Title: Chapter 8 — The Normal Distribution
Course: Introduction to Statistics (18-Week Generic Edition)

Course: Introduction to Statistics (18-week generic edition) The module's primary reading · ~30–40 minutes · Objective 5 (normal-distribution portion) · Read this before (or alongside) the week's lecture — everything else in the module builds on it, and the friendly z-table you'll use all week lives in Section 4.

Why you already care about this

Last week you watched something quietly remarkable. The binomial histogram — 100 free throws at p = 0.25, mean 25, SD ≈ 4.33 — kept arranging its bars into the same silhouette: high in the middle, sloping off symmetrically on both sides. A bell. That shape was not an accident, and it is not unique to free throws. Measure enough adult heights and you get it. Weigh enough machine-filled bottles, time enough batteries, track enough measurement errors — the same curve keeps showing up, in fields that have nothing to do with each other.

This week we stop admiring the bell and start using it. The normal distribution is a smooth curve completely described by just two numbers — a mean and a standard deviation — and it comes with an astonishing superpower: it converts distance from average into an exact percentage. By the end of this chapter you will be able to: read any density curve (area = proportion); apply the 68–95–99.7 rule to any normal model; compute a z-score and use it to compare values from completely different distributions; run forward calculations (value → percent) and inverse calculations (percent → value) with a small table or a spreadsheet; and — the professional habit — check whether the bell is actually there before trusting any of it.

One more reason to care this particular week: the normal model is the hinge of the course. It caps the first half (next week's midterm covers Weeks 1–8), and it powers the second half, where the same curve describes sample means and drives every inference tool we build. The one-line version to carry with you: the normal curve turns a ruler into a percentage.

Where we are

Weeks 1–4 taught you to collect data honestly and describe it — shape, center, spread, and the standard deviation as a ruler (Week 3 even introduced the z-score as "relative standing"). Weeks 5–7 taught you to quantify chance: probability rules, random variables, and the binomial model. This week those two threads braid together: a probability model (like the binomial, but continuous) whose spread is measured in SDs. Nothing here requires new arithmetic — just subtraction, division, and a table lookup.

1. Density curves: area is the whole story

A histogram of a large dataset is bumpy and jagged. A density curve is the smooth, idealized outline we use to model it — the shape the histogram is trying to be. Every density curve obeys two rules, and the two rules are the entire theory:

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

Notice what is not on that list: height. The curve's height at a value is not the percent of data there — only area carries proportion. In fact, for a continuous variable the probability of landing exactly on any single value is 0; only intervals have area.

Worked example. 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 running from 0 to 4. For the total area to equal 1, the height must be 1 ÷ 4 = 0.25. What's the probability the timer lands below 1? The area from 0 to 1: width × height = 1 × 0.25 = 0.25, or 25%. No bell in sight — a rectangle — and the area rules ran everything anyway.

Misconception check. "The curve is tall at my value, so lots of the data equals my value." Height is not proportion — area is. Train the habit now: every problem this week starts with a sketch of the curve and a shaded region. The shaded region is the answer; the rest of the chapter is just how to price it.

2. The normal model and the empirical rule

The star density curve — the one this whole week orbits — is the normal distribution: bell-shaped, symmetric, one central mound, tails thinning off forever in both directions. We write the model as

N(μ, σ) — "normal with mean μ and standard deviation σ."

Two numbers draw the entire curve. μ says where the center sits. σ — the same standard deviation you computed in Week 3 — says how spread out the curve is: small σ, tall and narrow; large σ, low and wide. Because the curve is symmetric, the mean equals the median, so half the area (0.50) always lies below μ.

For any normal model, three landmark facts hold. Together they are called the empirical rule (or the 68–95–99.7 rule), and they are the only numbers you memorize this week:

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

Worked example. 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 up to 65 + 2.5 = 67.5 inches → about 68% of women.
- Within 2 SDs: 65 − 5 = 60 up to 65 + 5 = 70 inches → about 95%.
- Within 3 SDs: 65 − 7.5 = 57.5 up to 65 + 7.5 = 72.5 inches → about 99.7%.

Symmetry lets you slice further, and the slicing is where the rule earns its keep. Outside 2 SDs lies 100% − 95% = 5%, split evenly by symmetry — so about 2.5% of women are taller than 70 inches, and 2.5% are shorter than 60. Between the mean and 1 SD above sits half of 68% = 34%; between 1 and 2 SDs above sits half of (95% − 68%) = 13.5%. Sketch the bell, label the slices, and most empirical-rule questions answer themselves.

