What does the central limit theorem say?

That the mean of independent samples from any distribution with finite variance is approximately normal, with standard error sigma / sqrt(n). The underlying distribution can be almost anything.

When does the central limit theorem fail?

When variance is infinite - heavy-tailed distributions like the Cauchy never converge. It also degrades badly under strong dependence, and convergence is slowest in the tails.

Statistics

The Central Limit Theorem in Interviews

NeetQuant · August 2026 · 4 min read

The statement

Take independent samples from a distribution with mean mu and finite variance sigma squared. The sample mean is approximately normal with mean mu and standard deviation sigma / sqrt(n), and the approximation improves as n grows.

The remarkable part is that the underlying distribution can be almost anything.

The sqrt(n) consequence

Standard error falls like 1 / sqrt(n). To halve your uncertainty you need four times the data.

This is the single most practically useful implication, and it recurs everywhere: to distinguish a Sharpe ratio of 1 from zero with confidence you need years of data, because the estimate's error shrinks so slowly. Interviewers ask "how much data do you need to know this strategy works?" precisely to see whether you have internalised it.

The conditions people forget

Finite variance. Without it the theorem simply does not apply. Cauchy-distributed samples never converge to normality - the sample mean of n Cauchy variables is distributed exactly like a single one, regardless of n. Mentioning this unprompted is a strong signal.

Independence, or at least weak dependence. Strongly correlated observations effectively reduce your sample size, so n observations with high autocorrelation carry the information of far fewer.

Why it matters for markets

Financial returns have heavy tails. The CLT still applies to the mean if variance is finite, but convergence is slow precisely in the tails - which is exactly where risk management operates.

So "returns are approximately normal by the CLT" is a statement that is technically defensible for aggregate averages and dangerously wrong for tail risk. Being able to draw that distinction cleanly is the mark of a good answer.

Common interview questions

"How large must n be?" The usual rule of thumb of 30 is for well-behaved distributions. Skewed or heavy-tailed ones need far more, and the honest answer is that it depends on the shape.

"Sum of 12 uniforms?" Approximately normal with mean 6 and variance 1 - historically used to generate normal random numbers.

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Frequently asked questions

What does the central limit theorem say?: That the mean of independent samples from any distribution with finite variance is approximately normal, with standard error sigma / sqrt(n). The underlying distribution can be almost anything.
When does the central limit theorem fail?: When variance is infinite - heavy-tailed distributions like the Cauchy never converge. It also degrades badly under strong dependence, and convergence is slowest in the tails.