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Week 5 · Data Lab

Week 5 — Data Lab · "The Long Run: 500 Rolls of a Virtual Die"

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objective: Objective 4 — probability as long-run relative frequency; the law of large numbers; independence · SLO A (reason quantitatively from data)
Worth 50 points · Data labs group = 15% of the grade · Data Lab 5 · Due: end of Week 5
Format: a hands-on data lab — this week you manufacture your own data: 500 virtual die rolls in a spreadsheet, watched in real time as chance settles toward its promise; then you catch the AI's mistakes when it interprets your run.

This is the course's signature weekly component. Every instructional week has a Data Lab: real, free, linked public data — nothing to buy, download, or sign up for — worked in your spreadsheet, finished with an AI-critique step. Plan on 60–90 minutes. (This week the "dataset" is one your own spreadsheet generates — 500 rolls that exist nowhere else in the world.)

Part 1 — The Big Picture

This week's chapter made a strange-sounding claim: nobody can predict your next die roll, but anybody can predict your next 10,000. The probability P(six) = 1/6 is a long-run promise — and today you'll watch that promise being kept, live, on your own screen. Your running proportion of sixes will thrash around wildly for the first dozen rolls, wander for a hundred, and then settle unmistakably toward 0.1667. That settling — the law of large numbers — is the engine that makes polls, quality checks, and every inference tool after the midterm possible.

You'll also meet the promise's fine print: your 500 rolls will not give exactly 0.1667, your six faces will not appear exactly 83.3 times each, and none of that means anything is broken. Telling chance variation from a real effect is the single judgment this course spends eighteen weeks sharpening; today is its purest form.

Optional 7-minute background read before you start: Probability (Math is Fun) 🔗 https://www.mathsisfun.com/data/probability.html

Part 2 — The Task

The guiding question: When does a probability's long-run promise kick in — how wild are 10 rolls, how tame are 500, and how do you tell ordinary chance variation from a die that's actually unfair?

The tools (nothing to buy or download): a spreadsheet — Google Sheets (free) or Excel. Its =RANDBETWEEN(1,6) function is a virtual die: every call produces 1, 2, 3, 4, 5, or 6, each with probability 1/6, with no memory of previous calls. You'll roll it 500 times in one column.

Part 3 — Materials & Procedure

You need: a spreadsheet and about 60–90 minutes. Nothing else.

Warm-up (5 min, no spreadsheet needed). Two small computations by hand or calculator — write both down; they're your benchmarks for the whole lab:

  1. The theoretical probability of rolling a six: 1 ÷ 6 = ______ (round to four decimal places).
  2. The number of sixes you'd expect in 500 rolls: 500 × (1/6) = ______ (round to the nearest whole number).

Now build the simulation:

  1. Set up headers. In a blank sheet, type into row 1: A1 = roll, B1 = result, C1 = running sixes, D1 = running proportion.
  2. Number the rolls. A2 = 1. A3 = =A2+1. Select A3, copy it, then paste into A4:A501. Column A should now count 1 to 500 (check: A501 shows 500).
  3. Roll the dice. B2 = =RANDBETWEEN(1,6). Copy B2 and paste into B3:B501. You now have 500 rolls. (They will re-roll every time the sheet recalculates — that's expected; see step 7.)
  4. Count sixes as you go. C2 = =COUNTIF($B$2:B2,6). Copy C2, paste into C3:C501. Each row now shows how many sixes have appeared so far. (The $ signs pin the start of the range; the end grows row by row — that's what makes it "running.")
  5. Compute the running proportion. D2 = =C2/A2. Copy down to D501. Format column D to show 4 decimal places (Format ▸ Number). D501 is your final proportion of sixes in 500 rolls.
  6. Chart the settling. Select D2:D501 → Insert ▸ Chart → line chart. You should see wild swings on the left flattening toward a value near 0.1667 on the right. (Excel: Insert ▸ Line chart. Add a title like "Running proportion of sixes.")
  7. Freeze your run. Any edit re-rolls all 500 dice. Once you like your run: select columns A:D → Copy → Paste special ▸ Values only onto the same cells (Excel: Paste Values). Your rolls are now frozen data, and your checkpoint numbers won't shift while you write.
  8. Record the checkpoints. Fill in Scaffold Table 1 from column D: the running proportion after 10 rolls (cell D11), 50 rolls (D51), 100 rolls (D101), 250 rolls (D251), and 500 rolls (D501).
  9. Tally the faces. In an empty area, list faces 1–6, and next to each: =COUNTIF($B$2:$B$501,1), then ,2), ,3), and so on. Divide each count by 500 for its relative frequency. Fill in Scaffold Table 2, and confirm the six counts sum to 500.

