The Central Limit Theorem is a profound concept in statistics that makes statistically significant inferences about the general population based on samples. For example, a variable has a normal distribution (or any other type of distribution). One repeatedly takes samples from this general population and calculates the mean of each sample and its standard deviation. The distribution of the variable varies between samples, and sample means also change. There may be both positive and negative differences between sample means and the mean of the general population. However, if one plots the sampling distribution of the sample means, it will look like a normal distribution. At the same time, the mean value of all samples will be close to the mean of the general population.
If one increases the sample size, the distribution will become more normal, and it will be fit tighter around the mean of the general population. In such a case, the estimated mean value of each sample will be more exact. Therefore, if one increases the sample size, significant deviations of sample mean from the general population’s mean will occur less frequently. If the sample size approaches infinity, one will get a perfectly normal distribution. If one takes more samples from the general population, the distribution of these sample means will approximate a normal distribution better.
Based on all the above-said, the Central Limit Theorem can be formulated in the following way. The distribution of sample means taken from the general population will be normal, with the mean being equal to the mean of the general population. The standard deviation of this normal distribution will be less than the standard deviation of the general population. The variance of the sample means decreases as the size and the number of samples increase. However, several criteria should be considered when using the Central Limit Theorem. Firstly, to obtain reasonable conclusions, the sample size should not be small. Secondly, a sample has to be representative, and its elements need to be chosen randomly.
There are essential inferences that can be made when using the Central Limit Theorem. The larger the sample size and the smaller the variable’s variance, the smaller is the standard deviation of the mean. The smaller the standard deviation of the mean, the more rarely sample means will deviate from the mean of the general population. It can be expected that the Central Limit Theorem can be used to predict the probability of any deviation from the mean of the general population. Also, one could note that this theorem may be helpful in hypothesis testing.