SD below)
6. True or false: half the area under any normal curve lies below the mean → true (symmetry).

Segment 5 — Forward Normal Calculations: The Friendly z-Table (25 min) · Session 2 opens

Hook back in: "Last session z told us how far. Today the table turns how far into what percent."

The tool. A z-table reports, for each z, the area to the LEFT — the proportion below that value. This course runs on a friendly six-value table (embedded in the chapter, the tutorial, and the assignment coach — and any values the midterm needs are provided with it):

z	area to the left	z	area to the left
−2.5	0.0062	0.5	0.6915
−2	0.0228	1	0.8413
−1.5	0.0668	1.25	0.8944
−1.25	0.1056	1.5	0.9332
−1	0.1587	2	0.9772
−0.5	0.3085	2.5	0.9938

(Symmetry pairs every negative row with a positive one: the area left of −z equals the area right of +z. z = 0 sits at 0.5000. Any z not on this table, technology supplies.)

The three-step recipe (drill it): ① compute z → ② look up the left area → ③ decide: below = read it; above = 1 − left area; between = bigger left area − smaller left area. Draw and shade before step ②.

Worked examples (all on N(65, 2.5), every step out loud):

Below: P(height < 61.25). z = (61.25 − 65) ÷ 2.5 = −3.75 ÷ 2.5 = −1.5 → left area 0.0668 → about 6.68% of women are shorter than 61.25 inches.
Above: P(height > 70). z = 2.0 → left area 0.9772 → above = 1 − 0.9772 = 0.0228 → 2.28%. (The empirical rule said "about 2.5%" — the table sharpens it.)
Between: P(62.5 < height < 70). z-scores −1 and 2 → 0.9772 − 0.1587 = 0.8185 → about 81.85% of women.

Misconception + cure (the classic wrong-tail error):
- ❌ Reporting 0.9772 when the question asked for the area ABOVE.
✅ Cure: the shaded sketch, plus a sanity check said out loud: "70 inches is tall — more than half the women below it? Yes, so 'below' is the big number and 'above' must be small." If your answer contradicts your sketch, you grabbed the wrong tail.

Segment 6 — Inverse Normal: From Percent Back to a Value (20 min)

Plain language first. Forward: value → percent. Inverse: someone hands you the percent — "the tallest 2.28%," "the 93.32nd percentile" — and asks what value sits at that cutoff. Run the table backwards: find the area inside, read the z, then un-standardize:

x = μ + z·σ — "start at the mean, walk z SDs."

Worked examples (every step):

A clothing designer wants sleeves to fit all but the tallest 2.28% of women. Tallest 2.28% ⇒ left area 0.9772 ⇒ z = 2 ⇒ x = 65 + 2(2.5) = 70 inches. Design to 70.
What height marks the 93.32nd percentile? Left area 0.9332 ⇒ z = 1.5 ⇒ x = 65 + 1.5(2.5) = 68.75 inches.

Percentile language (police it): "the 93.32nd percentile" means 93.32% of values sit below — it is a position, not a score. A student at the 84th percentile did not "get 84%."

Misconception + cure:
- ❌ Walking the wrong direction (subtracting when the cutoff is above the mean).
✅ Cure: ask "is this cutoff above or below the middle?" before touching the formula. Tallest-percent cutoffs sit above μ (z positive); shortest-percent cutoffs below (z negative). The sketch settles it.

Segment 7 — Is the Bell Actually There? Assessing Normality (15 min)

Plain language first. Every calculation this week carried a silent assumption: the data are approximately normal. Before using the model on real data, check it — three quick looks:

Histogram: one mound, roughly symmetric, no heavy skew or second bump.
Empirical-rule check: compute x̄ and s, then the actual percent within 1 and 2 SDs — close to 68 and 95?
Outliers/skew: a long tail or extreme values are the model's natural enemies.

Worked contrast: heights pass all three checks. Household incomes fail spectacularly — a long right tail (a few enormous values) drags the mean, and "95% within 2 SDs" collapses. Using z-based percentages on income data produces confident nonsense.

Misconceptions + cures:
- ❌ "It's numeric and continuous, so it's normal." / "My sample is large, so it's normal."
✅ Cure: shape is an empirical fact, not a birthright — and collecting more skewed data gives you a better picture of a skewed shape, not a bell. (Echo of Week 1: more of the wrong thing doesn't become the right thing.)
- ❌ "The percent check alone proves it."
✅ Cure: checks are evidence, not proof — a lumpy two-species mixture can pass one check and fail another (this week's data lab makes exactly this happen with the penguins). Judge all three looks together.

Tease the lab: "In Data Lab 8 you'll run these checks on real Antarctic penguins — and discover why which penguins you pool decides whether the bell appears."

