Lab 5: Central Limit Theorem and Sampling Distributions

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October 19, 2025

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Introduction: Central Limit Theorem

This laboratory explores the Central Limit Theorem (CLT) and sampling distributions through interactive simulation. We will use a Shiny application to investigate how sample means behave across different population distributions and sample sizes, examining when the CLT holds and when it fails.

Access the CLT simulation app here: https://math214.netlify.app/sampapp

Learning Objectives

Upon completion of this laboratory, students will be able to:

  • Understand and apply the Central Limit Theorem principles
  • Distinguish between population distributions and sampling distributions
  • Analyze how sample size affects the normality of sampling distributions
  • Identify when CLT assumptions are satisfied or violated
  • Compare sampling distributions of means and medians
  • Use interactive visualization tools for statistical exploration
  • Document analytical processes using R Markdown with professional formatting

Time Allocation (Total: 100 minutes)

  • Part 1: CLT Fundamentals and Normal Populations (35 minutes)
  • Part 2: CLT with Non-Normal Populations (35 minutes)
  • Part 3: When CLT Fails - Cauchy Distribution (30 minutes)

Part 1: CLT Fundamentals and Normal Populations

1.1 Central Limit Theorem Overview

The Central Limit Theorem states that when certain assumptions are satisfied, the sampling distribution of sample means will approach a normal distribution, regardless of the shape of the population distribution. The key assumptions are:

  1. The population has a finite mean (μ)
  2. The population has a finite standard deviation (σ)
  3. Either:
    • The population is normally distributed (sample size doesn’t matter), OR
    • The sample size is sufficiently large (typically n ≥ 30)

1.2 Using the CLT Shiny Application

Access the CLT simulation application at: https://math214.netlify.app/sampapp

App Interface Overview:

  • Population Distribution: Select from Normal, Uniform, Right Skewed, Left Skewed, Bimodal, or Cauchy
  • Sample Size: Number of observations in each sample (5-500)
  • Number of Samples: How many samples to generate (10-5000)
  • Random Seed: For reproducible results

What you’ll see in the app: The app provides interactive visualization of sampling distributions across multiple tabs:

  • Population tab: Shows the underlying population distribution with density curve
  • First 8 Samples tab: Displays dot plots of individual sample means
  • EDA of Samples tab: Shows summary statistics and histograms for the first 8 samples
  • Sampling Distribution of Means tab: Visualizes the distribution of sample means with histogram and Q-Q plot
  • Sampling Distribution of Medians tab: Shows the distribution of sample medians with histogram and Q-Q plot

1.3 Normal Population Analysis

What you’ll observe: When you select the Normal population distribution in the app, you’ll see that the sampling distribution of means appears normal even with small sample sizes. This demonstrates CLT assumption 3a - when the population is normally distributed, the sampling distribution is normal regardless of sample size.

Key observations to make:

  • The Q-Q plot in the “Sampling Distribution of Means” tab shows points closely following the reference line
  • The histogram of sample means appears symmetric and bell-shaped
  • The mean of the sampling distribution approximates the population mean
  • The standard deviation of the sampling distribution approximates σ/√n

Part 2: CLT with Non-Normal Populations

2.1 Uniform Distribution Analysis

What you’ll observe: Select the Uniform distribution in the app. With small sample sizes (n=5-10), you’ll see the sampling distribution is not perfectly normal, but as you increase sample size to n=30 or higher, the distribution becomes increasingly normal.

Key observations:

  • The Q-Q plot shows systematic deviations from normality with small samples
  • With larger samples, the Q-Q plot points align better with the reference line
  • The histogram becomes more symmetric and bell-shaped as sample size increases

2.2 Skewed Distributions Analysis

Right-Skewed observations: Select Right Skewed distribution. You’ll see that with small samples, the sampling distribution remains skewed, but with larger samples (n≥30), it approaches normality.

Left-Skewed observations: Select Left Skewed distribution. Similar pattern - small samples show skewness, larger samples show normality.

Key insights: Both skewed distributions demonstrate CLT assumption 3b - with sufficient sample size, sampling distributions approach normality regardless of population shape.

