Вопрос
What does the Central Limit Theorem say and why is it important in statistics and A/B testing?
Ответить самому
Сначала сформулируйте ответ как на собеседовании, затем откройте разбор и оцените себя.
Короткий ответ
Under common regularity conditions, the normalized sample mean tends to a normal distribution as sample size grows. This justifies normal approximations for confidence intervals, tests and experiment readouts.
Полный разбор
The Central Limit Theorem says that for independent identically distributed variables with finite variance, the distribution of the normalized sample mean approaches a normal distribution as sample size grows:
(mean - mu) / (sigma / sqrt(n)) -> N(0, 1).
This is important because many metrics are averages or proportions. Even if individual observations are not normally distributed, the sampling distribution of the mean can be approximately normal for large n. That gives practical formulas for standard errors, confidence intervals and hypothesis tests.
In A/B testing, CLT reasoning underlies z-tests and t-test approximations for many user-level metrics. The assumptions still matter: independence, finite variance, enough sample size and a metric definition that does not create severe dependence or heavy-tail instability.
Теория
CLT is about the distribution of an estimator, not about raw data becoming normal.
Типичные ошибки
- Say the original data becomes normally distributed.
- Ignore independence and finite variance assumptions.
- Use CLT blindly for heavy-tailed or dependent user events.
Как отвечать на собеседовании
- Say "sample mean" early.
- Connect to standard error sigma divided by square root of n.