Lesson 13 · Probability and Statistics in python

Understanding the Central Limit Theorem: Key Concepts and Practical Applications

In this hands-on Python session, we will explore the Central Limit Theorem (CLT), see why it is a cornerstone of statistics, and test its power with simple…

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CourseProbability and Statistics in python
Lesson13 of 35
Video14 min
FormatJupyter notebook · 15 code cells

NumPy pandas Matplotlib seaborn

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Welcome to Central Limit Theorem and Its Applications!#

In this hands-on Python session, we will explore the Central Limit Theorem (CLT), see why it is a cornerstone of statistics, and test its power with simple experiments.

Let us break down complex ideas and use real-world data so you see how theory meets practice.

# Suppress annoying warnings to keep things clean
import warnings; warnings.filterwarnings("ignore")

What is the Central Limit Theorem (CLT)?#

CLT says: No matter what your population looks like, if you take lots of samples and average them, their means will look like a bell curve (a normal distribution) as your sample gets big!

This is why averages of things (like average test scores, average heights) tend to follow a normal pattern, even if single measurements are not normal.

# Data setup
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

# To make everything look the same each time we run
np.random.seed(42)

# Create a skewed population: a fake set of 10000 people's incomes
population = np.random.exponential(scale=20000, size=10000)
print(f"Population mean: {np.mean(population):.2f}")
print(f"Population std: {np.std(population):.2f}")

Population mean: 19549.98
Population std: 19487.12

# Let us look at this population with a histogram
plt.figure(figsize=(8,4))
sns.histplot(population, bins=50, kde=False, color="skyblue")
plt.title("Histogram of Fake Income Population (not normal)")
plt.xlabel("Income")
plt.ylabel("Count")
plt.show()

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What happens if we take just one sample?#

If we picked just a few people, their average might be very far from the true average. Let us see how that works...

# Take a small random sample of 5 incomes
sample = np.random.choice(population, size=5, replace=False)
print("Sample incomes:", sample)
print(f"Sample mean: {np.mean(sample):.2f}")

Sample incomes: [27997.98460233 11324.87199283 31971.30952896 35234.21990535
 50688.17434222]
Sample mean: 31443.31

What does the Central Limit Theorem say?#

It says that as we take more samples of the same size and plot their means, the distribution of those means will look more and more like a normal curve.

# Take 5000 samples of 5, find the mean each time
means_small = [np.mean(np.random.choice(population, size=5, replace=False)) for _ in range(5000)]

plt.figure(figsize=(8,4))
sns.histplot(means_small, bins=40, kde=True, color="orange")
plt.title("Means of 5000 samples (size 5)")
plt.xlabel("Sample Mean")
plt.ylabel("Count")
plt.show()

# Try with bigger samples! Sample size 30 this time
means_bigger = [np.mean(np.random.choice(population, size=30, replace=False)) for _ in range(5000)]

plt.figure(figsize=(8,4))
sns.histplot(means_bigger, bins=40, kde=True, color="green")
plt.title("Means of 5000 samples (size 30)")
plt.xlabel("Sample Mean")
plt.ylabel("Count")
plt.show()

Summary of What We See#

When you take small samples from a weird population, the averages can jump around.
When you take more and bigger samples, the means act normal, like a bell.

This is the magic of the central limit theorem!

# Practice: Try your own sample size!
size = int(input("Enter a sample size (try 10, 50, or 100): "))
n_samples = 3000
experiment = [np.mean(np.random.choice(population, size=size, replace=False)) for _ in range(n_samples)]

plt.figure(figsize=(8,4))
sns.histplot(experiment, bins=40, kde=True, color="purple")
plt.title(f"Means of {n_samples} samples (size {size})")
plt.xlabel("Sample Mean")
plt.ylabel("Count")
plt.show()

# Compare means and standard deviations
print(f"Population mean: {np.mean(population):.2f}")
print(f"Sample means average: {np.mean(experiment):.2f}")
print(f"Population std dev: {np.std(population):.2f}")
print(f"Sample means std dev: {np.std(experiment):.2f}")

Population mean: 19549.98
Sample means average: 19569.41
Population std dev: 19487.12
Sample means std dev: 2755.03

# Let us apply CLT to a real dataset: the 'tips' dataset
tips = sns.load_dataset("tips")
print("Tips data shape:", tips.shape)
display(tips.head())

Tips data shape: (244, 7)

	total_bill	tip	sex	smoker	day	time	size
0	16.99	1.01	Female	No	Sun	Dinner	2
1	10.34	1.66	Male	No	Sun	Dinner	3
2	21.01	3.50	Male	No	Sun	Dinner	3
3	23.68	3.31	Male	No	Sun	Dinner	2
4	24.59	3.61	Female	No	Sun	Dinner	4

# What is the distribution of tips? Let us see if it is normal
plt.figure(figsize=(8,4))
sns.histplot(tips["tip"], bins=30, kde=True, color="salmon")
plt.title("Distribution of Tip Amounts")
plt.xlabel("Tip (dollars)")
plt.ylabel("Count")
plt.show()

# Try the CLT with tips: sample means
tips_means = [tips["tip"].sample(25, replace=False, random_state=i).mean() for i in range(2500)]
plt.figure(figsize=(8,4))
sns.histplot(tips_means, bins=35, kde=True, color="deepskyblue")
plt.title("Means of 2500 samples (size 25) from Tips")
plt.xlabel("Sample Mean Tip")
plt.ylabel("Count")
plt.show()

Best Practices and Troubleshooting#

If your samples seem odd, check these:

Make sure you are sampling randomly, not in order.
Use enough samples a few hundred at least.
If your code is slow, lower the number of samples.
When in doubt, plot what you are sampling.

Getting weird errors? Try running the cell again, or double-check your brackets and parentheses.

# Extra tip: Visualize how sample size affects normality
fig, axs = plt.subplots(1, 3, figsize=(18,4))
for i, size in enumerate([3, 10, 50]):
    means = [np.mean(np.random.choice(population, size=size, replace=False)) for _ in range(2000)]
    sns.histplot(means, bins=30, kde=True, ax=axs[i], color="steelblue")
    axs[i].set_title(f"Sample size = {size}")
    axs[i].set_xlabel("Sample Mean")
    axs[i].set_ylabel("Count")
plt.tight_layout()
plt.show()

# Mini-project: Simulate a game (rolling dice) and show the CLT in action
dice_pop = np.random.choice([1,2,3,4,5,6], size=10000, replace=True)
sample_size = 20
means = [np.mean(np.random.choice(dice_pop, size=sample_size, replace=False)) for _ in range(3000)]
plt.figure(figsize=(8,4))
sns.histplot(means, bins=25, kde=True, color="gold")
plt.title("Means of 3000 samples (size 20) from dice rolls")
plt.xlabel("Sample Mean (Dice Roll)")
plt.ylabel("Count")
plt.show()

# Challenge: Use CLT to check average total_bill in 'tips' for random samples
total_bill_means = [tips["total_bill"].sample(40, replace=False, random_state=i).mean() for i in range(1500)]
plt.figure(figsize=(8,4))
sns.histplot(total_bill_means, bins=30, kde=True, color="tomato")
plt.title("Means of 1500 samples (size 40) of Total Bill")
plt.xlabel("Sample Mean of Total Bill")
plt.ylabel("Count")
plt.show()

Recap: Key Takeaways#

The central limit theorem helps us trust averages.
No matter the original shape, averages of lots of samples turn bell-like.
This makes statistics powerful for science, business, and understanding daily life.

Keep practicing sampling and watch those bell curves appear!

Thank you for learning with us!#

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