Week 9 Data Lab: Does a Bell Fit the Penguins?

Part 1 — The Big Picture

This week you learned that a normal model is an assumption you check before you lean on it. In this lab you check it on real data. You open the penguin field log, pull out the body masses of the Adelie penguins, and run the week's checks yourself: the mean against the median, the empirical-rule shares within one, two, and three standard deviations, and a normal probability plot. Then you run the same checks on all 342 measured penguins together, use a normal model to predict a share and compare the prediction with the data, and decide where a bell is reasonable and where it is not. Last, you check a chatbot's answers about the same data.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: Is a normal model reasonable for the body masses of the Adelie penguins, and is it still reasonable when all three species are pooled together?

Workshop data: palmer-station-penguins

Get the data: palmer-station-penguins landing page or the direct data file — free, no login required.

Source: Measurements of 344 penguins of three species on three islands of Antarctica's Palmer Archipelago, recorded by field researchers and republished in the palmerpenguins package.

• Rows: 344 penguins
• Columns: species, island, bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g, sex, year
• Missing values: Numeric measurement columns and sex carry missing values for a small number of penguins

Part 3 — Materials & Procedure

Tools and materials

• Google Sheets (free) or Excel
• The palmerpenguins CSV file
• A document or notebook for your written answers

Ways of working

• interpret-real-data
• compute-and-plot
• structured-observation

Procedure

1. Open a blank sheet and name the first tab Penguins. In cell A1 type =IMPORTDATA("https://raw.githubusercontent.com/allisonhorst/palmerpenguins/main/inst/extdata/penguins.csv"). In Excel, save the file from that address, then use Data, then From Text/CSV, and load it starting in cell A1. Tool: Google Sheets (free) or Excel. Check: Row 1 shows the eight column names, the penguins fill rows 2 through 345, and body mass sits in column F.
2. Add a tab named Adelie. In cell A1 type =FILTER(Penguins!F2:F345, Penguins!A2:A345="Adelie", Penguins!F2:F345<>"NA") to list every Adelie body mass that was recorded. In C1 type =COUNT(A1:A200) to count them. Tool: Google Sheets (free) or Excel. Check: Column A fills with whole numbers of grams, and C1 shows 151, the number of Adelie penguins with a recorded body mass.
3. Fill the first scaffold table. For the Adelie column use =AVERAGE(A1:A151), =STDEV(A1:A151), and =MEDIAN(A1:A151) on the Adelie tab. For the all-penguins column use =AVERAGE(Penguins!F2:F345), =STDEV(Penguins!F2:F345), and =MEDIAN(Penguins!F2:F345); the text NA cells are skipped. Round each to two decimals only when you copy it into the table. Tool: Google Sheets (free) or Excel. Check: The Adelie mean and median are close together, while the all-penguins mean sits more clearly above its median.
4. Fill the second scaffold table. Put the Adelie mean in E1 and standard deviation in E2. For one standard deviation, the bounds are =E1-E2 and =E1+E2, and =COUNTIFS(A1:A151,">="&(E1-E2),A1:A151,"<="&(E1+E2)) counts the masses inside them; use 2*E2 and 3*E2 for two and three standard deviations. Divide each count by 151 for the share. Repeat on the Penguins tab with the all-penguins mean and standard deviation and the range F2:F345, dividing by 342. Tool: Google Sheets (free) or Excel. Check: Each count is a whole number no larger than its group size, and each share grows from the one-standard-deviation row to the three-standard-deviation row.
5. Build a normal probability plot of the Adelie masses. In G1 type =SORT(A1:A151) to list the masses from lightest to heaviest. In H1 type =NORM.S.INV((ROW()-0.5)/151) and fill it down to H151. Select both columns and use Insert, then Chart, and choose Scatter chart, with column H across the bottom and column G up the side. Save or sketch the chart. Tool: Google Sheets (free) or Excel. Check: The chart shows 151 dots rising from lower left to upper right; you judge whether they follow a straight line or bend steadily away from one.
6. Fill the third scaffold table. Treat the Adelie masses as following a normal model with the mean and standard deviation from your first table. In an empty cell type =NORM.DIST(4000, E1, E2, TRUE) for the share the model predicts below 4,000 g, and =COUNTIF(A1:A151,"<4000")/151 for the share the data actually show. Tool: Google Sheets (free) or Excel. Check: Both shares are decimals between 0 and 1, and you can compare the model's prediction with the real count side by side.

Fallback route

When to use it: Use this route if the penguin file will not import or no spreadsheet is available.

