Week 10 Data Lab: Sampling Penguin Body Masses

Part 1 — The Big Picture

This week you learned that a sample mean changes from sample to sample, that its sampling distribution has mean μ and standard deviation σ ⁄ √n, and that the Central Limit Theorem makes averages of large samples approximately normal even when the population is skewed. In this lab you test those claims on real data. You treat the 342 measured penguin body masses as a whole population, find its mean and standard deviation, then draw 100 random samples of 30 penguins and of 5 penguins and average each one. You compare the spread of your sample means with σ ⁄ √n, compare the shapes of one sample and of many sample means, and audit a chatbot's answers about the same data.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: If you draw many random samples of penguins from the field log and average their body masses, do the sample means center on the population mean, spread about as much as σ ⁄ √n predicts, and pile into a bell shape even though the body masses themselves are skewed?

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 Google Sheets, or in Excel use Data, then From Web, and paste the same address. Check that body_mass_g sits in column F. Tool: Google Sheets (free) or Excel. Check: Row 1 shows the eight column names, and the penguins fill rows 2 through 345; two rows show NA for body mass.
2. In empty cells type =COUNT(F2:F345) to count the measured body masses, =AVERAGE(F2:F345) for the population mean μ, =MEDIAN(F2:F345) for the median, and =STDEV.P(F2:F345) for the population standard deviation σ, which divides by N. Record all four in the first table. Tool: Google Sheets (free) or Excel. Check: The count is 342, because the two NA rows are skipped; the mean sits above the median.
3. Make a second tab named Samples. In cell A2 type =INDEX(Penguins!$F$2:$F$345, RANDBETWEEN(1,344)) to pick one penguin's mass at random. If a cell shows NA, that pick landed on an unmeasured penguin; wrap the formula as =IFERROR(1*INDEX(Penguins!$F$2:$F$345, RANDBETWEEN(1,344)), "") so it shows a blank instead. Fill the formula across to column AD and down to row 101, so each of the 100 rows holds one random sample of 30 picks, made with replacement. Tool: Google Sheets (free) or Excel. Check: A block of 100 rows by 30 columns of body masses, with an occasional blank where a pick landed on an unmeasured penguin.
4. In cell AF2 type =AVERAGE(A2:AD2) for the mean of the first sample of 30, and in AG2 type =AVERAGE(A2:E2) for the mean of its first 5 picks. Fill both down to row 101. Then copy columns AF and AG and paste them back as values only, so the sample means stop changing every time the sheet recalculates. Tool: Google Sheets (free) or Excel. Check: Two columns of 100 sample means each: one for samples of 30 and one for samples of 5.
5. Below each column of sample means, type =AVERAGE( ) and =STDEV.S( ) over its 100 values to get the mean and the standard deviation of your sample means. In the third table, compare them with μ and with σ ⁄ √n for n = 30 and for n = 5, computing σ ⁄ √n from your own σ with =SQRT( ). Tool: Google Sheets (free) or Excel. Check: The mean of each column lands near μ, and the samples of 30 spread much less than single penguins do.
6. Count how many of your 100 sample means of 30 fall within two standard errors of μ: type =COUNTIFS(AF2:AF101, ">=" & (μ − 2 × σ ⁄ √30), AF2:AF101, "<=" & (μ + 2 × σ ⁄ √30)), replacing the expressions with your own cell references. Record the count in the fourth table. Tool: Google Sheets (free) or Excel. Check: Most, but not all, of the 100 sample means fall inside the two-standard-error band.
7. Make three histograms with Insert, then Chart, and choose Histogram: one of the 30 masses in a single sample row, one of the 100 sample means of 5, and one of the 100 sample means of 30. Give each chart a title and labelled axes. Tool: Google Sheets (free) or Excel. Check: The single sample leans right like the population, the means of 5 still lean a little, and the means of 30 form a rough bell.

