Lab 16: Sampling Properties & Assessing Normality

BI 255 · Bethune-Cookman University · Fall 2026

Author

Dr. Rosie Stanbrook-Buyer

Published

September 30, 2026

Does Your Data Follow a Normal Distribution?

Most parametric statistical tests (t-tests, ANOVA, regression) assume that the data — or at minimum the residuals from a model — follow a normal distribution. In practice, biological data often deviate from normality: assay measurements can be right-skewed, count data are discrete, and small samples show irregular patterns.

This lab gives you the tools to formally assess normality, understand what happens when it fails, and choose appropriate transformations or alternatives.

Learning Objectives

By the end of this lab you will be able to:

Explain sampling variation and the role of the standard error
Construct and interpret a QQ plot with qqnorm() and qqline()
Apply the Shapiro-Wilk test with shapiro.test()
Recognise common patterns of non-normality: skewness, heavy tails, bimodality
Apply a log transformation to right-skewed biological data
Know when parametric tests remain valid despite non-normality

Part 1: Sampling Variation and Standard Error

When you measure a quantity from a sample, the sample mean is an estimate of the true population mean. Different samples from the same population will give different estimates. The standard error (SE) quantifies this variability.

set.seed(42)

# Simulate a population: serum CRP (C-reactive protein, mg/L) in healthy adults
# CRP is right-skewed — use log-normal approximation
crp_population <- rlnorm(100000, meanlog = 0.8, sdlog = 0.6)

cat("Population mean CRP:   ", round(mean(crp_population), 2), "mg/L\n")

Population mean CRP:    2.66 mg/L

cat("Population median CRP: ", round(median(crp_population), 2), "mg/L\n")

Population median CRP:  2.22 mg/L

cat("Population SD CRP:     ", round(sd(crp_population),   2), "mg/L\n")

Population SD CRP:      1.75 mg/L

# Draw 10 samples of n = 25 and show how means vary
sample_means <- replicate(10, mean(sample(crp_population, 25)))
round(sample_means, 2)

 [1] 2.29 2.54 3.37 2.58 2.53 2.90 3.75 2.60 2.85 2.76

cat("Mean of sample means:", round(mean(sample_means), 2), "\n")

Mean of sample means: 2.82

cat("SD of sample means (empirical SE):", round(sd(sample_means), 2), "\n")

SD of sample means (empirical SE): 0.44

cat("Theoretical SE (sigma/sqrt(n)):   ",
    round(sd(crp_population) / sqrt(25), 2), "\n")

Theoretical SE (sigma/sqrt(n)):    0.35

Three Different Quantities

Quantity	Formula	What it measures
Standard deviation (SD)	√[Σ(x−x̄)²/(n−1)]	Spread of individual observations
Standard error (SE)	SD / √n	Precision of the sample mean
95% confidence interval	x̄ ± 1.96 × SE	Plausible range for the true mean

A common error in published papers: using SE to make error bars look narrower than they should be. Always state whether your error bars represent SD or SE.

Part 2: QQ Plots — Visualising Normality

A quantile-quantile (QQ) plot compares the quantiles of your data to what those quantiles would be under a normal distribution. If the data are normal, the points fall on a straight line.

# Generate three datasets with different distributional shapes
set.seed(123)
normal_data    <- rnorm(80, mean = 50, sd = 10)
right_skewed   <- rexp(80, rate = 0.05)          # right-skewed (exponential)
heavy_tails    <- rt(80, df = 3)                 # heavy-tailed (t with low df)

par(mfrow = c(2, 3), mar = c(4, 4, 3, 1))

# Histograms
hist(normal_data,  col = "#2C5F8A", border = "white", main = "Normal", xlab = "")
hist(right_skewed, col = "#5B8DB8", border = "white", main = "Right-skewed", xlab = "")
hist(heavy_tails,  col = "#A8C4D9", border = "white", main = "Heavy tails", xlab = "")

# QQ plots
qqnorm(normal_data,  main = "QQ — Normal",       pch = 16, cex = 0.6, col = "#2C5F8A")
qqline(normal_data,  col = "#C8102E", lwd = 2)

qqnorm(right_skewed, main = "QQ — Right-skewed", pch = 16, cex = 0.6, col = "#5B8DB8")
qqline(right_skewed, col = "#C8102E", lwd = 2)

qqnorm(heavy_tails,  main = "QQ — Heavy tails",  pch = 16, cex = 0.6, col = "#A8C4D9")
qqline(heavy_tails,  col = "#C8102E", lwd = 2)

par(mfrow = c(1, 1))

Reading a QQ Plot

Pattern	What it means
Points on the line	Data are approximately normal
Curve bending upward (S-shape, top-right)	Right-skewed distribution
Curve bending downward (S-shape, top-left)	Left-skewed distribution
Points fan out at both ends	Heavy tails (more extreme values than normal)
Points cluster at the ends	Light tails

The QQ plot is subjective — you are looking for systematic deviations from the line, not random scatter around it.

