L-DS-1 Statistical Vocabulary & Sampling

Data & Study Design

Who gets to speak for the population?

A number inherits the limits of how it was collected.

Who? · What? · How? · Therefore?

One report · five evidence stages

“71% of voters support this.”

TargetRegistered voters
Selected2,000 mailedrandomly selected
Responded380 replies
Observedreported 71%among respondents
Headline71% of voters
A cycling-infrastructure survey evidence path.

Where did the headline outrun the evidence?

Network Poll 1 · opening diagnosis

Publish, rewrite, or reject?

Registered voters · 2,000 randomly mailed · 380 replies · reported 71% support

A Publish. Random selection makes the 380 replies representative. B Rewrite to describe the 380 respondents. C Reject the result. A low response rate makes the 71% useless. D Publish. A sample of 2,000 is large enough.

Vote by phone or A–D card. We will repair the exact headline after C5.

C1 · use the lesson definitions

Define the group before interpreting any number.

Population

The population is the complete set of all individuals, objects, or measurements of interest in a study. It is the group we want to draw conclusions about.

Notation: population size is written N (capital letter).

Sample

A sample is a subset of the population actually selected for study. We observe the sample and use it to make inferences about the population.

Notation: sample size is written n (lowercase letter).

Ask: “Who do I want my conclusions to apply to?” That is the population—not merely the people surveyed.

C1 + C2 · Figure 2 · population and sample

Figure 2 links population and sample.

Figure 2: The four-element framework across contexts. Population parameters describe the whole group and are usually unknown; sample statistics are computed from observed data and used to estimate them. Take another sample to see whether the statistic moves while the fixed parameter stays put; earlier statistics remain as smaller dots. The census scenario shows the important exception: measuring everyone reveals the parameter directly.

C2 · Figure 1 · sampling variability

A sample mean can vary from sample to sample while μ stays fixed.

Two rounded draws may occasionally match; a match is not guaranteed.

Figure 1: Where a statistic comes from. Draw a random sample of 10 people from the population and watch the selected values produce x̄. A new sample can produce a new x̄, while the population and its μ stay fixed.

C2 · name the number by its source

Now name the number that describes each group.

Parameter

A parameter is a numerical value that describes a characteristic of the population.

Common notation: mean μ · proportion p · standard deviation σ.

Statistic

A statistic is a numerical value computed from sample data. Statistics can vary from sample to sample.

Common notation: mean x̄ · proportion p̂ · standard deviation s.

Match the source: population → parameter · sample → statistic. Equal numerical values do not change their roles.

Network Poll 2 · census hinge · C1/C2

Every employee is in the payroll file.

The defined population contains all 4,500 employees. The recorded mean salary is $67,200.

A μ, a parameterB x̄, a statistic C Both, because the values could matchD Unknowable without random sampling

A · μ. The number came from the whole defined population.

A census removes sampling error—not measurement, coverage, nonresponse, or processing error.

C3 · Figure 3 · variable types

Classify what the values mean—not how they look.

Figure 3: The variable-type hierarchy. Select 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.

C3 hinge · one construct, two variables

The instrument can change what is recorded.

Unrounded elapsed timeAny real valueon a continuum before rounding
Form permits0, 1, 2, …, 24whole hours only
Stored response6recorded value

Which is correct: both continuous, both discrete, or the underlying variable continuous and the stored response discrete?

The unrounded duration is continuous; the stored response is discrete. The form stores only countable whole-hour values.

C4 · Figure 4 · sampling methods

Identify the selection action—not the study topic.

Figure 4: 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.

Network Poll 3 · C4 hinge

Every group—or some groups?

Plan A: randomly select 20 students within every academic program.
Plan B: randomly select three seminar sections; survey everyone in them.

A A is SRS; B is cluster B A is stratified; B is systematic C A is cluster; B is stratified D A is stratified; B is cluster

D. Some from every group; everyone from some groups.

C5 · common sources of sampling bias

Sampling bias distorts who is represented.

Common Sources of Sampling Bias

A sample is biased when it systematically favours certain outcomes over others—when it consistently misrepresents the population in one direction. Bias is not random error.

  • Undercoverage: Some segments of the population have a lower probability (or zero probability) of being selected.
  • Voluntary response bias: People who feel strongly about an issue are more likely to respond.
  • Non-response bias: People selected for the sample do not respond, and those who do not respond differ systematically from those who do.
  • Convenience-sampling risk: Whoever is easiest to reach receives a larger or unknown chance of inclusion.

A larger sample drawn through the same biased mechanism may reduce random variation, but it does not by itself remove the bias.

C5 · Figure 5 · bias and precision are independent

Bias asks where the centre lands. Precision asks how tightly.

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 process that remains systematically centred off target. Bias is not random error: a bigger sample shrinks the spread but never pulls a biased cloud back onto μ.

A larger sample can tighten the dots without moving their centre to μ. More data does not repair a systematically distorted selection or response process.

