Title: Chapter 1 — Statistics, Data & Study Design
Course: Introduction to Statistics (18-Week Generic Edition)

Course: Introduction to Statistics (18-week generic edition) The module's primary reading · ~25–35 minutes · Objective 1 · Read this before (or alongside) the week's lecture — everything else in the module builds on it.

Why you already care about this

Think about the last 24 hours. Did you rate anything — a driver, a video, a product? Did an app count anything about you — steps taken, minutes watched, money spent? Every one of those numbers is now data in somebody's spreadsheet, and somewhere, someone is using numbers exactly like yours to make a claim about millions of people they never met: "riders are satisfied," "viewers prefer the new format," "customers who join the loyalty program spend more."

This course is about the machinery behind those claims — how a handful of measurements becomes a statement about everyone, and how to tell the honest versions from the garbage. By the end of this chapter you will be able to: tell a population from a sample (and a parameter from a statistic); classify any variable by its level of measurement; name the standard sampling methods and spot the biased ones; and tell an observational study from an experiment — which is the difference between "these two things are linked" and "this causes that."

Here is the one-line version of the whole week, worth memorizing now: statistics is not really about the math — it's about trusting a number that describes people you didn't count.

Where we are

This is Week 1, so there is nothing to build on yet — this chapter is the foundation. Every later week leans on the vocabulary you learn here, so slow down and let the words land. None of it requires any math beyond one division problem.

1. Populations, samples, and the two most important words in the course

Start with the plain idea. When somebody makes a claim from data, there are always two groups in play:

- The population is everyone (or everything) the question is about.
- The sample is the part we actually measured.

We almost always measure a sample instead of the whole population, because measuring everyone is usually impossible, too slow, or too expensive. The rare exception — actually measuring the entire population — is called a census.

Two more words ride along with that split, and they are the two most important words in the course:

- A parameter is a number that describes the population — the true value we want.
- A statistic is the matching number computed from the sample — the measured value we have.

A memory hook that will serve you all term: Population → Parameter, Sample → Statistic. The letters line up.

Worked example. A city transit agency wants to know how satisfied its weekday riders are with a new schedule. There are 24,000 weekday riders — that whole group is the population. Surveying all of them is impractical, so the agency randomly selects 800 riders from the fare-card registry — those 800 are the sample. Of the 800, 592 say they are satisfied.

The proportion satisfied in the sample is 592 ÷ 800 = 0.74, or 74%. That 74% is a statistic: it came from the sample. The number the agency actually cares about — the satisfaction rate among all 24,000 riders — is the parameter. Nobody will ever see the parameter directly. The statistic, 74%, is our best estimate of it.

Once the idea is solid, the notation is easy: statisticians write the population proportion as p (parameter) and the sample proportion as p̂, read "p-hat" (statistic). The hat means "measured," not "true." You do not need to do anything with these symbols yet; just recognize them when they appear.

One subtlety that trips people up: population does not mean "a large group." The population is whoever the question is about — it could be every voter in a country, or the 28 students in one course section, or every order placed at one coffee shop last month. Population and sample are roles defined by the question, not sizes. This term's lab submissions could be a sample of all submissions ever, or the entire population of this term's submissions — same data, different question.

Misconception check. "The study measured two million people, so it must be accurate." Not necessarily — and this misconception is so important it gets its own section (Section 4). Size is not the same thing as quality.

2. What kind of variable is that? The four levels of measurement

A variable is any characteristic we record about the individuals in our data — a person's major, an order's delivery distance, a room's temperature. Before you can summarize a variable (Week 2's topic), you have to know what kind of thing it is, because the kind determines which math is meaningful.

There are two big families, each with two levels — four levels of measurement in all.

Categorical (qualitative) variables are labels or group names:

- Nominal — names with no natural order. Examples: a student's major, blood type, a delivery zone (North / East / South / West), yes/no answers, a driver's ID number.
- Ordinal — categories with a real order, but where the gaps between categories aren't equal or measurable. Examples: letter grades, T-shirt sizes (S / M / L / XL), a 1-to-5 star rating.