Work from the model data below, which reports summaries and counts from the real file. Copy them into the first two scaffold tables, compute every share by hand, and work the third table with a printed standard normal table: standardize 4,000 g with the Adelie mean and standard deviation, round z to two decimals, and read the area to its left. For the normal probability plot, describe in words what the mean-versus-median and empirical-rule checks predict the plot would show. Answer every analysis question and the critique step from these records, and submit the same items for equal credit.

This route earns the same credit as the primary route.

These summaries and counts from the real file give you every number the lab needs without importing it.

Fallback model data

Quantity from the file	Adelie penguins	All measured penguins
Number with a recorded body mass	151	342
Mean (g)	3,700.66	4,201.75
Standard deviation (g)	458.57	801.95
Median (g)	3,700	4,050
Count within 1 standard deviation	99	231
Count within 2 standard deviations	146	333
Count within 3 standard deviations	151	342
Adelie masses below 4,000 g	112
Share within 1 standard deviation (percent)

Part 4 — The Scaffold

Summarize each group so you can compare its mean with its median before you trust a bell.

Student scaffold table 1

Summary	Adelie penguins	All measured penguins
Number with a recorded body mass
Mean, x̄ (g)
Standard deviation, s (g)
Median (g)

Run the empirical-rule check: count the masses within one, two, and three standard deviations and compare each share with the rule.

Student scaffold table 2

Group and range	Lower bound (g)	Upper bound (g)	Count inside	Share inside (percent, one decimal)	Empirical rule
Adelie, within 1 standard deviation					about 68%
Adelie, within 2 standard deviations					about 95%
Adelie, within 3 standard deviations					about 99.7%
All penguins, within 1 standard deviation					about 68%
All penguins, within 2 standard deviations					about 95%
All penguins, within 3 standard deviations					about 99.7%

Put a normal model's prediction beside what the Adelie data actually show, to test the model on one share.

Student scaffold table 3

Share of Adelie penguins below 4,000 g	Value (four decimals)
Predicted by the normal model, NORM.DIST
Counted in the data, COUNTIF divided by 151

Part 5 — Analysis Questions

1. Using your second table, how far is each Adelie share from 68%, 95%, and 99.7%? Do the same for all 342 penguins. Which group comes closer to the empirical rule?
2. Describe your normal probability plot of the Adelie masses in words: do the points follow a straight line with small wiggles, or bend steadily away from it at one or both ends? What does that suggest about the shape of the Adelie masses?
3. From your first table, compare the mean with the median for the Adelie penguins and for all 342 penguins. For each group, give a verdict, reasonable or not reasonable, for using a normal model, and explain it with at least two checks.
4. From your third table, how close is the normal model's predicted share below 4,000 g to the share you counted? What does the size of the gap tell you about using the Adelie normal model for questions like this one, and what z-score does 4,000 g have under that model?
5. Choose one and answer it in 3–4 sentences. (a) All 342 penguins can come close to the empirical-rule shares and still not suit a normal model: explain how that can happen and what checks catch it. (b) Your Adelie verdict comes from 151 penguins logged on three islands over three seasons: what can and cannot it tell you about Adelie penguins that were never weighed?

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I've pasted my completed Week 9 Data Lab tables above. In the lab I used the Palmer penguins file: 151 Adelie penguins have a recorded body mass, and so do 342 penguins of all three species. Please do five things. First, tell me what share of the 151 Adelie body masses lie within one standard deviation of their mean, and whether that share proves the masses are normally distributed. Second, describe the shape of the body masses of all 342 penguins and name any skew. Third, say whether living on a particular island makes Adelie penguins heavier. Fourth, suppose each of two Adelie penguins picked at random independently has a 0.25 chance of weighing more than 4,000 g; give the chance that exactly one of the two does. Fifth, as a stretch question for later in the course, a 95% confidence interval for the mean Adelie body mass runs from 3,626.92 g to 3,774.40 g; does that mean 95% of Adelie penguins weigh between those two values?

Test the response for these likely failures:

• Chatbots often state a share they never counted, such as exactly 68% of the Adelie masses within one standard deviation, and then call the masses normal because the share looks right.
• Chatbots sometimes call the 342 body masses left-skewed because most penguins are on the lighter side, naming the skew by where the pile is.
• Chatbots often slide from a difference in average mass between islands to a cause, saying conditions on one island make Adelie penguins heavier.
• Chatbots often compute 0.25 × 0.75 = 0.1875 for exactly one heavy penguin and stop, forgetting that the heavy one could be the first pick or the second.
• On the stretch question, chatbots often say that 95% of Adelie penguins weigh between 3,626.92 g and 3,774.40 g.

