What is the Central Limit Theorem?

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The sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population's shape. The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size n increases, regardless of the shape of the underlying population distribution. This applies even when the population is skewed or non-normal, provided the sample size is sufficiently large (typically n ≥ 30). The mean of the sampling distribution equals the population mean, and the standard deviation equals σ/√n.

The claim that the mean is always at the center of the distribution confuses the CLT with properties of symmetric distributions; the CLT is about the distribution of sample means, not individual data. Stating that large samples always produce the same mean as the population confuses the expected value (which equals the population mean) with individual sample results, which still vary from sample to sample. Saying the center is always the most likely value confuses central tendency with the mode and is not what the CLT addresses.

The Central Limit Theorem is one of the most important theorems in statistics and is heavily tested on the DSST exam. Students should understand why it matters: it allows us to make inferences about population means using normal distribution methods even when the population is not normal, provided the sample size is large enough.

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