Misconception check. "So about 95% of any dataset lies within 2 SDs of the mean." This is the week's most important error. The empirical rule is a property of the normal curve, not of data in general — a strongly skewed dataset can break it badly. Memorize the fine print as a password: IF bell-shaped, THEN 68–95–99.7. Section 5 shows how to check for the bell before you trust the rule.

3. z-scores: one ruler for everything

The empirical rule only speaks in whole SDs — 1, 2, 3. For everything in between, we first ask the universal question: how many standard deviations from the mean is my value? The answer is the z-score:

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

A z-score of 2 means the value sits two SDs above the mean; z = −1.5 means one-and-a-half SDs below; z = 0 means exactly average. The sign is pure direction, and the number is pure distance — a z-score has no units at all, which is exactly what makes it powerful.

Worked example. How unusual is a 70-inch woman under N(65, 2.5)? z = (70 − 65) ÷ 2.5 = 5 ÷ 2.5 = 2.0. Two standard deviations above the mean — by the empirical rule, taller than roughly 97.5% of women. (Section 4's table sharpens that to 97.72%.)

Worked example — comparing across different rulers. Who is relatively taller: a 70-inch woman, where women's heights run N(65, 2.5), or a 74-inch man, where men's heights run N(70, 4)? Her z = (70 − 65) ÷ 2.5 = 2.0. His z = (74 − 70) ÷ 4 = 1.0. She stands two SDs above her distribution's mean; he stands only one above his. She is relatively taller, even though he has four raw inches on her. That is the signature move of the z-score: it puts values from different distributions onto one shared ruler, so apples and oranges finally compare.

Standardizing also explains a small miracle of bookkeeping: subtract the mean and divide by the SD, and any normal model becomes the standard normal distribution, N(0, 1). Heights in inches, lifetimes in hours, volumes in milliliters — after standardizing they are literally the same curve. That is why one small table (next section) serves every normal problem ever posed.

Misconception check. Three quick ones. "A negative z is bad news." No — it's a direction (below the mean), and half of all z-scores are negative; for a "lower is better" variable, negative z is good news. "z = 2 means twice the average." No — z counts SD steps, not multiples; say "two standard deviations above the mean" out loud. "Divide by the variance." No — z divides by σ, the SD, never σ². A correct z-score is unit-free; if yours still has units attached, you divided by the wrong quantity.

4. Forward and inverse: the friendly z-table

A z-table answers one question: for a given z, what is the area to the LEFT — the proportion of values below? This course runs on a deliberately small, friendly version. The same table appears in your tutorial and your lab, and any values needed on the midterm are provided with it.

The table below lists the left-tail area for the six z-values this course uses (plus zero), with their negative mirrors.

z | Area to the left |  | z | Area to the left
−2.5 | 0.0062 |  | 0 | 0.5000
−2 | 0.0228 |  | 0.5 | 0.6915
−1.5 | 0.0668 |  | 1 | 0.8413
−1.25 | 0.1056 |  | 1.25 | 0.8944
−1 | 0.1587 |  | 1.5 | 0.9332
−0.5 | 0.3085 |  | 2 | 0.9772
 |  |  | 2.5 | 0.9938

Symmetry ties the halves together: the area to the left of −z equals the area to the right of +z. (Check one: left of −1 is 0.1587, and right of +1 is 1 − 0.8413 = 0.1587. Same number.)

The three-step recipe for any forward calculation (value → percent):

1. Standardize: compute z = (x − μ)/σ.
2. Look up the area to the left of z.
3. Decide: below → read it off; above → 1 − (left area); between → (bigger left area) − (smaller left area).

And before step 2, always: sketch and shade.

Worked examples (all on women's heights, N(65, 2.5)).

- Below. What proportion of women are shorter than 61.25 inches? z = (61.25 − 65) ÷ 2.5 = −3.75 ÷ 2.5 = −1.5. Left area: 0.0668. About 6.68% of women.
- Above. What proportion are taller than 70 inches? z = (70 − 65) ÷ 2.5 = 2.0. Left area 0.9772, so above = 1 − 0.9772 = 0.0228 → 2.28%. (The empirical rule said "about 2.5%" — the table sharpens it.)
- Between. What proportion are between 62.5 and 70 inches? The two z-scores are (62.5 − 65)/2.5 = −1 and 2. Between = 0.9772 − 0.1587 = 0.8185 → about 81.85%.

