The Central Limit Theorem justifies an approximate normal distribution for a mean of sufficiently many independent observations from a common population with finite mean and variance. It does not assert that the population is normal. For a sufficiently large sample, \[\overline{X}\approx N\left(\mu,\frac{\sigma^2}{n}\right).\] Keep the word “approximately”. The size needed depends on the population shape; \(n\ge30\) is a common H2 working example, not a universal accuracy guarantee. Bridge the idea Scenario What is justified? 100 independent observations from a right-skewed population An approximate normal sample-mean model may be justified by CLT. The individual observations remain right-skewed. 8 observations; population shape unknown CLT does not supply a reliable large-sample justification from this information. 100 repeated readings affected by the same disturbance Sample size alone does not establish the independence needed for this simple CLT model. Exam wording: “Since the independent random sample is sufficiently large, by the Central Limit Theorem the sample mean is approximately normally distributed. No normality assumption for the population is required.” If the population is already normal, use the exact result instead. Increasing sample size does not repair sampling bias or make dependent observations independent. This card concerns sample means; it is not a justification for adding a normal approximation to a binomial count.

Using the Central Limit Theorem Correctly

Junior College 2

TGM Original Questions

The Central Limit Theorem justifies an approximate normal distribution for a mean of sufficiently many independent observations from a common population with finite mean and variance. It does not assert that the population is normal.

For a sufficiently large sample, \[\overline{X}\approx N\left(\mu,\frac{\sigma^2}{n}\right).\] Keep the word “approximately”. The size needed depends on the population shape; \(n\ge30\) is a common H2 working example, not a universal accuracy guarantee.

Bridge the idea

Scenario	What is justified?
100 independent observations from a right-skewed population	An approximate normal sample-mean model may be justified by CLT. The individual observations remain right-skewed.
8 observations; population shape unknown	CLT does not supply a reliable large-sample justification from this information.
100 repeated readings affected by the same disturbance	Sample size alone does not establish the independence needed for this simple CLT model.

Exam wording: “Since the independent random sample is sufficiently large, by the Central Limit Theorem the sample mean is approximately normally distributed. No normality assumption for the population is required.”

If the population is already normal, use the exact result instead. Increasing sample size does not repair sampling bias or make dependent observations independent. This card concerns sample means; it is not a justification for adding a normal approximation to a binomial count.

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