What is the Central Limit Theorem, and why does it matter?
The Central Limit Theorem says that the distribution of sample means approaches a normal distribution as the sample size grows, whatever the shape of the original data (given finite variance). That is why normal-based tests and confidence intervals work on messy real data.
Explain it simply
Rolling one die gives a flat distribution: each face is equally likely. Now roll 30 dice and take the average, and repeat that many times. The averages cluster around 3.5 in a bell shape, even though a single die is nowhere near bell-shaped.
Averaging cancels out extremes. Some values come out high, some low, and the mean sits near the centre. The spread of those averages shrinks as the sample grows.
Key facts:
- The mean of the sample means equals the population mean.
- Their standard deviation, the standard error, is population SD divided by the square root of n.
- A common rule of thumb is n of about 30 or more, but heavily skewed data needs more.
Example
import numpy as np
rng = np.random.default_rng(1)
means = [rng.integers(1, 7, 30).mean() for _ in range(2000)]
print(abs(np.mean(means) - 3.5) < 0.05) # TrueA histogram of means looks like a bell curve centred near 3.5.
What to say in the interview
"The CLT states that sample means become approximately normal with mean mu and standard deviation sigma over root n, for large enough samples, regardless of the population shape. It justifies z-tests, t-tests and confidence intervals even when the raw data is not normal."
Common mistakes
- Saying the data itself becomes normal. Only the distribution of the sample mean does.
- Applying it to tiny samples from very skewed or heavy-tailed data.