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Module 1 · Descriptive Statistics · DS-1

Statistical Vocabulary
& Sampling

Follow one journalist’s investigation — and meet the language behind every statistical claim.

Live session · join with code OTTO-DS1 · ottolearn.org/join

How today works

Predicting is part of the method

You’ll often vote before I explain. Predicting incorrectly is doing it right — committing first is what makes the idea stick.

Every poll is anonymous. No one sees who answered what — not even me.

Changing your mind after talking it over is the goal, not a failure.

Today · Maya’s investigation

One question: how much do CEGEP students sleep?

Each step is a real decision Maya has to make — and a poll you’ll answer first.

Warm-up · 60 seconds · anonymous

Keep that instinct — by the end you’ll know exactly what went wrong, and how Maya fixes it.

C1 · Predict first · ConcepTest · vote → discuss → re-vote

Core concept 1 · C1 · now let’s name it

Population vs. Sample

Population

The complete set Maya wants conclusions about — all 48,000.

size = N

Sample

The subset she actually studies — her 300.

size = n

If you picked “the 300” — that’s the sample. The population is who Maya wants to generalize to: “Who do I want my conclusions to apply to?”

See it · a sample drawn from a population

Different samples, different x̄

Predict first: will two of Maya’s samples give the same average? Then I’ll draw a few.

Figure 1: A population of 50 people (left) and a random sample (right). Click Draw Sample to select a new group — x̄ shifts every time, but μ stays fixed. Switch sample sizes (n = 5 / 10 / 20) and watch the history strip: larger n means x̄ clusters more tightly around μ.

Practice · population vs sample · new version anytime

Core concept 2 · C2

Parameter vs. Statistic

Parameter

Describes the population. Usually unknown — Maya can’t reach all 48,000.

μ   σ

Statistic

Computed from the sample. Known, but varies sample to sample.

x̄   s

Parameter → Population · Statistic → Sample. Greek for the population (μ, σ); Roman for the sample (x̄, s).

Model · watch me label all four · think aloud

The four elements of Maya’s own study

Maya wants the true mean nightly sleep of all 48,000 CEGEP students. She surveys 300 and gets x̄ = 6.8 h.

Population

Who she wants to conclude about — all 48,000. (Not the 300.)

Sample

Who she actually measured — the 300 surveyed.

Parameter

The number she wants but can’t see: true mean sleep of all 48,000 → μ. Unknown.

Statistic

The number she computed from the 300 → x̄ = 6.8 h. Known — her best estimate of μ.

Two groups, two numbers: x̄ is what we know; μ is what we’re after.

Together · we label one · your turn on the two numbers

Now we label one together

New study: a campus gym wants the average weekly visits of all 5,000 members. A random sample of 120 averages 2.4 visits/week.

  1. Population  = all 5,000 gym members (who they want to conclude about).
  2. Sample  = the 120 members actually measured.

  3. Your turn: name the parameter and the statistic — with notation.

    Parameter = μ, the true mean weekly visits of all 5,000 (unknown). Statistic = x̄ = 2.4, computed from the 120 (known — it estimates μ).

See it · the estimation bridge

A statistic estimates an unknown parameter

Predict: as Maya’s sample grows, does x̄ drift toward the true μ — or wander away?

Figure 2: The four-element framework of statistical inference. The left side lives in the population — values that exist but are usually unknown. The right side lives in the sample — values we compute from data. The arrow captures the entire purpose of sampling: use what you know (x̄) to estimate what you can't observe (μ).

Activity · classify it · single anonymous vote

Practice · parameter vs statistic · new version anytime

Core concept 3 · C3

Types of Variables

Qualitative (categorical)

Labels, not numbers you do arithmetic with.

Nominal — no order. program of study Ordinal — ordered, unequal gaps. poor / fair / good

Quantitative (numerical)

Numbers where differences and ratios make sense.

Discrete — counted. cups of coffee Continuous — measured. hours of sleep

The deciding question is never “is it written with digits?” — it’s “does arithmetic make sense?”

Model · watch me classify · think aloud

How I’d type a variable — out loud

Maya’s form also records each respondent’s phone number. What type is it?

1 · Label or number?

It’s written with digits — but I don’t stop there. Digits alone never decide it. On to the real test.

2 · Does arithmetic work?

Averaging two phone numbers is meaningless; …0199 isn’t “more” than …0188. Arithmetic fails → qualitative.

3 · Any natural order?

No number ranks above another → no order → nominal.

Phone number → qualitative, nominal. The digits were a decoy; the arithmetic test settled it.

Together · we finish this one · your turn on the last step

Now we classify one together

Next field: each respondent’s number of caffeinated drinks yesterday (0, 1, 2, …).

  1. 1 · Label or number?  A real count — drinks. A number.
  2. 2 · Does arithmetic work?  Yes — 3 drinks is one more than 2, and averages mean something → quantitative.

  3. 3 · Your turn: counted or measured? Name the sub-type — then I’ll reveal.

    Counted in whole units — you can’t log 2.5 drinks → quantitative, discrete.

See it · trace a variable to its type

Follow the path to a classification

Pick a ⚠ trap example — predict its type before you trace the path.

Figure 2: The variable-type hierarchy. Click any example below the tree to trace its classification path. Each leaf node shows a distinct colour used consistently in this course. The ⚠ examples are traps — variables that look like one type but are actually another.

