What is the central limit theorem and why does it matter for everyday analysis work?

What the interviewer wants to hear

• Sample means are approximately normal for large samples
• Holds even when the raw data is skewed, given finite variance
• Justifies confidence intervals and z or t tests on averages
• Needs larger samples for very skewed or heavy-tailed data

Model answer

The central limit theorem says that if you take many random samples and compute the mean of each, those means form an approximately normal distribution, even when the raw data is not normal, as long as the variance is finite and the sample is reasonably large. Its spread is the standard error, sigma divided by the square root of n. This matters because most of our metrics are averages: average order value, conversion rate, time spent. The CLT is what lets us build confidence intervals and run z or t tests on them without knowing the exact shape of the data. The caveat is that very skewed or heavy-tailed data, like revenue with a few whales, needs a much larger sample before the approximation is good. In those cases I check with a bootstrap, cap extreme values, or use a transformation.

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