Part 3: The Shapiro-Wilk Test

For a formal test of normality, use the Shapiro-Wilk test. It tests H₀: the data come from a normal distribution.

# Apply Shapiro-Wilk to our three datasets
shapiro.test(normal_data)


    Shapiro-Wilk normality test

data:  normal_data
W = 0.99471, p-value = 0.9867

shapiro.test(right_skewed)


    Shapiro-Wilk normality test

data:  right_skewed
W = 0.85333, p-value = 2.097e-07

shapiro.test(heavy_tails)


    Shapiro-Wilk normality test

data:  heavy_tails
W = 0.50398, p-value = 5.393e-15

Limitations of the Shapiro-Wilk Test

The Shapiro-Wilk test has two important limitations:

Large samples always reject H₀ — with n > 200, even tiny, biologically irrelevant deviations from normality produce significant p-values. The test is too sensitive for large datasets.
Small samples rarely reject H₀ — with n < 20, the test has low power and will fail to detect even moderately non-normal data.

Best practice: always look at the QQ plot as well as the test. The combination of visual assessment and formal test gives a more complete picture than either alone.

Biological Data: Cytokine Concentrations

Cytokines (immune signalling proteins) are typically measured at very low concentrations with a few extremely high values — a classic right-skewed distribution seen in immunology research.

set.seed(77)
# Simulated IL-6 concentrations (pg/mL) in patients with varying inflammation
il6 <- rlnorm(50, meanlog = 2.1, sdlog = 1.0)

par(mfrow = c(1, 2))
hist(il6, col = "#2C5F8A", border = "white",
     xlab = "IL-6 (pg/mL)", main = "Raw IL-6 Concentrations")
qqnorm(il6, pch = 16, cex = 0.7, col = "#2C5F8A", main = "QQ Plot — Raw IL-6")
qqline(il6, col = "#C8102E", lwd = 2)

par(mfrow = c(1, 1))

shapiro.test(il6)


    Shapiro-Wilk normality test

data:  il6
W = 0.77044, p-value = 1.949e-07

Part 4: Log Transformation

When biological data are right-skewed (as is common for concentrations, enzyme activities, and species abundances), a log transformation often produces a more symmetric, approximately normal distribution.

log_il6 <- log(il6)   # Natural logarithm; use log10() for base-10

par(mfrow = c(1, 2))
hist(log_il6, col = "#5B8DB8", border = "white",
     xlab = "log(IL-6) (pg/mL)", main = "Log-Transformed IL-6")
qqnorm(log_il6, pch = 16, cex = 0.7, col = "#5B8DB8",
       main = "QQ Plot — log(IL-6)")
qqline(log_il6, col = "#C8102E", lwd = 2)

par(mfrow = c(1, 1))

shapiro.test(log_il6)


    Shapiro-Wilk normality test

data:  log_il6
W = 0.97406, p-value = 0.3357

Why Log-Transform?

Many biological quantities follow log-normal distributions — i.e., their logarithms follow a normal distribution. This is common for:

Protein and cytokine concentrations
Bacterial colony counts
Gene expression values (RNA-seq log2 CPM)
Pharmacokinetic parameters (Cmax, AUC)
Species abundance data

After log transformation, you can apply parametric tests normally and then back-transform results to the original scale for reporting.

Back-transform with exp() if you used log(), or 10^x if you used log10().

Part 5: When Can You Ignore Non-Normality?