Segment 8 — Technology + AI-Critique · Midterm Review Kickoff · Hand-off (15 min) · Session 2 closes (~75)

Technology workflow (exact steps, live):
1. Forward, in a spreadsheet: =NORM.DIST(x, mean, sd, TRUE) returns the area to the LEFT. Demo: =NORM.DIST(67, 65, 2.5, TRUE) → 0.7881 — off our six-value table, and that's the point: the table teaches the logic; technology handles any z. (Google Sheets and Excel are identical here.)
2. Inverse: =NORM.INV(0.9, 65, 2.5) → ≈ 68.20 — the 90th percentile of heights. (Behind the scenes that's z ≈ 1.28.)
3. Desmos-class tool: type normaldist(65, 2.5), shade the region, and read the area live — the fastest way to see forward vs. inverse.

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

Paste this to your chatbot: "Without using a calculator or code, what is the area to the left of z = 1.25 under the standard normal curve? And the area within 1 SD of the mean for a skewed distribution?"
Then check it against the embedded table: the correct left area is 0.8944 — chatbots quoting tables from memory routinely drift by a digit or invent precision. And the second question is a trap you now catch: for a skewed distribution the 68% figure doesn't apply — a confident "68%" is the wrong answer. The tool drafts, you judge.

MIDTERM REVIEW KICKOFF (the week's second job — give it real airtime):
- Next week is Midterm Week. The midterm covers Weeks 1–8, it's 50 multiple-choice, closed to AI, and it's a low-stakes checkpoint — 5% of the grade. The weekly work you've already done is the grade engine; the exam is a temperature reading.
- The review roadmap (draw it as one line on the board): W1 where data come from → W2–3 pictures and summaries → W4 two variables at once → W5 probability → W6 random variables → W7 the binomial → W8 the normal. One story: get data honestly, describe it, quantify chance, model it.
- The exam bundle lives in the Week 9 module: a study guide, a practice exam, and an exam-prep tutorial (that tutorial is next week's graded tutorial). Tell students: start with the study guide's self-diagnosis, not with rereading.
- Point out the bridge: the normal model learned this week is exactly what Week 10 uses to describe sample means — the inference half runs on it.

Callback + tease:
- Callback: "Week 3 gave us the SD as a ruler; Week 7's binomial kept sprouting bells. This week the ruler and the bell merged into a machine that converts distance into percent."
- Tease: "After the midterm: the second half. We stop describing data we have and start concluding things about data we don't — and the bell curve drives every bit of it."

Hand-off (the week's work):
- Chapter 8 (primary reading, with the friendly z-table) — then Lecture Tutorial 8 (AI tutor; share link + summary).
- Data Lab 8 (penguins vs. the bell) · Quiz 8 (end of week) · Discussion 8 ("Designed for the Middle") · Assignment 8 (AI-coached).

Instructor FAQ — Common Stumbles

Student says / does	Quick cure
"Is a negative z-score bad?"	It's only direction — below the mean. Half of all values have negative z. For "lower is better" variables, negative z is good news.
Reports 0.9772 when asked for the area above.	Draw-and-shade first, then sanity-check aloud: "should the answer be more or less than half?" Above = 1 − left area, always.
Applies 68–95–99.7 to obviously skewed data.	The rule's password is the shape: IF bell-shaped, THEN 68–95–99.7. Show incomes as the counterexample.
Divides by the variance instead of the SD.	z = (x − μ)/σ. Quick tell: a correct z is unit-free; dividing by σ² leaves stray units.
"z = 2 means twice the average."	z counts SD steps: "2 SDs above the mean." Force the words, not just the digits.
"84th percentile = scored 84%."	A percentile is a position: 84% of values sit below. Tie it to the table: left area 0.8413 ↔ z = 1.
"My value isn't on the six-value table."	By design — use technology (`=NORM.DIST`) for off-table z. The table teaches the logic; the software does the lookup.
"Is every bell-ish histogram normal?" / "Big samples become normal?"	Bell-ish is necessary, not sufficient — run all three checks. And more data sharpens the true shape, whatever it is; size never straightens skew.
"What do I memorize for the midterm?"	Just 68–95–99.7 and the z recipe — any needed table values are provided. The midterm is a low-stakes checkpoint; the study guide in the Week 9 module is the starting gun.

Scope flag

This outline stays within Objective 5's normal-distribution portion. The uniform "rectangle" density curve, the technology NORM.INV demo, and the depth of the assessing-normality segment are added context (not strictly required by the objective) — kept because they cure the week's misconceptions; cut them for a leaner session. The normal approximation to the binomial is mentioned only as the Week 7 hook, not computed — and sampling distributions are deliberately deferred to Week 10.