The inverse problem runs the same machine backwards: you're handed the percent — "the tallest 2.28%," "the 93.32nd percentile" — and asked for the value at the cutoff. Find the area inside the table, read off its z, then un-standardize:

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

Worked examples.

- A clothing designer wants sleeves to fit all but the tallest 2.28% of women. Tallest 2.28% means 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.

A vocabulary guardrail while we're here: a percentile is a position, not a score. "The 93.32nd percentile" means 93.32% of values sit below that point. A student at the 84th percentile did not "get 84%" — they landed at z = 1 (left area 0.8413), wherever that falls in raw units.

Misconception check. The classic table error is grabbing the wrong tail — reporting 0.9772 when the question asked for the area above. The cure is the sketch plus one out-loud sanity check: "70 inches is tall — are more than half of women below it? Yes. So 'below' should be the big number, and 'above' must be small." If your final answer contradicts your shaded sketch, you took the wrong tail. The inverse version of the same error is walking the wrong direction — subtracting z·σ for a cutoff that sits above the mean. Ask "above or below the middle?" before touching the formula.

In practice (your tools). The friendly table teaches the logic; technology handles any z. In Google Sheets or Excel, =NORM.DIST(x, mean, sd, TRUE) returns the area to the left — =NORM.DIST(67, 65, 2.5, TRUE) gives 0.7881, a z of 0.8 that our table doesn't carry. The inverse function is =NORM.INV(percent, mean, sd) — =NORM.INV(0.9, 65, 2.5) returns about 68.20 inches, the 90th percentile of heights. In a Desmos-class tool, type normaldist(65, 2.5), shade the region on screen, and read the area live — the fastest way to see forward versus inverse. You'll use both in this week's data lab.

5. Is the bell actually there? Assessing normality

Every calculation above leaned on a silent assumption: the data really are approximately normal. Real data are under no obligation to cooperate. Before you deploy the model — and especially before you announce a z-based percentage to anyone — run three quick checks:

1. Look at the histogram. One mound, roughly symmetric, no heavy skew, no second bump.
2. Run the empirical-rule check. Compute x̄ and s, then count the actual percent of values within 1 and 2 SDs of the mean. Approximately-normal data land near 68% and 95%.
3. Hunt skew and outliers. A long tail or a handful of extreme values are the model's natural enemies — they stretch s and drag x̄, and the advertised percentages quietly stop being true.

Heights pass all three checks, which is why they're this chapter's running example. Household incomes fail spectacularly: a long right tail (a few enormous values) drags the mean above the typical household, "95% within 2 SDs" collapses, and any z-based claim about incomes becomes confident nonsense.

Two myths to retire on the spot. "It's numeric and continuous, so it's normal" — shape is an empirical fact about the data, not a birthright of numbers. And "my sample is large, so it's normal" — collecting more skewed data paints a sharper picture of a skewed shape; size never straightens skew. (Echo of Week 1: more of the wrong thing doesn't become the right thing.)

Misconception check. "My data passed the 68% check, so they're normal." One passed check is evidence, not proof. A lumpy mixture of two groups can land near 68% within 1 SD while its histogram shows two mounds and no bell — this week's data lab engineers exactly that surprise with real penguin data. Judge all three looks together, and treat "approximately normal" as a judgment call you defend, not a box you tick.

6. Where the first half lands: your review roadmap

Next week is Midterm Week. The exam covers Weeks 1–8, it's 50 multiple-choice questions, closed to AI, and — keep the temperature down — it's a low-stakes checkpoint worth 5% of the grade. Your weekly work is, and remains, the grade engine.

Here is the whole first half as one line: W1 where trustworthy data come from → W2–3 pictures and summaries (shape, center, spread) → W4 two variables at once → W5 probability → W6 random variables → W7 the binomial → W8 the normal model. One story: get data honestly, describe it, quantify chance, model it. The Week 9 module holds the full exam bundle — a study guide, a practice exam, and an exam-prep tutorial. Start with the study guide's self-diagnosis rather than rereading chapters front to back. And notice where this week sits: the normal model is the bridge — Week 10 will point this same curve at sample means, and the inference half of the course drives across it.