# Demonstrate: t-test is robust to moderate non-normality due to the CLT
set.seed(500)

# Two right-skewed groups (enzyme activity under two growth conditions)
group_A <- rexp(40, rate = 0.1)   # Mean = 10
group_B <- rexp(40, rate = 0.08)  # Mean = 12.5

par(mfrow = c(1, 2))
hist(group_A, col = "#2C5F8A50", breaks = 15, main = "Group A (skewed)", xlab = "")
hist(group_B, col = "#C8102E50", breaks = 15, main = "Group B (skewed)", xlab = "")

par(mfrow = c(1, 1))

# t-test on non-normal data — still reasonable at n = 40 due to CLT
t.test(group_A, group_B)


    Welch Two Sample t-test

data:  group_A and group_B
t = -1.3029, df = 74.95, p-value = 0.1966
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -7.662497  1.602788
sample estimates:
mean of x mean of y 
 9.916134 12.945989

When Parametric Tests Are Still Valid

Parametric tests (t-tests, ANOVA) require that the sampling distribution of the mean is approximately normal — not necessarily that the raw data are normal. Due to the CLT:

For n ≥ 30, parametric tests are generally robust to skewness
For n < 20, normality is more important to check carefully
For heavily non-normal data at small n: use non-parametric alternatives (Lab 20)

The key insight: it is the distribution of means, not raw values, that needs to be approximately normal.

3-Minute Knowledge Check

Close your notes. Answer these on your own — you have 3 minutes. We’ll go through the answers together after.

Knowledge Check Questions

1. On a QQ plot, you observe that the points form a curve that bends upward to the right (the top-right points fall above the line). This indicates:

a) Left-skewed data    b) Right-skewed data    c) Heavy-tailed data    d) Normal data with outliers

2. You apply shapiro.test() to a dataset of 500 observations and get p = 0.02. What is the most important caution about interpreting this result?

3. A researcher measures serum triglycerides (mg/dL) and finds the data are right-skewed (W = 0.87, p = 0.003). What transformation should they try first?

a) Square root    b) Reciprocal (1/x)    c) Log    d) Squaring (x²)

4. You have two samples of n = 35 from a skewed population. A colleague says you cannot run a t-test because the data are not normal. Are they correct?

5. True or False: The standard error of the mean will always be smaller than the standard deviation of the same sample.

Answers (Only reveal if you have answered the above questions)

1. b) Right-skewed data — In a right-skewed distribution, there are more extreme large values than the normal distribution predicts. On a QQ plot these appear as points above the reference line in the top-right, creating an upward curve. A “banana” shape bending upward = right skew.

2. With n = 500, the Shapiro-Wilk test is so sensitive that it detects trivially small deviations from normality that have no practical consequence for parametric tests. A significant p-value here does not mean parametric tests are inappropriate — always inspect the QQ plot and consider whether the deviation is biologically meaningful.

3. c) Log transformation — The log (or log10) transformation is the standard first attempt for right-skewed biological data such as concentrations, counts, and ratios. It compresses the right tail and often produces an approximately normal distribution.

4. No, they are not correct — with n = 35 per group, the Central Limit Theorem ensures that the sampling distribution of the mean is approximately normal even from a skewed population. The t-test is robust to this degree of non-normality at this sample size. They should still inspect QQ plots and consider a non-parametric alternative if skewness is extreme.

5. True — SE = SD / √n. Since √n > 1 for any n > 1, the SE is always smaller than the SD. The more observations you have, the smaller the SE — reflecting greater precision in estimating the population mean.

Lab 16 Checklist

Before you leave, make sure you can:

Explain the difference between SD, SE, and a confidence interval
Produce a QQ plot with qqnorm() and qqline() and interpret its shape
Run shapiro.test() and state the null hypothesis being tested
Recognise the limitations of the Shapiro-Wilk test at large and small sample sizes
Apply log() to right-skewed data and verify the transformation improved normality
Explain why parametric tests are often valid for n ≥ 30 even with non-normal data

Bonus Challenge

Find a real dataset of your choice (CDC public health data, a published biology paper’s supplementary table, or the datasets package in R: try data(airquality)). Choose a continuous variable, assess its normality with a QQ plot and Shapiro-Wilk test, apply an appropriate transformation if needed, and reassess. Write a paragraph describing your findings and whether a parametric test would be appropriate.

