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Understanding the Central Limit Theorem (CLT) is much easier when you see it in action. Visualization allows you to watch how the distribution of sample means changes as you increase your sample size. By plotting both the original population and the sample means for different sample sizes, you can see how the CLT works and why it is so important in statistics.


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import numpy as np
import matplotlib.pyplot as plt

# Set random seed for reproducibility
np.random.seed(42)

# Generate a non-normal population: exponential distribution (skewed)
population = np.random.exponential(scale=2, size=10000)

fig, axes = plt.subplots(1, 4, figsize=(20, 4))

# Plot the original population distribution
axes[0].hist(population, bins=50, color="skyblue", edgecolor="black")
axes[0].set_title("Population (Exponential)")
axes[0].set_xlabel("Value")
axes[0].set_ylabel("Frequency")

sample_sizes = [2, 10, 30]
for i, n in enumerate(sample_sizes, 1):
    sample_means = []
    for _ in range(1000):
        sample = np.random.choice(population, size=n)
        sample_means.append(np.mean(sample))
    axes[i].hist(sample_means, bins=30, color="salmon", edgecolor="black")
    axes[i].set_title(f"Sample Means (n={n})")
    axes[i].set_xlabel("Sample Mean")
    axes[i].set_ylabel("Frequency")

plt.tight_layout()
plt.show()


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import seaborn as sns

sample_sizes = [2, 10, 30]
sample_means_dict = {}

for n in sample_sizes:
    sample_means = []
    for _ in range(1000):
        sample = np.random.choice(population, size=n)
        sample_means.append(np.mean(sample))
    sample_means_dict[n] = sample_means

plt.figure(figsize=(10, 6))
colors = ["#FFA07A", "#20B2AA", "#9370DB"]
for n, color in zip(sample_sizes, colors):
    sns.histplot(sample_means_dict[n], bins=30, kde=True, color=color, label=f"n={n}", stat="density", alpha=0.6)

plt.title("Overlay of Sample Means Distributions")
plt.xlabel("Sample Mean")
plt.ylabel("Density")
plt.legend()
plt.show()

As you saw in the visualizations, the original population distribution was highly skewed and not normal. However, as you increased the sample size, the distribution of the sample means became more symmetric and bell-shaped. Even when the underlying population is not normal, the sample means approach a normal distribution as sample size grows. This is the essence of the Central Limit Theorem: regardless of the population's shape, the distribution of sample means will tend toward normality as the sample size increases.

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