Plausibility note (read before judging your run): for 500 rolls of a fair die, the proportion of sixes almost always lands between about 0.13 and 0.20 (roughly 67–100 sixes) — and the same range applies to every other face. In the first 10 rolls, anything from 0 to about 5 sixes is unsurprising. Wild early, tame late: that's the whole show.

Fallback (no spreadsheet available): use a real six-sided die. Roll it 60 times, tallying results in ten-roll blocks; compute the running proportion of sixes after each block (after 10, 20, …, 60 rolls) for Table 1, and total face counts (out of 60) for Table 2. Note in your submission that you used the physical-die fallback — the analysis questions work identically (your "500" becomes "60," and the plausible band is wider).

Part 4 — The Scaffold (complete both tables)

Table 1 tracks your running proportion of sixes at five checkpoints, next to the theoretical value.

Checkpoint Your running proportion (col. D) Theoretical value Distance from theoretical (absolute)
After 10 rolls ______ 0.1667 ______
After 50 rolls ______ 0.1667 ______
After 100 rolls ______ 0.1667 ______
After 250 rolls ______ 0.1667 ______
After 500 rolls ______ 0.1667 ______

Table 2 records how often each face appeared across your 500 rolls.

Face Count (=COUNTIF) Relative frequency (count ÷ 500)
1 ______ ______
2 ______ ______
3 ______ ______
4 ______ ______
5 ______ ______
6 ______ ______
Total ______ (should be 500) ______ (should be 1.00)

Part 5 — Analysis Questions

Answer in 1–3 sentences each:

  1. Compare your Table 1 rows: which checkpoint sits closest to 0.1667, and which sits farthest? Describe what your line chart does from left to right, and name the law this behavior demonstrates.
  2. Your 500-roll proportion (D501) is almost certainly not exactly 0.1667. Is your virtual die broken? Using the plausibility note in Part 3, explain how you'd decide — and what result would make you suspicious.
  3. A classmate scrolls their column B, finds a stretch of 30 rolls with no sixes at all, and concludes the next rolls "must be six-heavy to catch up." Using independence and this week's swamping-not-compensating idea, write the two-sentence reply you'd give them.
  4. Look at Table 2: one face "won" your run. Which one, and by how much over the expected count? If you re-roll the whole simulation (undo your paste-values freeze, or rebuild column B), would you bet on the same face winning again? What does this teach about reading meaning into small gaps in random data?
  5. (Connect forward.) Suppose a game paid you $6 every time your die showed a six and nothing otherwise. Using your long-run reasoning, about how much would you expect to collect per roll across many rolls — and which idea from this week justifies treating that per-roll value as meaningful? (Next week makes this exact move official: random variables and expected value.)

Part 6 — AI-Critique Moment (required — the BYOAI step)

Now bring in your chatbot and be the analyst who checks its work.

  1. Paste your completed Table 2 (the face counts) and your final proportion of sixes into your chatbot and ask: "Here are the face counts from my 500 simulated die rolls. Is my die fair? Also, my last 30 rolls contained only 2 sixes — is a six more likely on my next roll?"
  2. Check everything it says against your own work. This week's three named AI failure modes:
    - The gambler's-fallacy nod — agreeing (or half-agreeing, to be agreeable) that a six is "more likely now." It isn't: rolls are independent, and P(six) = 1/6 every time.
    - Noise read as bias — declaring your die "possibly unfair" because one face beat another by a couple dozen counts, when everything sits inside the ordinary 0.13–0.20 band.
    - Arithmetic slips — face counts that don't sum to 500, or a botched count ÷ 500 division. Re-add and re-divide yourself.
  3. Write 2–3 sentences reporting what the AI got right and at least one thing you had to correct, question, or verify. If it happened to get everything right, say how you verified each claim — that's the skill.

The habit all term: the tool drafts, you judge. A chatbot will confidently call a fair die biased — or soothe you that a six is due. Catching it is the point.

Part 7 — What to Submit

One document or text entry containing: your two warm-up values, your completed Table 1 and Table 2, a screenshot or one-sentence description of your line chart, your Part 5 answers (1–5), and your Part 6 AI-critique paragraph. Due at the end of Week 5 · 50 points (rubric below the key).

Instructor answer key & model values — REMOVE BEFORE PUBLISHING TO STUDENTS

Every student's 500 rolls are their own, so simulated proportions and face counts are graded on method and interpretation, not on matching any printed value. The theoretical values and plausibility bands below are pre-computed and verified in the week's math-check script; no dataset-wide claim here depends on an unverified number.

