The central limit theorem (CLT), in very broad terms, tells us that the “magical” bell curve of normal distributions happens (and it does, a lot!) in the real world, often regardless of the data’s original shape. But have you wondered why?
The bottom line behind CLT is a fundamental statistical rule: if you take enough samples from any data and calculate their averages, these averages will approach a normal distribution — no matter what the original data’s form was.
To build your intuition, this article shows three visual proofs that the classic bell curve appears in myriad situations.
# 1. Rolling Multiple Dice
When rolling a single, six-sided die many times, we intuitively end up with a flat, uniform distribution. However, the story changes when rolling multiple dice simultaneously — for example, five at once. If we calculate the average of those dice results and plot them, a bell curve emerges, peaking around the intermediate scores of 3 and 4. These average scores tend to become much more frequent than those near the extreme values of 1 and 6, which require all dice to score low (or high, respectively): something intuitively more difficult to achieve than having mixed scores compensate each other.
import numpy as np
import matplotlib.pyplot as plt
# Simulating 100,000 averages of 5 dice rolls: upper bound of 7 is excluded in randint()
dice_averages = [np.mean(np.random.randint(1, 7, 5)) for _ in range(100000)]
plt.hist(dice_averages, bins=20, edgecolor="black", color="skyblue")
plt.title("Distribution of 5-Dice Averages")
plt.show()
Distribution of averages from rolling 5 dice 100,000 times
# 2. Averaging a Divided Crowd: Bimodal Screen Time
Suppose a video-watching app in which users fall into two sharply distinct categories: they either stay for 1–2 minutes only, or they are “deep scrollers” who stay watching for 30 minutes or more — no users in between. Plotted raw data would, of course, yield a bimodal distribution with a central, deep valley. However, if you repeatedly grab random groups of, say, 40 users, calculate the average screen time, and plot it across many samplings, things change. The distribution of averages utterly ignores the valley between the two peaks and turns into a bell curve.
# Creating a bimodal distribution (quick checkers and deep scrollers)
quick = np.random.normal(2, 0.5, 50000)
deep = np.random.normal(30, 5, 50000)
bimodal_population = np.concatenate([quick, deep])
# Taking 10,000 samples of 40 users each, and averaging screen times
screen_time_averages = [np.mean(np.random.choice(bimodal_population, 40)) for _ in range(10000)]
plt.hist(screen_time_averages, bins=30, edgecolor="black", color="mediumpurple")
plt.title("Averages of Bimodal Screen Times for 10K 40-user samples")
plt.show()
Distribution of averages from sampling a bimodal screen time population
# 3. Smoothing Chaotic Skewness: Average Incomes
Another real-world example of CLT at its finest. Wealth distribution is, let’s face it, remarkably right-skewed, with a vast majority of the population leaning closer to the lower bound. The wealthy minority, therefore, forms a long tail stretching toward the right.
This picture changes when randomly picking, for instance, 50 people and averaging their incomes. If we repeat this sampling process a few thousand times and plot the averages obtained, the result is — what else could we expect at this point? — a pristine, almost perfectly symmetric bell curve: rich financial outliers get smoothed out by the humbler crowd.
# Generating heavily skewed "income" data (exponential distribution)
population = np.random.exponential(scale=50000, size=100000)
# Taking 10,000 samples of 50 people and averaging them
income_averages = [np.mean(np.random.choice(population, 50)) for _ in range(10000)]
plt.hist(income_averages, bins=30, edgecolor="black", color="salmon")
plt.title("Averages of Skewed Wealth Samples")
plt.show()
Distribution of averages from sampling a right-skewed income population
Iván Palomares Carrascosa is a leader, writer, speaker, and adviser in AI, machine learning, deep learning & LLMs. He trains and guides others in harnessing AI in the real world.