Before Next Class (Quiz Friday, then Week 8)

Friday: Quiz 4 — covers the normal distribution, CLT, sampling properties, and assessing normality (Labs 15–16)
After the quiz, Week 8 covers t-tests and ANOVA — be confident with the CLT and QQ plots before then
Read R for Data Science Ch. 9–10

--- title: "Lab 16: Sampling Properties & Assessing Normality" subtitle: "BI 255 · Bethune-Cookman University · Fall 2026" author: "Dr. Rosie Stanbrook-Buyer" date: "September 30, 2026" format: html: theme: cosmo toc: true toc-location: left toc-title: "In This Lab" toc-depth: 3 number-sections: false code-fold: false code-tools: true highlight-style: github smooth-scroll: true embed-resources: true callout-appearance: default execute: warning: false message: false echo: true --- ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE) ``` ------------------------------------------------------------------------ ## Does Your Data Follow a Normal Distribution? Most parametric statistical tests (t-tests, ANOVA, regression) assume that the data — or at minimum the residuals from a model — follow a normal distribution. In practice, biological data often deviate from normality: assay measurements can be right-skewed, count data are discrete, and small samples show irregular patterns. This lab gives you the tools to formally assess normality, understand what happens when it fails, and choose appropriate transformations or alternatives. ::: callout-note ## Learning Objectives By the end of this lab you will be able to: 1. Explain sampling variation and the role of the standard error 2. Construct and interpret a QQ plot with `qqnorm()` and `qqline()` 3. Apply the Shapiro-Wilk test with `shapiro.test()` 4. Recognise common patterns of non-normality: skewness, heavy tails, bimodality 5. Apply a log transformation to right-skewed biological data 6. Know when parametric tests remain valid despite non-normality ::: ------------------------------------------------------------------------ ## Part 1: Sampling Variation and Standard Error When you measure a quantity from a sample, the sample mean is an **estimate** of the true population mean. Different samples from the same population will give different estimates. The **standard error** (SE) quantifies this variability. ```{r} set.seed(42) # Simulate a population: serum CRP (C-reactive protein, mg/L) in healthy adults # CRP is right-skewed — use log-normal approximation crp_population <- rlnorm(100000, meanlog = 0.8, sdlog = 0.6) cat("Population mean CRP: ", round(mean(crp_population), 2), "mg/L\n") cat("Population median CRP: ", round(median(crp_population), 2), "mg/L\n") cat("Population SD CRP: ", round(sd(crp_population), 2), "mg/L\n") ``` ```{r} # Draw 10 samples of n = 25 and show how means vary sample_means <- replicate(10, mean(sample(crp_population, 25))) round(sample_means, 2) ``` ```{r} cat("Mean of sample means:", round(mean(sample_means), 2), "\n") cat("SD of sample means (empirical SE):", round(sd(sample_means), 2), "\n") cat("Theoretical SE (sigma/sqrt(n)): ", round(sd(crp_population) / sqrt(25), 2), "\n") ``` ::: callout-note ## Three Different Quantities | Quantity | Formula | What it measures | |----------|---------|-----------------| | Standard deviation (SD) | √[Σ(x−x̄)²/(n−1)] | Spread of individual observations | | Standard error (SE) | SD / √n | Precision of the sample mean | | 95% confidence interval | x̄ ± 1.96 × SE | Plausible range for the true mean | A common error in published papers: using SE to make error bars look narrower than they should be. Always state whether your error bars represent SD or SE. ::: ------------------------------------------------------------------------ ## Part 2: QQ Plots — Visualising Normality A **quantile-quantile (QQ) plot** compares the quantiles of your data to what those quantiles would be under a normal distribution. If the data are normal, the points fall on a straight line. ```{r} # Generate three datasets with different distributional shapes set.seed(123) normal_data <- rnorm(80, mean = 50, sd = 10) right_skewed <- rexp(80, rate = 0.05) # right-skewed (exponential) heavy_tails <- rt(80, df = 3) # heavy-tailed (t with low df) par(mfrow = c(2, 3), mar = c(4, 4, 3, 1)) # Histograms hist(normal_data, col = "#2C5F8A", border = "white", main = "Normal", xlab = "") hist(right_skewed, col = "#5B8DB8", border = "white", main = "Right-skewed", xlab = "") hist(heavy_tails, col = "#A8C4D9", border = "white", main = "Heavy tails", xlab = "") # QQ plots qqnorm(normal_data, main = "QQ — Normal", pch = 16, cex = 0.6, col = "#2C5F8A") qqline(normal_data, col = "#C8102E", lwd = 