Warm-up (fixed — every student should match): 1 ÷ 6 = 0.1667 (four decimal places; 0.16667 to five — accept 0.1666 only if they truncated and say so). Expected sixes in 500 rolls: 500 × 1/6 = 83.33… → 83 (accept "about 83").

Simulation plausibility bands (theory, verified in script): the standard error of a sample proportion at p = 1/6, n = 500 is √((1/6)(5/6)/500) ≈ 0.0167, so ~95% of honest runs land within 0.1667 ± 0.033 → 0.133 to 0.200 (≈ 67–100 sixes). The same band applies per face. For the first 10 rolls, 0–5 sixes is ordinary; note for the debrief: P(zero sixes in 10 rolls) = (5/6)¹⁰ ≈ 0.16, so roughly 1 student in 6 will see a six-free first ten — call that moment out in class; it's the law of large numbers' opening argument.

Table 1 grading: all five checkpoint cells filled from the correct cells (D11, D51, D101, D251, D501) with distances computed correctly from their own numbers (re-do one subtraction per submission — this is the arithmetic actually graded). The distance column does not need to shrink monotonically — real runs wobble — but the 500-roll distance should almost always be the smallest or near it; a student whose 500-roll proportion sits outside ~0.11–0.22 most likely broke the formula range (common: COUNTIF over B2:B501 unanchored) — check their formulas rather than their luck.

Table 2 grading: six counts summing to exactly 500 (this is the self-check that catches range errors) and relative frequencies = count ÷ 500, summing to 1.00. Each relative frequency typically lands in ~0.13–0.20; flag only formula errors, never luck.

Part 5 expected answers:
1. Farthest is (almost always) the 10-roll checkpoint, closest the 500-roll one; the chart thrashes on the left and flattens toward ≈ 0.1667 on the right; the law of large numbers.
2. Not broken — chance variation. A fair 500-roll run almost always lands in ≈ 0.13–0.20; suspicion is warranted only well outside that band (e.g., 0.05 or 0.30 — and the honest next step would be more rolls, not a verdict).
3. Any two sentences hitting both halves: rolls are independent — the die has no memory, so the next roll is still P(six) = 1/6 — and the long run repairs proportions by swamping early gaps under thousands of ordinary rolls, not by compensating with extra sixes.
4. Full credit names their winning face, compares to ≈ 83, recognizes a re-roll would likely crown a different face, and lands the lesson: small gaps in random data carry no meaning — don't read a story into noise. (This is the lab's error-source/limitation question: the "pattern" is sampling noise, not signal.)
5. About $1 per roll ($6 × 1/6 = $1): in the long run a six arrives on ~1/6 of rolls, so the average payout per roll settles at $1 — justified by probability as long-run relative frequency (and it previews Week 6's expected value).

AI-critique full credit: names a specific claim checked or corrected — most commonly the chatbot hedging toward "due" (gambler's fallacy), over-reading a face gap as bias, or a count/division slip — OR documents verification of each claim (counts re-added to 500, band checked, independence stated). Zero credit for "the AI agreed with everything" with no evidence of checking.

Grading rubric — 50 points

The rubric below maps each part to its points; bands are full / partial / none.

Criterion Full Partial None
Warm-up + build — 0.1667 and 83 correct; columns A–D constructed with the specified formulas (8) 8 4–6 0–3
Simulation + checkpoint table — run frozen; Table 1 filled from the right cells; distances correct from their own values (12) 12 6–10 0–5
Chart + face tally — line chart present and read correctly; Table 2 counts sum to 500 with correct relative frequencies (10) 10 5–8 0–4
Analysis questions — Q1–Q5 hit the expected ideas, especially law of large numbers (Q1), chance variation (Q2), and independence/swamping (Q3) (15) 15 8–12 0–7
AI-critique — specific catch or verification reported in 2–3 sentences (5) 5 3 0–2

Quality gate (self-checked): warm-up arithmetic re-verified (1/6 = 0.1667 to 4 dp; 500/6 → 83); every theoretical value and band in this key (0.0167 SE; 0.133–0.200; 67–100; (5/6)¹⁰ ≈ 0.16; $1 per roll) re-computed in the week's math script; no assertion about any student's random results beyond method and plausibility bands; simulation uses =RANDBETWEEN per the course's simulation-lab convention with expected ranges, not exact values; no graded-item leakage (the quiz, practice, assignment, and chapter self-checks use different surfaces and numbers); optional background link verified live; rubric sums 8 + 12 + 10 + 15 + 5 = 50. ✓