What is the difference between the Law of Large Numbers and the Central Limit Theorem?
The Law of Large Numbers says the sample mean converges to the true mean as n grows. The Central Limit Theorem describes the shape and spread of the sample mean around that true value: approximately normal with SD sigma over root n.
Explain it simply
Think of a bowler's average over a career. After 3 matches it jumps around; after 300 it settles near the true skill. That settling is the Law of Large Numbers (LLN): where the average ends up.
The CLT is about the small wobbles that remain: if many bowlers each averaged over 300 matches, those averages would form a bell curve around the true skill, with a known width. That is how the average behaves on the way there.
| LLN | CLT | |
|---|---|---|
| Says | sample mean gets close to the true mean | sample mean is about normal around it |
| About | where it converges | how it is distributed |
| Gives | consistency | confidence intervals and tests |
Example
import numpy as np
rng = np.random.default_rng(7)
flips = rng.integers(0, 2, 100000)
print(abs(flips.mean() - 0.5) < 0.01) # TrueWith 100,000 fair coin flips the proportion of heads is very close to 0.5, as the LLN predicts.
What to say in the interview
"LLN gives consistency: averages converge to the expected value. CLT adds the distribution: the error of the average is approximately normal with standard deviation sigma over root n. LLN tells me a bigger sample is better; CLT tells me how much better and lets me build intervals."
Common mistakes
- Using LLN to claim a coin is "due" for heads after a streak of tails. That is the gambler's fallacy.