Central Limit Theorem

The Central Limit Theorem (CLT) states that if you repeatedly take sufficiently large random samples from any population and calculate their means, the distribution of those sample means will approximate a Normal distribution - regardless of the shape of the original population.

Why This Is So Important

The CLT is the reason so much of statistics can rely on the Normal distribution, even when the underlying data isn't normally distributed at all. It underpins confidence intervals and hypothesis testing across nearly every field that uses statistics.

A Simple Simulation

import numpy as np
import matplotlib.pyplot as plt

# Original population: a strongly skewed exponential distribution
population = np.random.exponential(scale=2, size=100000)

# Take 1000 samples of size 50, and record each sample's mean
sample_means = [np.mean(np.random.choice(population, size=50)) for _ in range(1000)]

plt.hist(sample_means, bins=30)
plt.title("Distribution of Sample Means")
plt.show()   # this comes out approximately bell-shaped

Key Conditions

• Samples should be drawn independently and randomly
• Sample size should generally be at least 30 for the approximation to hold well
• The theorem applies to the distribution of the sample mean, not to individual data points

Coming Up Next

Next, you'll move into hypothesis testing, starting with the Null and Alternate Hypothesis.