The Central Limit Theorem

Customer spending isn't a bell curve. It's skewed, with a long tail. Yet statistics is full of bell curves. How can that work? There's a remarkable fact at the centre of the whole subject, and we can see it happen.

• Spending isn't a bell curve
• Statistics is full of them
• A remarkable fact

Bell curves from anything

We start with very skewed data. Now take averages of groups. Just five, and then fifty. Run it. Look at the three charts. The first is lopsided. Averages of five are less so. And averages of fifty make a clear bell curve. It happens for nearly any data. That's the central limit theorem, and it's why you can use the same tools across almost all business problems.

import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(0)
skewed = rng.gamma(2, 2, 100000)

fig, ax = plt.subplots(1, 3, figsize=(10, 3.2))
ax[0].hist(skewed, bins=60); ax[0].set_title("One customer")
for a, n in zip(ax[1:], [5, 50]):
    means = rng.gamma(2, 2, (20000, n)).mean(axis=1)
    a.hist(means, bins=60); a.set_title(f"Average of {n}")
plt.tight_layout(); plt.savefig("clt.png", dpi=110)
print("Skewed data, yet the averages form a bell curve")

Skewed data, yet the averages form a bell curve

Summary