Statistics · easy

What is the law of large numbers, and how is it different from the Central Limit Theorem?

Asked in Data Scientist interviews, in the Statistics round.

Short answer

The law of large numbers says the sample mean converges to the true mean as n grows. The CLT says how it gets there: the sampling distribution of the mean becomes normal with spread sigma over root n.

How to answer it

Two statements about the sample mean, one about where it goes and one about how it is spread on the way.

The law of large numbers: average enough independent draws and the sample mean settles on the population mean. It says nothing about the shape of the errors, only that they shrink.

The Central Limit Theorem: for large n the sample mean is approximately normal around the true mean with standard deviation sigma divided by the square root of n, whatever the shape of the underlying data. That is what lets you put a confidence interval or a p-value on a mean of skewed revenue numbers.

The interview version: LLN is why more data makes the estimate closer, CLT is why you can say how close, and both need independence. Autocorrelated daily metrics or one user contributing many rows break the "root n" part first.

• What they are checking: whether you can separate convergence from the shape of the error, and whether you mention independence.
• Common mistake: saying the CLT makes the data normal. It is the mean that becomes normal, the data stays whatever it was.

Related questions

• When would you use a t-test vs a chi-square test?
• What is Simpson's paradox?
• Explain p-value in plain language.
• How do you determine sample size for an A/B test?
• What is the difference between Type I and Type II error?
• Explain the Central Limit Theorem and why it matters.

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