# Central Limit Theorem

The Central Limit Theorem is a fundamental theorem of probability and describes the characteristics of the population of the means. According the Central Limit Theorem, for simple random samples from any population with finite mean and variance, as n becomes increasingly large, the sampling distribution of the sample means is approximately normally distributed. Here n represents the size of each sample.

The sample mean will be normally distributed regardless of the population distribution, i.e., regardless of the distribution of parent.

The Central Limit Theorem tells us what happens when we have the sum of a large number of independent random variables each of which contributes a small amount to the total.

The central limit theorem has the following characteristics:

• The mean of the sample means is equal to the mean of the population (μ) from which the samples were drawn.
• The standard deviation of the distribution of sample means is s divided by the square root of n (σ/Sqrt(n)). This is also called the standard error of the sample mean.
• For the sampling distribution to approximate a normal distribution, the sample size must be large (n >=30).

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