Fallback route

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

Build a small skewed population by hand and sample from it. Roll two dice, or use two slips drawn with replacement from six slips numbered 1 to 6, and record the smaller of the two faces; the model data below gives the chance of each smaller face and the population's μ and σ. One sample is 10 such records. Draw 20 samples of 10, average each one, and also average the first 2 records of each sample. Fill the same four tables using μ and σ from the model data in place of the penguin values, with n = 10 in place of 30 and n = 2 in place of 5, and sketch the three histograms on paper. 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.

This model population of the smaller face of two dice gives you a skewed population to sample from without the penguin file.

Fallback model data

Smaller face of two dice	Chance (out of 36)
1	11
2	9
3	7
4	5
5	3
6	1
Population mean μ	91 ⁄ 36, about 2.53
Population standard deviation σ	about 1.40
Your predicted standard error for n = 10

Part 4 — The Scaffold

Summarize the population of measured body masses so you know what the sample means should center on and how much single penguins vary.

Student scaffold table 1

Quantity	Your value
Measured body masses, N
Population mean μ (grams, two decimals)
Median (grams)
Population standard deviation σ, dividing by N (grams, two decimals)

Record a few of your sample means so a reader can see the wobble from one sample to the next.

Student scaffold table 2

Sample number	Mean of 5 picks (grams)	Mean of 30 picks (grams)
1
2
3
4
5

Compare the center and spread of your sample means with what the sampling distribution of x̄ predicts.

Student scaffold table 3

Quantity	Samples of 5	Samples of 30
Mean of the 100 sample means (grams)
Standard deviation of the 100 sample means (grams)
Predicted standard error σ ⁄ √n (grams)
Ratio: your standard deviation ÷ the predicted standard error

Check how many sample means of 30 land within two standard errors of the population mean, the range the empirical rule says should hold about 95% of them.

Student scaffold table 4

Quantity	Your value
Lower end, μ − 2 × σ ⁄ √30 (grams)
Upper end, μ + 2 × σ ⁄ √30 (grams)
Sample means of 30 inside that range, out of 100

Part 5 — Analysis Questions

1. Using your third table, how close is the mean of your 100 sample means to μ for samples of 5 and for samples of 30, and how close is each standard deviation of the sample means to its predicted standard error σ ⁄ √n?
2. Describe your three histograms in words: where the peak sits, which way any long tail runs, and how wide each one is. Name the shape of each, naming any skew by its tail, and say which one looks most like a bell.
3. Explain why the standard deviation of your sample means of 30 is so much smaller than σ, the standard deviation of single penguins. Use the words standard deviation and standard error, and say which one describes one penguin and which describes an average of 30.
4. Using your fourth table, how many of your 100 sample means of 30 fell within two standard errors of μ, and how does that compare with the empirical rule's about 95%? Then use the Central Limit Theorem and the course's rule of thumb to explain why a normal model is reasonable for the means of 30 but less so for the means of 5.
5. Choose one reflection and answer it in 3–5 sentences. (a) This lab treats 342 penguins as the whole population, but in the field study they are themselves a sample from much larger colonies: what can your sample means tell you, and what can they not tell you, about penguins that were never weighed? (b) Your 100 samples were drawn with replacement from three species mixed together: how might that choice, or running only 100 samples, change what your histograms and standard deviations show?

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 10 Data Lab tables above. In the lab I treated the 342 measured body masses in the Palmer penguins file as a whole population, with mean 4,201.75 g and median 4,050 g, and I drew 100 random samples of 30 penguins each, with replacement, and averaged each sample. Please do five things. First, describe the shape of the 342 body masses and name the skew. Second, tell me what the standard deviation of my 100 sample means should be close to, and show how you got it. Third, the Gentoo penguins in the file are much heavier on average than the Adelie penguins, and every Gentoo was logged on Biscoe island: does living on Biscoe make a penguin heavy? Fourth, if I pick 3 penguins at random from the file, with replacement, what is the chance that exactly 2 of them are Gentoo, given that 124 of the 344 penguins are Gentoo? Fifth, as a stretch question for later in the course, about 95% of my sample means landed within 2 standard errors of the population mean: does that mean there is a 95% chance that the true mean is inside the interval from one sample?

