# CENTRAL LIMIT THEOREM - Research Methodology

When sampling is from a normal population, the means of samples drawn from such a population are themselves normally distributed. But when sampling is not from a normal population sample plays a critical role. When n is small, the shape of the distribution will depend largely on the shape of the parent population, but as n gets large (n > 30), the shape of the sampling distribution will become more and more like a normal distribution, irrespective of the shape of the parent population. The theorem which explains this sort of relationship between the shape of the population distribution and the sampling distribution of the mean is known as the central limit theorem. This theorem is by far the most important theorem in statistical inference. It assures that the sampling distribution of the mean approaches normal distribtion as the sample size increases. In formal terms, we may say that the central limit theorem states that “the distribution of means of random samples taken from a population having mean m and finite variance s 2 approaches the normal distribution with mean m and variance s2 /n as n goes to infinity.”
“The significance of the central limit theorem lies in the fact that it permits us to use sample statistics to make inferences about population parameters without knowing anything about the shape of the frequency distribution of that population other than what we can get from the sample.”

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