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Identify the random object
Imagine taking a fresh sample of n observations repeatedly under the same model. Each repetition produces a mean. The CLT addresses the distribution of those means after centering by the population mean and scaling by their standard error. It does not turn the original observations into normal data.
State the version being used
Here the observations are independent and identically distributed with finite, positive variance. There are other CLTs with different conditions, but they require separate justification. Dependent financial time series and infinite-variance models are not automatically covered by this version.
Keep exact identities distinct from convergence
Under the stated independent common-distribution model, the sample mean has expectation mu and variance sigma²/n for each n. Approximate normal shape is a separate asymptotic conclusion. An unbiased mean can have a very nonnormal sampling distribution at a small n.
| Property | Status under this model |
|---|---|
| Expected sample mean equals mu | Exact |
| Variance equals sigma²/n | Exact |
| Standardized mean approximately normal | Approximation whose quality depends on model and n |
Evaluate a concrete approximation
No single count of thirty observations provides a distribution-free accuracy guarantee. A rare-event example can remain concentrated at zero at n = 30. Look at skewness, tail behavior, dependence and the task’s required precision rather than treating a mnemonic as a theorem.