2.3 Bimodal Distribution Analysis

What you’ll observe: Select Bimodal distribution. This is the most challenging case for CLT. You’ll need larger sample sizes (n≥50-100) to see the sampling distribution approach normality.

Key observations:

  • With small samples, the sampling distribution may show bimodality
  • As sample size increases, the distribution becomes unimodal and symmetric
  • The Q-Q plot shows significant deviations with small samples that improve with larger n

Part 3: When CLT Fails - Cauchy Distribution

3.1 Cauchy Distribution Properties

The Cauchy distribution violates CLT assumptions 1 and 2 - it has undefined mean and infinite variance. This makes it an excellent test case for understanding CLT limitations.

3.2 Sampling Distribution Analysis

What you’ll observe: Select the Cauchy distribution in the app. You’ll see that no matter how large you make the sample size, the sampling distribution of means never approaches normality.

Key observations:

  • The Q-Q plot shows extreme deviations from the reference line
  • The histogram of sample means remains heavy-tailed and non-normal
  • The distribution of means doesn’t stabilize or become symmetric
  • Even with very large samples (n=500), the distribution remains pathological

Why CLT fails: The Cauchy distribution violates CLT assumptions 1 and 2 - it has undefined mean and infinite variance, making the CLT inapplicable.

3.3 Alternative: Sampling Distribution of Medians

When CLT fails for means, we can sometimes use the sampling distribution of medians, which may have better properties for certain heavy-tailed distributions.

Assessment (Total: 50 points)

Section A: Formative Understanding (15 points)

A1. Explain the three key assumptions of the Central Limit Theorem and why each is necessary for the theorem to hold. Provide examples of distributions that violate each assumption. (5 points)

A2. Describe how sample size affects the sampling distribution of means for non-normal populations. Include discussion of why n ≥ 30 is often recommended and situations where larger samples might be needed. (4 points)

A3. Compare and contrast the sampling distributions of means and medians. When would you prefer to use medians instead of means, and what additional assumptions might be needed? (3 points)

A4. Discuss the practical implications of the Central Limit Theorem in statistical inference. How does it enable us to make inferences about population parameters? (3 points)

Section B: Application and Analysis (20 points)

B1. Using the CLT app with a normal population distribution, sample size n=5, and 500 samples: - Report the population mean and standard deviation - Report the mean and standard deviation of the sampling distribution - Compare the theoretical standard error (σ/√n) with the observed standard deviation - Create and interpret the QQ-plot for normality (5 points)

# Your code and analysis here

B2. Repeat B1 with sample sizes n=30 and n=100. Analyze how increasing sample size affects the sampling distribution properties and normality. (5 points)

# Your code and analysis here

B3. Choose a non-normal distribution (Uniform, Right Skewed, or Left Skewed) and analyze the sampling distribution with n=5, n=30, and n=100 samples. Discuss how the CLT manifests with increasing sample size. (5 points)

# Your code and analysis here

B4. Analyze the Cauchy distribution with various sample sizes. Explain why the CLT fails and discuss alternative approaches for this distribution. (5 points)

# Your code and analysis here

Section C: Statistical Synthesis (15 points)

C1. Write a comprehensive report (200-250 words) discussing the practical importance of the Central Limit Theorem in statistical practice. Include examples from different fields where CLT applications are crucial and discuss limitations and alternatives when CLT assumptions are violated. (7 points)

C2. Design a simulation study to investigate the minimum sample size needed for the sampling distribution to approximate normality for a specific non-normal distribution of your choice. Outline the methodology, evaluation criteria, and expected results. (4 points)

C3. Reflect on the pedagogical value of interactive simulations like the CLT app for teaching statistical concepts. Discuss how such tools enhance understanding compared to traditional mathematical derivations alone. (4 points)

Submission Guidelines:

  • Complete R Markdown document with all code and analysis
  • Detailed descriptions of app observations and outputs for each analysis
  • Professional writing with clear analytical narrative
  • Knitted HTML document submitted via designated platform
  • Use the app at https://math214.netlify.app/sampapp for all interactive explorations