Week 3 — Quiz (auto-graded) · Numerical Summaries: Center & Spread

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 2 — numerical summaries: center, spread, five-number summaries, outliers, z-scores.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 3 · Closed to AI.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-03-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file.

Blueprint

No trick questions; distractors target the Week 3 misconceptions named in the lecture outline (unsorted medians, the n-vs-(n−1) divisor, variance-as-SD, feelings-as-outlier-rules, z-as-percentage).

Questions, key, and feedback

Q1 (MC). Six days of one student's bus commute took 18, 20, 22, 24, 26, and — on a day with a road closure — 70 minutes. Which statement is correct?
- A. Mean = 23 minutes; median = 30 minutes
- B. Mean = 30 minutes; median = 23 minutes ✅
- C. Mean = 30 minutes; median = 22 minutes
- D. Mean = 22 minutes; median = 23 minutes
Feedback: Sum = 180, so the mean = 180 ÷ 6 = 30. Sorted, the two middle values are 22 and 24, so the median = 23. The road-closure day drags the mean 7 minutes above the median — the mean chases the tail. (D silently deleted the outlier; C forgot to average the middle two.)

Q2 (MC). A city's apartment rents are strongly right-skewed by a small number of luxury units. Which summary best represents a typical rent, and why?
- A. The mean, because it uses the exact size of every rent in the city
- B. The mode, because the most common rent is by definition the typical one
- C. The range, because it shows the full span from cheapest to priciest unit
- D. The median, because it resists the pull of the few luxury units in the tail ✅
Feedback: In right-skewed data the tail drags the mean upward, so the resistant median is the honest "typical." (Using every value's exact size — option A — is precisely how the luxury units get their oversized vote.)

Q3 (Multiple answer — select all that apply). Which of the following summaries are resistant to outliers?
- A. The median ✅
- B. The mean
- C. The interquartile range (IQR) ✅
- D. The standard deviation
- E. The range
Feedback: Median and IQR are built from positions in the sorted data, so extremes barely move them. Mean, SD, and range use every value's size — the range most fragile of all, since one outlier owns an endpoint.

Q4 (MC). Five weekly grocery bills are $30, $30, $40, $50, $50; the mean is $40. What is the sample standard deviation?
- A. $10 ✅
- B. $100
- C. $8.94
- D. $20
Feedback: Deviations −10, −10, 0, 10, 10 → squares sum to 400 → 400 ÷ (n − 1) = 400 ÷ 4 = 100 = s² → s = √100 = $10. (B stopped at the variance; C divided by n = 5, the population divisor; D is the range's half-sibling.)

Q5 (MC). The sample variance of a set of exam scores is 49 (in squared points). The sample standard deviation is —
- A. 49 points
- B. 24.5 points
- C. 7 points ✅
- D. 2,401 points
Feedback: SD = √variance = √49 = 7, back in the data's own units. Variance lives in squared units (points²) — which is why it's the stepping stone, not the headline. (D squared instead of rooting; B halved.)

Q6 (Matching). Match each summary to what it measures.
| Summary | Correct description |
|---|---|
| Mean | The balance point — the sum of the values divided by how many there are |
| Median | The middle value once the data are sorted |
| Standard deviation | The typical distance of the values from the mean |
| IQR | The width of the middle 50% of the data |
Feedback: The pairing rule follows directly: the mean travels with the SD (both use every value's size); the median travels with the IQR (both use positions).

Q7 (MC). Twenty recorded commute times have the five-number summary min = 22, Q1 = 28, median = 34, Q3 = 40, max = 75 (minutes). The IQR is —
- A. 53 minutes
- B. 12 minutes ✅
- C. 6 minutes
- D. 34 minutes
Feedback: IQR = Q3 − Q1 = 40 − 28 = 12 — the width of the middle 50%. (A is the range, max − min; D is the median itself.)

Q8 (MC). For those same commute times (Q1 = 28, Q3 = 40, IQR = 12), apply the 1.5×IQR rule. Which values are formally flagged as outliers?
- A. Only the 75-minute commute — it lies above the upper fence of 58 ✅
- B. Both the 22-minute and the 75-minute commutes are flagged
- C. Nothing is flagged — every commute lies inside the fences
- D. Every commute longer than Q3 = 40 minutes counts as an outlier
Feedback: 1.5 × 12 = 18, so the fences sit at 28 − 18 = 10 and 40 + 18 = 58. The min (22) is inside; the max (75) is beyond 58 → flagged. Being above Q3 just means "upper quarter" — the fence, not the quartile, defines an outlier.

Q9 (MC). Your score on a certification practice test was 82; the group mean is 70 with a standard deviation of 8. Your z-score is —
- A. 12
- B. 0.67
- C. 1.2
- D. 1.5 ✅
Feedback: z = (82 − 70) ÷ 8 = 12 ÷ 8 = 1.5 — you sit one and a half standard deviations above the group mean. (A is the raw gap before dividing; it isn't a z-score until the SD scales it.)

Q10 (True / False). "In a strongly right-skewed distribution, the mean is typically larger than the median."
- True ✅
- False
Feedback: True. The long right tail's extreme values drag the balance point (mean) up past the middle person (median) — the mean chases the tail. That gap is why incomes and rents report medians.

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item's two resistant summaries (median, IQR) are the only resistant options listed; every computation re-verified in the week's math script (mean 30 / median 23; s = $10 from Σd² = 400; √49 = 7; IQR 12; fences 10 and 58; z = 1.5); no positional pattern in the key (B D A C B A D) and no letter appears in more than two of the seven MC items; option lengths within each item are comparable (no length giveaway); no item asserts a fact outside the Week 3 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces.

Item-bank entries (for variants + the midterm/final)

All ten items are tagged week=3 · objective=2 · topic=numerical-summaries-center-spread and deposited in Item Bank: Week 3 — Numerical Summaries: Center & Spread with idents w03q1–w03q10. The midterm (Week 9), the final (Week 18), and per-term variant updates draw fresh variants from this bank's concepts — never these live stems. (Tags: w03q1 mean-vs-median-outlier, w03q2 resistant-center-choice, w03q3 resistant-measures, w03q4 sd-computation, w03q5 variance-vs-sd, w03q6 summary-definitions, w03q7 iqr, w03q8 outlier-rule, w03q9 z-score, w03q10 skew-mean-median.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 3 Quiz — Numerical Summaries: Center & Spread"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true