Anchor return · write before the reveal

Repair the same evidence path.

TargetRegistered votersp remains the target
Selected2,000 mailedSRS: roster-based random sample
Responded380 replies380 / 2,000 = 0.19 = 19% · 1,620 missing
Observedp̂ ≈ 0.71respondent statistic only
RepairLimit the claim“Among the 380 respondents, a reported 71% supported the project.”
The original cycling-survey path, now annotated with the response rate, nonresponse gap, respondent statistic, and defensible claim.

Direction is unknown without response-by-opinion evidence. Follow up the selected sample and compare respondents with the frame.

Independent transfer · keep answer-free

Audit the report before it becomes a headline.

Target Canadian adultsFrame 5,000 newsletter subscribers Invited all 5,000 subscribersResponded 1,847 Reported mean 6.4 h/day, rounded to 1 decimal Form whole hours from 0 to 24Draft claim “Canadians average 6.4 h/day”
  1. Who? Name the population and observed sample.
  2. What? Name the parameter/statistic and both variable types.
  3. How? Locate coverage and response risks; limit direction.
  4. Therefore? Write the strongest defensible sentence.

Delayed transfer model

Compare your audit with the evidence path.

Target adults · 5,000 newsletter subscribers all invited · 1,847 respondents · reported mean 6.4 h, rounded to 1 decimal · whole-hour form

Who?population: Canadian adults · sample: 1,847 respondents

What?μ is the target · x̄ ≈ 6.4 h describes respondents · underlying duration continuous · stored response discrete

How?newsletter undercoverage + possible nonresponse; direction unknown

Therefore?“Among the 1,847 respondents, the reported mean was 6.4 h/day.”

Response rate: 1,847 / 5,000 = 36.94% ≈ 37%.

C6 enrichment · sources of response error in surveys

A good sample can still receive a bad question.

Sources of Response Error in Surveys

  • Leading questions: Wording pushes respondents toward a particular answer.
  • Double-barrelled questions: Two questions are combined into one.
  • Social desirability bias: Respondents give socially acceptable rather than honest answers.
  • Ambiguous wording: Different respondents interpret the question differently.
  • Order effects: Question order influences answers.

Sampling bias changes who is represented. Response error changes what is recorded. A probability sample—and even a census—can still suffer response error.

C6 enrichment · Figure 6 · response error

Locate the problematic phrase, then compare the repair.

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.

Synthesis · four lenses

Every study gets the same audit.

  1. Who?target, selected, responded
  2. What?variable, parameter, statistic
  3. How?frame, selection, response
  4. Therefore?the strongest supported claim

Network Poll 4 · objective exit · C1/C2

Which role map is correct?

Population 4,000 meal-plan students · 200 distinct random selections · all respond · 136 top-two ratings

A Population 200; sample 4,000; p = 136/200 B Population 4,000; sample 200; p = 0.68; p̂ unknown C Population 4,000; sample 200; p unknown; p̂ = 136/200 = 0.68 D Population 4,000; sample 136; p = 0.68; p̂ = 200/4,000

C. Source determines role; 136/200 = 0.68 is the sample proportion.

Network Poll 5 · objective exit · C3–C5

Which complete statement is defensible?

Complete roster · random start + every 20th student · all respond · main dining location coded 1, 2, or 3 · 62% choose Dining Hall

A Systematic; nominal; p = 0.62 because everyone selected replied B Systematic; nominal; p̂ = 0.62 is a sample estimate; sampling variability remains C Multistage; nominal; p = 0.62 because everyone selected replied D SRS; ordinal; selection bias must be upward

B. A random start plus a fixed stride is systematic; numeric codes are unordered labels; p̂ estimates p, while sampling variability remains.

Network Poll 6 · delayed retrieval · C4

Name the method from the action—not the label.

Name all three methods before choosing a sequence.

1

Number the full roster; randomly select 80 distinct students.

2

Randomly choose campuses, then programs, then students.

3

Survey the first 80 students leaving the cafeteria.

A SRS · multistage · convenience B Systematic · cluster · voluntary response C Stratified · multistage · SRS D SRS · cluster · convenience

A. One random subset from the full roster; successive random stages; then easiest to reach.

Anonymous exit reflection · confidence, not mastery

How confident are you with each idea?

C1

I can distinguish a population from a sample.

C2

I can distinguish a parameter from a statistic.

C3

I can classify a recorded variable by type.

C4

I can identify a sampling method from its selection process.

C5

I can recognize sampling bias and limit a conclusion appropriately.

C6

I can recognize survey wording and response errors. Enrichment

0 Not yet 1 Very unsure 2 Beginning 3 Mostly understand 4 Confident 5 I could explain it

One habit to carry forward

A number inherits the process.

Who?→What?→How?→Therefore?

Next: choose a graph and summary that respect what was recorded.

L-DS-2 · Data Visualization