Activity · classify it · single anonymous vote

Practice · variable types · new version anytime

Core concept 4 · C4 · sampling methods (1/2)

Probability-based methods

Simple random (SRS)

Every sample of size n equally likely. Gold standard — but needs a full list.

Stratified

Split into homogeneous strata; sample from each. Guarantees representation.

Systematic

Random start, then every k-th. Easy — biased if the list has a periodic pattern.

All three give every student a known chance of selection — the basis for valid inference.

Core concept 4 · C4 · sampling methods (2/2)

Cluster, multistage & convenience

Cluster

Split into heterogeneous clusters; sample a few entirely. Cheap when spread out.

Multistage

Combine methods in stages. Practical for big national surveys; errors can compound.

Convenience

Whoever is easiest to reach. Fast, cheap — and almost always biased.

Cluster ≠ stratified. Strata are similar inside (sample all); clusters are diverse inside (sample some).

See it · who actually gets selected?

Six methods, one population

Before I switch methods — predict which dots stratified will pick.

Figure 3: A population of 56 people arranged in a grid. Each tab shows which people get selected under that sampling method. For Stratified sampling, dot colour shows stratum membership; for Cluster, it shows cluster membership.

Model · watch me choose · think aloud

How I’d actually pick a method — out loud

1 · Start from the question

Maya wants the typical sleep of all CEGEP students — so every student needs a real chance of being picked. That alone kills convenience: the cafeteria crowd isn’t all students.

2 · Inventory what I have

She has a full enrolment list. With a complete list, simple random is on the table — the gold standard. Systematic is easier, but I pause: is the list ordered by program or cohort? If it repeats, every k-th could lock onto a pattern.

3 · Name the worry, then commit

Could a small program vanish in a random draw? If that risk mattered I’d switch to stratified to guarantee each appears. Here it doesn’t — so I commit to SRS, and I can say why I rejected the rest.

Let’s run those three steps together on a new constraint →

Together · we pick one · your turn on the final call

Now we choose a method together

New constraint: Maya has one list of all 9,000 students at her campus, ordered by student ID (effectively random order). She wants a quick, evenly spread sample of 300 — and has no software to draw random numbers.

  1. 1 · Start from the question.  She wants a representative read on all students — every student needs a real chance.
  2. 2 · Inventory what she has.  One full list, no random-number tool, order is arbitrary (by ID, not program or cohort).

  3. 3 · Your turn: which method fits — and why is the “periodic pattern” risk low here?

    Systematic — random start, then every k-th (k = 9000 / 300 = 30). No software needed, and with no meaningful order in the list there’s no repeating pattern to bias it. → systematic sampling.

Activity · which tool applies? · single anonymous vote

Practice · sampling methods · new version anytime

C5 · Predict first · ConcepTest · vote → discuss → re-vote

Core concept 5 · C5 · now let’s name them

Bias in Sampling

Undercoverage

Some groups have little or no chance of selection. an online-only survey skips the offline

Voluntary response

Only those who feel strongly reply. Maya’s gaming-Discord poll

Non-response

Selected people don’t answer — and differ from those who do.

Convenience

Easiest-to-reach people are systematically different. the 11 p.m. library crowd

Bias is directional, not random — it does not shrink with sample size. A biased survey of 10,000 can beat nothing.

See it · bias vs. precision

Why a bigger biased sample won’t save you

Predict: does a bigger biased sample land closer to the truth, or just tighter around the wrong answer?

Figure 5: Each dot is one sample's estimate of μ (mean weekly exercise hours). The four quadrants cross bias (centred on μ vs. off-target) with precision (tight vs. spread) — because the two are independent. Read the columns: the left (unbiased) column centres on μ whether tight or spread; the right (biased) column misses μ either way. The top-right cell is the trap — a precise, confident-looking estimate that is reliably wrong. Bias is not random error: a bigger sample shrinks the spread but never pulls a biased cloud back onto μ.

1936 — the original cautionary tale. The Literary Digest mailed 10 million ballots and got 2.4 million back, then predicted the wrong U.S. president by a landslide. Its list came from car and phone owners — wealthier than most in the Depression. A giant sample, sunk by who it left out.

Practice · spot the bias · new version anytime

Build it · with your neighbour · constructed response

Beyond the sample · C6

Even a good sample can ask bad questions

Leading

Wording that pushes an answer.

Double-barrelled

Two questions in one.

Social desirability

People answer to look good.

Ambiguous

“Do you sleep regularly?”

Order effects

Earlier questions sway later ones.

See it · bad-question autopsy

Same question, fixed wording

Spot the loaded word in each question first — then compare the original against the fix.

Figure 6: Four common survey design flaws. For each, the problematic word or phrase is highlighted in the original question. Click a flaw type to examine it — then compare the original against the corrected version.

Activity · spot the flaw · tap one line

Exit ticket · shapes next class

Maya files her story

Three things to keep

Population vs. sample — Maya studies 300 to learn about 48,000.

Statistic estimates parameter — her x̄ → the unknown μ, never assume they’re equal.

Bias ≠ small sample — direction, not size, is the danger.

Maya’s headline: a stratified sample of 300 → x̄ ≈ 6.8 h (vs. 8 h recommended). Her friend’s Discord poll said 5.1 — voluntary response, don’t trust it.   Next: DS-2 — Graphs & numerical summaries.

Sleep figures are inspired by real college-sleep research — illustrative, not a specific study.

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