The central limit theorem establishes that when independent random variables are summed in many situations, their sum tends to a normal distribution, even if the original variables are not normally distributed. The theorem is a critical concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applied to many problems related to other types of distributions. The larger the sample, the better the replacement of one by the other occurs. Most importantly, this tells us that the average value is equal to the actual average of the population you are trying to estimate. With a small sample, random variables are independent and equally distributed and also have finite variance. A large selection suggests that the distribution, especially of natural biological populations, is often approximately normal since the variation combines many minor effects.

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The central limit theorem is widely applied throughout science – in most empirical sciences in physics, psychology, and economics; this theorem resorts. Every time you see the results of a survey covered in the news along with confidence intervals, this is followed by an appeal to the central limit theorem. The central limit theorem tests the probability of an estimate that anyone makes. As a basis, you can take polls during elections when it is necessary to find out how many people will vote for a particular candidate. It is essential to determine whether it is possible to find out from the result of a small sample whether the result is valid for the entire population. The central limit theorem tells us that if the survey were conducted repeatedly, the assumptions obtained would usually be distributed around the actual value of the people. It follows from this that if candidate A won in the first poll, this candidate would most likely succeed in subsequent polls.