Write 3–5 sentences naming one thing the chatbot got right and at least two places it went wrong or overreached, quoting its words and correcting each with evidence from your own tables, your normal probability plot, or the course's pinned values.

Equivalent route (no AI required)

No AI account is required for this lab. Instead of the chatbot, critique this flawed analysis, printed here in full: "Exactly 68% of the Adelie body masses lie within one standard deviation of the mean, so the masses are normally distributed. The body masses of all 342 penguins are left-skewed, because most penguins are on the light side. Adelie penguins on one island are heavier because that island's food makes them grow. If each of two Adelie penguins has a 0.25 chance of weighing over 4,000 g, the chance that exactly one does is 0.25 × 0.75 = 0.1875. The 95% confidence interval from 3,626.92 g to 3,774.40 g means 95% of Adelie penguins weigh between those values." Mark it up: quote each flawed sentence, name what is wrong (an uncounted or overclaimed statistic, a wrong skew claim, cause read from association, a dropped number of ways, or a misread confidence statement), cite the evidence from your tables, your plot, or the course's pinned values, and write a corrected sentence beneath it; for the last sentence, say what the course will check later. This route assesses the same objective (Objective 5 — use the normal model and sampling distributions, including the Central Limit Theorem, to reason about variability in individual values and in sample statistics) as the chatbot critique, takes comparable effort (about 30 minutes and 200–300 words), is scored by the same scoring criteria for the same credit, and its own submission artifact is the marked-up flawed analysis, posted in place of the chatbot share link and report.

Part 7 — What to Submit

• All three completed scaffold tables
• Your normal probability plot of the Adelie masses, or a labelled sketch or written description of it
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Data Lab report on whether a bell fits the penguins: three completed tables (summaries of the Adelie and all-penguin body masses; the empirical-rule check within one, two, and three standard deviations for both groups; and a normal model's predicted share below 4,000 g beside the counted share), a normal probability plot of the Adelie masses or a labelled sketch or description of it, written answers to five analysis questions, and a critique of a chatbot's answers about the same data, or, on the equivalent route, a marked-up correction of the printed flawed analysis. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Importing and summarizing the body masses	The Adelie and all-penguin body masses are summarized with the right group sizes, means, standard deviations, and medians, from the file or the fallback model data.	The summaries are present, but one or two values are missing, use the wrong group, or include the missing NA entries.	No summaries are recorded, or the values are invented rather than computed.
Running the empirical-rule check	All six rows of the empirical-rule table have bounds built from the right mean and standard deviation, counts, and shares divided by the right group size, and analysis question 1 compares each share with 68%, 95%, and 99.7%.	The table is mostly complete, but some bounds, counts, or shares are missing or wrong, or the comparison with the rule is left out.	The empirical-rule table is missing or not built from the data.
Reading the normal probability plot	A normal probability plot of the Adelie masses, or a labelled sketch or description of it, has both axes labelled, and analysis question 2 says whether the points follow a straight line or bend and what that suggests about shape.	The plot or description is present, but an axis is unlabelled or the reading of straight versus bending is missing.	No plot, sketch, or description of the normal probability plot is submitted.
Verdicts and the model-versus-data test	Analysis questions 3 and 4 give a reasonable-or-not-reasonable verdict for each group backed by at least two checks, compare the model's predicted share below 4,000 g with the counted share, and state the z-score for 4,000 g; no answer calls the data normal.	The verdicts or the comparison are attempted, but a verdict rests on one check, calls the data normal, or the z-score is missing.	Analysis questions 3 and 4 are missing or do not use the student's own tables.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors are quoted and corrected with evidence from the student's own tables, plot, or the course's pinned values, and one correct point is acknowledged; both routes earn this band equally.	On either route, errors are named but corrected without evidence, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.
Reflection on what a normal model can and cannot show	The chosen reflection option is answered in 3–4 sentences that name a specific limit, such as pooled groups passing one check while still skewed, or 151 logged penguins not speaking for unweighed ones.	The reflection is answered but stays general or does not name a specific limit.	No reflection is submitted.

Quality Gate

• Every count, summary, and share in the tables comes from your own formulas or the fallback model data, never from memory or a chatbot.
• Each share is its count divided by the group size, 151 for the Adelie penguins or 342 for all penguins.
• Each bound is the mean minus or plus one, two, or three standard deviations of the same group.
• The normal probability plot, or its description, has the masses on one axis and the normal z-scores on the other, both labelled.
• Every verdict is reasonable or not reasonable, backed by at least two checks, and never says the data are normal.
• No statistic, count, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence from your tables, your plot, or the course's pinned values.