Quantitative (numerical) variables are genuine amounts you can do arithmetic on:

- Interval — ordered, with equal gaps between values, but no true zero: zero is just a mark on the scale, not "none of it." Examples: temperature in °F or °C (0° does not mean "no heat"), the calendar year a phone model was released (year zero is an arbitrary marker, not "no time").
- Ratio — ordered, equal gaps, and a true zero, so ratios genuinely make sense. Examples: height, distance, income, and the count of anything. Zero dollars really is "no money," and a 4-mile delivery really is twice as far as a 2-mile one.

The memory hook: N-O-I-R — Nominal, Ordinal, Interval, Ratio, in order of how much math they permit. (Noir is the French word for "black," if that helps it stick.)

Worked example. A food-delivery app records six things about every order. Classify each:

1. Driver ID number → nominal. It's written in digits, but it names a driver. Averaging ID numbers is nonsense.
2. Customer star rating (1–5) → ordinal. Five beats four, but the jump from 4 to 5 stars isn't a measured, equal-sized gap.
3. Food temperature at drop-off (°F) → interval. Ordered, equal gaps, but 0°F is not "no heat."
4. Delivery distance (miles) → ratio. Zero means none, and 4 miles is genuinely twice 2 miles.
5. Tip (dollars) → ratio. A true zero (as every driver knows).
6. Delivery zone → nominal. Four names, no order.

When you're unsure, run this one-question test, in order: Does zero mean "none"? → ratio. Equal gaps but zero is arbitrary? → interval. Ordered labels with fuzzy gaps? → ordinal. Just names? → nominal.

Misconception check. "If it's a number, it's quantitative." No — zip codes, phone numbers, bus route numbers, and ID numbers are numbers that label. The test is never "is it written in digits?" It is "does arithmetic mean anything?" You can't average bus routes.

In practice (your tools). When you open a dataset in Google Sheets or Excel — as you will in this week's Data Lab — the first thing a statistician does is walk the columns and classify each variable. Spreadsheets happily compute the "average" of a column of ID numbers if you ask; the software never protects you from a meaningless calculation. Classifying first is the protection.

3. How the sample gets picked: sampling methods

Section 1 said the statistic is our estimate of the parameter. Whether that estimate deserves trust depends almost entirely on how the sample was chosen.

The gold standard is the simple random sample (SRS): a method where every individual has an equal chance of selection, and every possible group of that size is equally likely — like drawing names from a hat. Every other method is judged against it.

There are four standard probability methods (methods that use genuine chance):

- Simple random — names in a hat. Fair, but by luck it can miss small subgroups.
- Stratified — divide the population into meaningful groups (strata) first, then draw a random sample within each group. Use it when you want every subgroup guaranteed representation.
- Cluster — divide the population into natural groups (clusters), randomly pick whole clusters, and measure everyone in them. Use it to save cost when the population is spread out.
- Systematic — put the population in a list, pick a random starting point, then take every k-th individual. Easy when you have a list or a line of people.

The classic mix-up is stratified versus cluster, so say the hook twice: stratified = sample within every group; cluster = sample whole groups.

And two methods to distrust on sight:

- Convenience sample — measuring whoever is easiest to reach. Cheap, and almost always biased.
- Voluntary response — letting people opt themselves in (online polls, "tap to rate"). The angry and the passionate over-reply; the indifferent middle stays silent.

Worked example — one goal, four designs. A public library system wants the average number of weekly visits across its 60,000 cardholders.

- Survey people walking into the main branch one morning → convenience — and doubly biased, because sampling at the library over-represents frequent visitors.
- Email a random 500 cardholders drawn from the full database → SRS — the trustworthy move.
- Want every branch's community represented → draw a random sample within each branch's cardholders → stratified.
- Need a cheaper field option → randomly pick 2 of the 5 branches and survey every visitor there this week → cluster (accepting the risk that those 2 branches may not represent all 5).

