Central limit theorem (or CLT) is a subject in the probability theory concerned with the conditions and number of independent random variables with finite mean and variance. According to the central limit theorem is any of a set of weak-convergence theories. They all express the fact that a sum of many independent random variables will tend to be distributed according to one of a small set of “attractor” (i.e. stable) distributions. When the variance of the variables is finite, the “attractor” distribution is the normal distribution. Specifically, the sum of a number of random variables with power law tail distributions decreasing as 1/|x|a + 1 where 0 < a < 2 (and therefore having infinite variance) will tend to a stable distribution with stability parameter (or index of stability) of a as the number of variables grows. But here we will discuss on the classical (i.e. finite variance) central limit theorem.
The simplest example of the central limit theorem is the problems of rolling dice, in which each is weighted unfairly and in unknown way. The CLT provides explanation for the prevalence of general probability distribution as well as to justify the approximation of large sample statistics to the normal distribution especially in general experiment.
The central limit theorem is also used to evaluate the reasonable approximation by using the asymptotic distribution and requires a large number of tests and observations. There are alternative statements of the Central limit theorem available which help to solve the problems with the help of variables, samples in a large numbers and various mathematical elements such as characteristic functions, multidimensional central limit theorem and Lindeberg condition.