2) qqnorm(right_skewed, main = "QQ — Right-skewed", pch = 16, cex = 0.6, col = "#5B8DB8") qqline(right_skewed, col = "#C8102E", lwd = 2) qqnorm(heavy_tails, main = "QQ — Heavy tails", pch = 16, cex = 0.6, col = "#A8C4D9") qqline(heavy_tails, col = "#C8102E", lwd = 2) par(mfrow = c(1, 1)) ``` ::: callout-important ## Reading a QQ Plot | Pattern | What it means | |---------|--------------| | Points on the line | Data are approximately normal | | Curve bending upward (S-shape, top-right) | Right-skewed distribution | | Curve bending downward (S-shape, top-left) | Left-skewed distribution | | Points fan out at both ends | Heavy tails (more extreme values than normal) | | Points cluster at the ends | Light tails | The QQ plot is subjective — you are looking for systematic deviations from the line, not random scatter around it. ::: ------------------------------------------------------------------------ ## Part 3: The Shapiro-Wilk Test For a formal test of normality, use the **Shapiro-Wilk test**. It tests H₀: the data come from a normal distribution. ```{r} # Apply Shapiro-Wilk to our three datasets shapiro.test(normal_data) ``` ```{r} shapiro.test(right_skewed) ``` ```{r} shapiro.test(heavy_tails) ``` ::: callout-warning ## Limitations of the Shapiro-Wilk Test The Shapiro-Wilk test has two important limitations: 1. **Large samples always reject H₀** — with n > 200, even tiny, biologically irrelevant deviations from normality produce significant p-values. The test is too sensitive for large datasets. 2. **Small samples rarely reject H₀** — with n < 20, the test has low power and will fail to detect even moderately non-normal data. Best practice: **always look at the QQ plot** as well as the test. The combination of visual assessment and formal test gives a more complete picture than either alone. ::: ### Biological Data: Cytokine Concentrations Cytokines (immune signalling proteins) are typically measured at very low concentrations with a few extremely high values — a classic right-skewed distribution seen in immunology research. ```{r} set.seed(77) # Simulated IL-6 concentrations (pg/mL) in patients with varying inflammation il6 <- rlnorm(50, meanlog = 2.1, sdlog = 1.0) par(mfrow = c(1, 2)) hist(il6, col = "#2C5F8A", border = "white", xlab = "IL-6 (pg/mL)", main = "Raw IL-6 Concentrations") qqnorm(il6, pch = 16, cex = 0.7, col = "#2C5F8A", main = "QQ Plot — Raw IL-6") qqline(il6, col = "#C8102E", lwd = 2) par(mfrow = c(1, 1)) ``` ```{r} shapiro.test(il6) ``` ------------------------------------------------------------------------ ## Part 4: Log Transformation When biological data are right-skewed (as is common for concentrations, enzyme activities, and species abundances), a **log transformation** often produces a more symmetric, approximately normal distribution. ```{r} log_il6 <- log(il6) # Natural logarithm; use log10() for base-10 par(mfrow = c(1, 2)) hist(log_il6, col = "#5B8DB8", border = "white", xlab = "log(IL-6) (pg/mL)", main = "Log-Transformed IL-6") qqnorm(log_il6, pch = 16, cex = 0.7, col = "#5B8DB8", main = "QQ Plot — log(IL-6)") qqline(log_il6, col = "#C8102E", lwd = 2) par(mfrow = c(1, 1)) ``` ```{r} shapiro.test(log_il6) ``` ::: callout-tip ## Why Log-Transform? Many biological quantities follow **log-normal distributions** — i.e., their logarithms follow a normal distribution. This is common for: - Protein and cytokine concentrations - Bacterial colony counts - Gene expression values (RNA-seq log2 CPM) - Pharmacokinetic parameters (Cmax, AUC) - Species abundance data After log transformation, you can apply parametric tests normally and then back-transform results to the original scale for reporting. Back-transform with `exp()` if you used `log()`, or `10^x` if you used `log10()`. ::: ------------------------------------------------------------------------ ## Part 5: When Can You Ignore Non-Normality? ```{r} # Demonstrate: t-test is robust to moderate non-normality due to the CLT set.seed(500) # Two right-skewed groups (enzyme activity under two growth conditions) group_A <- rexp(40, rate = 0.1) # Mean = 10 group_B <- rexp(40, rate = 0.08) # Mean = 12.5 par(mfrow = c(1, 2)) hist(group_A, col = "#2C5F8A50", breaks = 15, main = "Group A (skewed)", xlab = "") hist(group_B, col = "#C8102E50", breaks = 15, main = "Group B (skewed)", xlab = "") par(mfrow = c(1, 1)) ``` ```{r} # t-test on non-normal data — still reasonable at n = 40 due to CLT t.test(group_A, group_B) ``` ::: callout-note ## When Parametric Tests Are Still Valid Parametric tests (t-tests, ANOVA) require that the **sampling distribution of the mean** is approximately normal — not necessarily that the raw data are normal. Due to the CLT: - For **n ≥ 30**, parametric tests are generally robust to skewness - For **n < 20**, normality is more important to check carefully - For **heavily non-normal data at small n**: use non-parametric alternatives (Lab 20) The key insight: it is the distribution of **means**, not raw values, that needs to be approximately normal. ::: ------------------------------------------------------------------------ ## 3-Minute Knowledge Check *Close your notes. Answer these on your own — you have 3 minutes. We'll go through the answers together after.