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The Central Limit Theorem (CLT) states that if you have a population with mean \( \mu \) and standard deviation \( \sigma \) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30. In fact, even if the population is binomial, as long as the minimum of np and n(1-p) is at least 5, then the sample distribution of the population can also be considered approximately normal.

For instance, if we take a population of individuals with ages ranging from 1 to 99, the distribution of ages may not be a normal distribution. There might be more young people than old people, or vice versa. Now, if we take samples of 50 individuals repeatedly and calculate the mean age of these samples, the distribution of these sample means will form a pattern that looks roughly like a normal distribution. The mean of the sample means will be very close to the mean age of the population. This holds true no matter what the original age distribution looked like.

The CLT is the reason why many statistical procedures work. It forms the backbone of hypothesis testing and the creation of confidence intervals, and it allows us to make inferential statements about a population based on information from a sample. In practical terms, it means that we can use attributes of a sample to make inferences about the population from which it was drawn.