AppliedMultiple choiceFree solution

Which statement about the central limit theorem is correct?

  1. AWith enough data, the observations become normally distributed
  2. BThe standardised sample mean converges to a standard normal
  3. CIt applies to absolutely any distribution, with no conditions at all
  4. DIt guarantees the sample mean equals the true mean for large nn
Show the answer and worked solution

Answer: B – The standardised sample mean converges to a standard normal

The theorem is about the sampling distribution of an average, not about the data. Collect a million daily equity returns and they remain fat-tailed and skewed; what converges is n(Xˉ−μ)/σ\sqrt{n}(\bar X - \mu)/\sigma, towards a standard normal. The finite-variance condition matters and is not a technicality: the sample mean of Cauchy draws has exactly the same distribution as a single draw, however many you take. Convergence of the mean to the truth is the law of large numbers, a different and weaker statement.

Worked solution

  1. Formula
    n Xˉn−μσ→dN(0,1)\sqrt{n}\,\frac{\bar X_n - \mu}{\sigma} \xrightarrow{d} \mathcal{N}(0,1)
  2. Substitute
    requires Var⁡(X)<∞\text{requires } \operatorname{Var}(X) < \infty
  3. Solve
    the data keep their own distribution\text{the data keep their own distribution}
    Collecting more never changes it.
  4. Cauchy: Xˉn∼X1 for every n\text{Cauchy: } \bar X_n \sim X_1 \text{ for every } n
    The variance is infinite, so nothing converges.
  5. Answer
    It is about the sampling distribution of the mean\text{It is about the sampling distribution of the mean}

Sanity check. Convergence of the mean to the truth is the law of large numbers, which is a different and weaker statement.

  • A. The data keep whatever distribution they have; collecting more does not change it.
  • B. Correct, and the finite-variance condition is what excludes Cauchy-like tails.
  • C. It needs a finite variance. A Cauchy sample mean has the same distribution as a single draw.
  • D. That is closer to the law of large numbers, and even then it is convergence, not equality.

Takeaway: The CLT is about the distribution of an average, not about the data.

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