The Shapiro Wilk Test for Normality (Limited Samples)

This section demonstrates the Shapiro–Wilk Test for Normality using R.
We will generate two datasets: - One from a normal distribution - One from a skewed exponential distribution

We then: 1. Perform the Shapiro–Wilk test on each dataset.
2. Visualize the data using histograms and Q–Q plots.
3. Interpret the test results and visual diagnostics.

The Shapiro–Wilk test checks whether the data are consistent with a normal distribution, using:

\[ H_0: \text{Data are normally distributed} \] \[ H_1: \text{Data are not normally distributed} \]

If the p-value < 0.05, we reject \(H_0\) and conclude that the data are not normally distributed.

R Code: Normality Testing and Visualization

The following code block: - Generates normally distributed and exponential data samples, - Applies the Shapiro–Wilk test to each, - Produces histograms and Q–Q plots for visual inspection.

# Set seed for reproducibility
set.seed(123)

# Generate normally distributed data
normal_data <- rnorm(50, mean = 10, sd = 2)

# Generate skewed (non-normal) exponential data
skewed_data <- rexp(50, rate = 0.5)

# Perform Shapiro–Wilk tests
shapiro_normal <- shapiro.test(normal_data)
shapiro_skewed <- shapiro.test(skewed_data)

# Display results
cat("Shapiro–Wilk Test Results:\n")
## Shapiro–Wilk Test Results:
cat("Normal Data: W =", round(shapiro_normal$statistic, 4), 
    ", p-value =", round(shapiro_normal$p.value, 4), "\n")
## Normal Data: W = 0.9893 , p-value = 0.9279
cat("Skewed Data: W =", round(shapiro_skewed$statistic, 4), 
    ", p-value =", round(shapiro_skewed$p.value, 4), "\n\n")
## Skewed Data: W = 0.8249 , p-value = 0
# Plot layout: 2 rows x 2 columns
par(mfrow = c(2, 2))

# Plot histogram and Q-Q plot for normal data
hist(normal_data, breaks = 10, col = "blue",
     main = "Histogram: Normal Data", xlab = "Values", border = "white")
qqnorm(normal_data, main = "Q-Q Plot: Normal Data")
qqline(normal_data, col = "red", lwd = 2)

# Plot histogram and Q-Q plot for skewed data
hist(skewed_data, breaks = 10, col = "blue",
     main = "Histogram: Skewed Data", xlab = "Values", border = "white")
qqnorm(skewed_data, main = "Q-Q Plot: Skewed Data")
qqline(skewed_data, col = "red", lwd = 2)

# Reset plot layout
par(mfrow = c(1, 1))