The Central Limit Theorem states that the distribution of sample means approximates a normal distribution if:

The classical Central Limit Theorem (CLT) states that for a sequence of independent, identically-distributed observations with finite mean µ and variance σ², the standardized sample mean ( x̄ – µ )/(σ/√n) converges in distribution to N(0,1) as the sample size n → ∞. Thus, regardless of the parent population’s shape, the sampling distribution of x̄ becomes approximately normal provided the sample size is sufficiently large. None of the other listed conditions is required by the theorem itself, though random sampling/independence is usually assumed separately.

A. Population normality is unnecessary; CLT works for any distribution with finite variance.
B. “Sample is normally distributed” confuses the statistic x̄ with individual observations; CLT concerns x̄, not each xᵢ.
C. Random selection assures independence but is not the CLT’s defining condition; size, not randomness, drives normality.
E. Accurate arithmetic is assumed but irrelevant to the theorem’s probabilistic result.

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§2.6

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Lecture Notes “The Central Limit Theorem

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Berger

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§7.2

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§8.6

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” Supplemental Notes

Central Limit Theorem section

slide 7.