In practice (your tools). Drawing a real SRS takes about thirty seconds in a spreadsheet: put your list in column A, type =RAND() in column B and fill down, then sort both columns by column B and take the top rows. Re-sorting reshuffles the sample — that's the randomness working. You'll do exactly this in Data Lab 1.

4. Bias: why "how you pick" beats "how many you pick"

Bias is error baked into the method — it pushes results in the same wrong direction no matter how big the sample gets. That last clause is the whole point: you cannot fix a biased method by collecting more data with it. Four kinds to recognize:

- Undercoverage — part of the population never had a chance to be included (the sampling frame missed them).
- Nonresponse bias — the people who don't respond differ systematically from those who do.
- Response bias — the wording, setting, or questioner pushes answers (leading questions, sensitive topics, a boss asking).
- Voluntary-response bias — the opt-in crowd isn't typical of the population.

Here is the story that made this lesson famous. In 1936, the magazine Literary Digest ran the largest presidential poll ever attempted: it mailed about 10 million ballots and received 2.4 million back, and it confidently predicted that Alf Landon would defeat Franklin Roosevelt. Roosevelt won in one of the biggest landslides in American history.

What went wrong? The mailing list was built from car registrations and telephone directories — and in the depths of the Depression, people with cars and telephones skewed wealthy (undercoverage). On top of that, only the most motivated recipients mailed their ballots back (nonresponse). Meanwhile, a young pollster named George Gallup surveyed only a few thousand people — chosen carefully — and called the election correctly.

The lesson, in one line you should carry all term: 2.4 million badly-chosen people lost to a few thousand well-chosen ones. Method beats size.**

5. Watching versus doing: observational studies, experiments, and the causation trap

There is one more design question, and it decides what kind of conclusion your data can ever support.

- In an observational study, researchers watch and record — they measure what people already do, changing nothing. (Survey people about their habits; track what happens next.)
- In an experiment, researchers deliberately impose a treatment and compare the results. (Randomly assign some customers the new app layout and others the old one; compare usage.)

The payoff for running an experiment is enormous: only a randomized experiment can support a cause-and-effect claim. An observational study can show that two things are linked — never that one causes the other.

The reason is the confounding variable: a third variable tangled up with both of the things you measured, so you can't tell which is doing the work.

Worked example. Headline: "People who use a fitness tracker take more daily steps." This is observational — nobody assigned trackers; people bought their own. Now ask: who buys a fitness tracker? Largely, people who already exercise. "Already active" is a confounder: it drives tracker-buying and step counts, so the data cannot separate "tracker → steps" from "already-active → both." The tracker may help, or it may be along for the ride — this study can't say.

What could say? An experiment: randomly assign trackers to some people and not others, then compare steps. Random assignment breaks the confounder's grip, because active and inactive people land in both groups by chance.

The memory hook: "Correlation is a handshake, not a push." Two things moving together is a handshake — an association. It takes a randomized experiment to show a push.

Misconception check. "The study found a strong correlation, so X causes Y." Before accepting any causal claim, ask two questions: Could a third variable explain both? and Was anything randomly assigned? If nothing was assigned, you are looking at a link, not a cause — no matter how strong the correlation or how confident the headline.

