Central Limit Theorem

Visual demonstration of the Central Limit Theorem - why averages are normally distributed.

Statement

The sum (or average) of many independent random variables tends toward a normal distribution, regardless of the original distribution.

$$ \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0, 1) \quad \text{as } n \to \infty $$

Visual Demonstration

Why It Matters

Demonstration

 1import numpy as np
 2import matplotlib.pyplot as plt
 3
 4# Sample from uniform distribution (not normal!)
 5n_samples = 1000
 6sample_size = 30
 7
 8means = []
 9for _ in range(n_samples):
10    sample = np.random.uniform(0, 1, sample_size)
11    means.append(np.mean(sample))
12
13# Distribution of means is approximately normal!
14plt.hist(means, bins=50, density=True)
15plt.title('Distribution of Sample Means (CLT)')
16plt.show()

Practical Importance

• Justifies using normal distribution in many applications
• Foundation for confidence intervals and hypothesis testing
• Explains why measurement errors are often normal

Further Reading

• Central Limit Theorem - Wikipedia

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