* ::: callout-caution ## Knowledge Check Questions **1.** On a QQ plot, you observe that the points form a curve that bends upward to the right (the top-right points fall above the line). This indicates: a) Left-skewed data b) Right-skewed data c) Heavy-tailed data d) Normal data with outliers **2.** You apply `shapiro.test()` to a dataset of 500 observations and get p = 0.02. What is the most important caution about interpreting this result? **3.** A researcher measures serum triglycerides (mg/dL) and finds the data are right-skewed (W = 0.87, p = 0.003). What transformation should they try first? a) Square root b) Reciprocal (1/x) c) Log d) Squaring (x²) **4.** You have two samples of n = 35 from a skewed population. A colleague says you cannot run a t-test because the data are not normal. Are they correct? **5.** True or False: The standard error of the mean will always be smaller than the standard deviation of the same sample. ::: ::: {.callout-caution collapse="true"} ## Answers (Only reveal if you have answered the above questions) **1.** b) Right-skewed data — In a right-skewed distribution, there are more extreme large values than the normal distribution predicts. On a QQ plot these appear as points above the reference line in the top-right, creating an upward curve. A "banana" shape bending upward = right skew. **2.** With n = 500, the Shapiro-Wilk test is so sensitive that it detects trivially small deviations from normality that have no practical consequence for parametric tests. A significant p-value here does **not** mean parametric tests are inappropriate — always inspect the QQ plot and consider whether the deviation is biologically meaningful. **3.** c) Log transformation — The log (or log10) transformation is the standard first attempt for right-skewed biological data such as concentrations, counts, and ratios. It compresses the right tail and often produces an approximately normal distribution. **4.** No, they are not correct — with n = 35 per group, the Central Limit Theorem ensures that the sampling distribution of the mean is approximately normal even from a skewed population. The t-test is robust to this degree of non-normality at this sample size. They should still inspect QQ plots and consider a non-parametric alternative if skewness is extreme. **5.** **True** — SE = SD / √n. Since √n > 1 for any n > 1, the SE is always smaller than the SD. The more observations you have, the smaller the SE — reflecting greater precision in estimating the population mean. ::: ------------------------------------------------------------------------ ## Lab 16 Checklist Before you leave, make sure you can: - [ ] Explain the difference between SD, SE, and a confidence interval - [ ] Produce a QQ plot with `qqnorm()` and `qqline()` and interpret its shape - [ ] Run `shapiro.test()` and state the null hypothesis being tested - [ ] Recognise the limitations of the Shapiro-Wilk test at large and small sample sizes - [ ] Apply `log()` to right-skewed data and verify the transformation improved normality - [ ] Explain why parametric tests are often valid for n ≥ 30 even with non-normal data ::: callout-tip ## Bonus Challenge Find a real dataset of your choice (CDC public health data, a published biology paper's supplementary table, or the `datasets` package in R: try `data(airquality)`). Choose a continuous variable, assess its normality with a QQ plot and Shapiro-Wilk test, apply an appropriate transformation if needed, and reassess. Write a paragraph describing your findings and whether a parametric test would be appropriate. ::: ------------------------------------------------------------------------ ## Before Next Class (Quiz Friday, then Week 8) - **Friday: Quiz 4** — covers the normal distribution, CLT, sampling properties, and assessing normality (Labs 15–16) - After the quiz, **Week 8** covers t-tests and ANOVA — be confident with the CLT and QQ plots before then - Read **R for Data Science Ch. 9–10** ------------------------------------------------------------------------