Key terms

- Population — everyone (or everything) the question is about.
- Sample — the part of the population we actually measured.
- Census — measuring the entire population.
- Parameter — a number describing the population (the value we want).
- Statistic — the matching number computed from the sample (the value we have).
- Variable — any characteristic recorded about the individuals in a dataset.
- Categorical variable — a label or group name; nominal (no order) or ordinal (ordered, unequal gaps).
- Quantitative variable — a genuine amount; interval (equal gaps, no true zero) or ratio (equal gaps and a true zero).
- Simple random sample (SRS) — every individual, and every group of that size, equally likely; the gold standard.
- Stratified sample — random samples drawn within every meaningful group.
- Cluster sample — whole groups randomly selected, everyone in them measured.
- Systematic sample — every k-th individual from an ordered list, after a random start.
- Convenience sample — whoever is easiest to reach; usually biased.
- Voluntary response sample — people opt themselves in; usually biased.
- Bias — error baked into the method, pushing results the same wrong direction regardless of sample size (undercoverage, nonresponse, response, voluntary-response).
- Observational study — watch and record; supports claims of association only.
- Experiment — deliberately impose a treatment, ideally with random assignment; the only design that supports cause-and-effect claims.
- Confounding variable — a third variable tangled with both measured variables, making the cause impossible to isolate.

Summary

- Statistics turns a measured sample into a claim about an unmeasured population; the statistic (measured) estimates the parameter (true but unseen). P→P, S→S — the letters line up.
- Classify every variable before computing: N-O-I-R — nominal, ordinal, interval, ratio — settled by the tests "is there order?", "are the gaps equal?", and "does zero mean none?"
- Trustworthy samples come from chance: SRS (names in a hat), stratified (within every group), cluster (whole groups), systematic (every k-th). Distrust convenience and voluntary response on sight.
- Bias is baked into the method and does not shrink as the sample grows: method beats size (the 1936 Literary Digest poll is the forever-example).
- Observational studies show links; only randomized experiments show causes, because random assignment breaks confounding. Correlation is a handshake, not a push.

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.

1. A podcast app wants to know the average listening time of its 2.1 million subscribers. It computes the average for a random 1,000 of them and gets 47 minutes per day. Name the population, the sample, and say whether "47 minutes" is a parameter or a statistic — and why.
2. Classify each variable by its level of measurement, with a one-line reason: (a) an apartment's floor number in a high-rise; (b) a marathon runner's finishing time in minutes; (c) a phone's operating system (the platform it runs); (d) water temperature in °C.
3. A gym manager wants member opinions on new hours. Design A: poll the 6 a.m. spin class. Design B: post a QR code by the door and tally whoever scans it. Design C: pull a random 100 members from the full membership list. Name each design's sampling method and rank them from least to most trustworthy.
4. A streaming service observes that subscribers who turn on subtitles finish more series than those who don't. A manager concludes subtitles cause higher completion. What design produced this data, why can't it support the causal claim, and what study would?

Answers (worked reasoning — read after attempting)

1. Population: all 2.1 million subscribers (the group the question is about). Sample: the 1,000 randomly chosen subscribers (the part measured). "47 minutes" is a statistic — it was computed from the sample. The matching parameter is the true average across all 2.1 million subscribers, which we never observe directly; 47 minutes estimates it.
2. (a) Floor number → ordinal. Floor 30 is above floor 3, but floors aren't equal-sized steps of anything we measured (and floor numbering is a label with order, not an amount). (b) Finishing time → ratio. Equal gaps and a meaningful zero; 240 minutes is genuinely twice 120. (c) Operating system → nominal. Platform names, no order. (d) Water temperature in °C → interval. Ordered with equal gaps, but 0 °C is the freezing mark, not "no temperature."
3. A = convenience sample (whoever is in one class — and 6 a.m. attendees are hardly typical members). B = voluntary response (people opt in; the strongly-opinionated over-reply). C = simple random sample (every member equally likely). Least to most trustworthy: A and B are both untrustworthy — A measures a narrow slice, B lets opinions self-select — with C clearly the most trustworthy. (Ranking A vs. B is debatable, which is fine; what matters is that both lose to C.)
4. The data are observational — nobody assigned subtitles; viewers chose them. A confounding variable could drive both choices: for example, highly engaged viewers may both turn on subtitles and finish more series. So the link cannot be read as a cause. The design that could support the claim: a randomized experiment — randomly assign subtitles on-by-default to some new viewers and not others, then compare completion rates.

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 1 turns your chatbot into your personal Week 1 tutor.