Test the response for these likely failures:

• Chatbots often call the body masses left-skewed, or call them normal, because most penguins sit in a central pile, naming the shape by the pile instead of the tail.
• Chatbots sometimes report the standard deviation of single masses, about 800 g, as the spread of the sample means, or quote simulation results as if they had seen your sheet.
• Chatbots sometimes slide from the file to a cause, saying Biscoe's food or climate makes penguins heavy because the heavy Gentoos were all logged there.
• Chatbots often multiply the chance of one arrangement and stop, reporting (124 ⁄ 344)² × (220 ⁄ 344), about 0.0831, as the chance that exactly 2 of 3 picks are Gentoo.
• On the stretch question, chatbots often say that one sample's interval has a 95% chance of holding the true mean.

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 histograms, or the population values you computed from the file.

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: "The body masses are left-skewed, because most penguins sit in the middle. The sample means of 30 should have a standard deviation of about 800 g, the same as single penguins, and my simulation shows exactly 801.95 g. Gentoos are heavier and all live on Biscoe, so Biscoe makes penguins heavy. The chance that exactly 2 of 3 random picks are Gentoo is (124 ⁄ 344)² × (220 ⁄ 344), about 0.0831. About 95% of my sample means were within 2 standard errors, so there is a 95% chance the true mean is inside any one sample's interval." Mark it up: quote each flawed sentence, name what is wrong (a skew named by the pile, a spread that ignores σ ⁄ √n or an invented simulation result, cause read from association, a dropped number of ways, or a misread confidence statement), cite the evidence from your tables, histograms, or the population values, and write a corrected sentence beneath it; for the last sentence, say what the course will check next week. This route assesses the same objectives (Objective 1 and Objective 5) 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 four completed scaffold tables
• Your three histograms, or labelled sketches of them
• 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 sampling penguin body masses: four completed tables (the population summary; a few of your sample means; the center and spread of your 100 sample means of 5 and of 30 beside σ ⁄ √n; and the count of sample means of 30 inside two standard errors), three histograms or labelled sketches of them, 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
Summarizing the population from the real file	The count of measured masses, μ, the median, and σ come from the file or the fallback model data, with σ dividing by N and the values reported to two decimals.	The summaries are present, but one is missing, σ divides by the wrong count, or a value is rounded too early.	No population summary is recorded, or the numbers are invented rather than produced.
Drawing samples and recording sample means	100 random samples of 30, or the fallback samples, are drawn and averaged; the means of the first 5 picks are recorded too; the sample means are frozen as values; and the second table shows sample means that differ from one another.	Samples are drawn, but one column of means is missing, the means were not frozen, or fewer samples were used without saying so.	No sample means are recorded, or the means are invented rather than produced.
Comparing the sample means with the sampling distribution	The third and fourth tables compare the mean and standard deviation of the sample means with μ and σ ⁄ √n for both sample sizes and count the means inside two standard errors, and analysis questions 1 and 4 interpret the match and the gaps.	The comparison is made for only one sample size, the predicted standard error is computed with the wrong spread, or the count inside two standard errors is not interpreted.	No comparison with μ or σ ⁄ √n appears.
Reasoning about shape, the standard error, and limits	Analysis questions 2, 3, and 5 are answered accurately: the histograms are described with any skew named by its tail, the standard deviation of single masses is kept apart from the standard error of a mean, the normal model is applied to the sample means rather than the data, and the chosen reflection option names a specific limit.	The questions are attempted, but one or two answers name skew by the pile, mix up the standard deviation and the standard error, say the data become normal, or give a general reflection.	The reasoning questions are missing or rest on the misconceptions listed above throughout.
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, histograms, or the population 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.

Quality Gate

• Every count, mean, and standard deviation in the tables comes from your own formulas, your own samples, or the fallback model data, never from memory or a chatbot.
• σ is computed from the measured body masses dividing by N, and each predicted standard error is σ divided by the square root of that sample size.
• Your sample means are frozen as values before you summarize them, and each column holds 100 sample means.
• Each histogram has a title and labelled axes, and any skew is named by its tail.
• The verdict about a normal model is about the sampling distribution of the sample mean, never a claim that the body masses themselves 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, histograms, or the population values.