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Say what the central limit theorem states and what follows from it

CFA® Level I · Quantitative Methods · Estimation and Hypothesis Testing · QM.5.b

The idea

The central limit theorem states that the distribution of a sample mean approaches a normal distribution as sample size grows, whatever the shape of the underlying population, provided its variance is finite. A sample of thirty is the conventional rule of thumb. This is what makes inference possible in practice: an analyst can build a confidence interval or run a hypothesis test about a mean without knowing the population's distribution. The theorem concerns the distribution of the mean, not of the data — a skewed population stays skewed however large the sample.

Formulas

Standard error of the sample mean

σx̄ = σ ÷ √n

σ
the population standard deviation
n
the sample size

The central limit theorem gives this shape whatever the population's own distribution, once n is about 30.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does the central limit theorem describe?

  1. The distribution of the underlying populationWhy not: Sampling does not reshape the population — it is whatever it was before anyone drew from it.
  2. The distribution of the sample mean as sample size grows — correct
  3. The shape of any large datasetWhy not: This is the misreading that matters. A large sample from a skewed population is still skewed; it is the *mean* across repeated samples that converges.

The theorem applies to the sampling distribution of the mean, which tends toward normal regardless of the population's shape. The population itself is unchanged by how it is sampled.

A harder one

Why does the central limit theorem matter for practical inference?

  1. It lets analysts use normal-based tests on a mean without knowing the population's distribution — correct
  2. It guarantees that sample data will be normally distributedWhy not: It guarantees nothing about the data. Believing it does leads directly to underestimating tail risk in return series.
  3. It removes the need to estimate a standard errorWhy not: The standard error is still needed and still estimated. The theorem tells you the *shape* of the sampling distribution, not its width.

Confidence intervals and t-tests about a mean rely on the sampling distribution, not on the data being normal. The theorem supplies exactly that, which is why these methods are so widely applicable.

The hardest tier

An analyst takes a large sample from a heavily skewed population and concludes the data are now approximately normal. What is the error?

  1. The theorem applies only to small samplesWhy not: It applies as the sample grows — large samples are where it works best, not where it stops.
  2. Skewed populations are excluded from the theoremWhy not: Skewed populations are included, provided the variance is finite. They simply need a larger sample before the mean's distribution looks normal.
  3. The theorem concerns the distribution of the sample mean, not of the observations — correct

Sampling does not reshape the population. A large sample from a skewed distribution is still skewed; it is the distribution of its *mean* across repeated samples that approaches normality.

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