Key terms

- Density curve — a smooth curve modeling a distribution; total area 1, and area over an interval = proportion of the data there.
- Normal distribution (normal model) — the bell-shaped, symmetric density curve written N(μ, σ); completely determined by its mean and SD.
- N(μ, σ) — notation for a normal model with mean μ and standard deviation σ.
- Empirical rule (68–95–99.7) — in a normal model, about 68% of values lie within 1 SD of the mean, 95% within 2, 99.7% within 3.
- z-score — z = (x − μ)/σ; how many SDs a value sits from the mean (sign = direction, size = distance); unit-free.
- Standardizing — converting a value to its z-score, which maps any normal model onto the standard normal.
- Standard normal distribution — the normal model N(0, 1); the shared ruler every normal problem is translated into.
- Left-tail area — what a z-table reports: the proportion of values below a given z.
- Percentile — the position below which the stated percent of values fall (a position, never a score).
- Inverse normal calculation — from a given percent back to the value at that cutoff: x = μ + z·σ.
- Assessing normality — checking (histogram, empirical-rule check, skew/outlier hunt) whether a normal model fits before trusting its percentages.

Summary

- A density curve carries proportion as area: total area 1; area between two values = fraction of the data there. Height is never proportion.
- The normal model N(μ, σ) is fully drawn by two numbers; symmetry puts half the area below μ (mean = median).
- Empirical rule: 68% within 1 SD, 95% within 2, 99.7% within 3 — memorize these three, and their password: IF bell-shaped.
- z = (x − μ)/σ counts SD-steps from the mean and puts values from different distributions on one ruler; standardizing turns every normal model into N(0, 1).
- Forward: z → left-tail area → below / above (1 − area) / between (difference). Inverse: percent → z → x = μ + z·σ. Sketch and shade first, every time.
- Assess normality before trusting the model: histogram, actual 68/95 check, skew-and-outlier hunt — a judgment call you defend.

Check your understanding

Work these before looking at the answers — they're ungraded, and the answers (with reasoning) are at the bottom of the page. The friendly z-table from Section 4 is fair game.

1. A bakery's sourdough loaves come out of the oven with weights approximately N(800 g, 20 g). (a) Between what two weights do the middle 95% of loaves fall? (b) About what percent of loaves weigh more than 840 g?
2. At a veterinary clinic, adult house-cat weights run approximately N(4.5 kg, 0.8 kg). A patient named Biscuit weighs 5.5 kg. Compute Biscuit's z-score and use the table to say what percent of cats weigh less than Biscuit.
3. An orchard grades its apples by weight, approximately N(190 g, 12 g). The heaviest 2.28% earn a "premium" sticker. What weight is the premium cutoff?
4. A friend says: "Every dataset follows the 68–95–99.7 rule — that's why it's a rule." In two or three sentences, explain what your friend has wrong and name one concrete check you'd run on a dataset before using the rule on it.

Answers (worked reasoning — read after attempting)

1. (a) The middle 95% sits within 2 SDs of the mean: 800 − 2(20) = 760 g up to 800 + 2(20) = 840 g. (b) Above 840 g is beyond +2 SDs: outside ±2 SDs lies 5%, split evenly by symmetry, so about 2.5% of loaves. (The table's sharper answer: 1 − 0.9772 = 0.0228, i.e., 2.28%.)
2. z = (5.5 − 4.5) ÷ 0.8 = 1 ÷ 0.8 = 1.25 — Biscuit is 1.25 SDs above the mean. Left-tail area for z = 1.25 is 0.8944, so Biscuit outweighs about 89.44% of adult house-cats. (Sanity check: above the mean, so the "below" percent must beat 50% — it does.)
3. "Heaviest 2.28%" means 97.72% of apples sit below the cutoff — left area 0.9772 → z = 2. Un-standardize: x = 190 + 2(12) = 214 g. Apples at 214 g and up get the sticker.
4. The 68–95–99.7 rule is a property of the normal (bell-shaped) curve, not of data in general — skewed or multi-mound data can miss those percentages badly, so "every dataset" is exactly the wrong claim. Before using the rule: look at the histogram for one symmetric mound (and/or compute the actual percent of values within 1 and 2 SDs of the mean and compare to 68/95, while hunting for skew and outliers). If the shape check fails, the rule — and every z-based percentage — is off the table.

Next up in this module: the slide deck if you want the lecture's visual walk-through, then the Readings & Resources page for optional deeper dives — and when you're ready to practice, Lecture Tutorial 8 hands your chatbot the friendly z-table and turns it into your personal Week 8 tutor.