Understanding central limit theorem definition:
Let us define central limit theorem. The arithmetic mean of very large number of unique variables will be distributed in normal way according to the central limit theorem definition. Consider that a particular sample is generated by many observations with each observation being independent of the another one as they are created in random manner. Even if we carry out this procedure frequently, the mean will not be same every time and will be normally distributed.
Central limit theorem formula to find the mean:
The central limit theorem formula is given by µ_{x }= µ and σ_{x} = σ/√n where µ_{x }being the mean of sample and µ_{ }being the mean of population. Also σ_{x} is the standard deviation of given sample while σ_{ }is the standard deviation of population taken for n i.e. size of a sample. This states that if we have a mean and standard deviation of particular population, the following facts are true according to the central limit theorem if a big sample is taken out of whole population,

The sample mean will be equal to population mean

The ratio of standard deviations of sample and population is equal to squareroot of sample size.
Discussion on central limit theorem proof:
We have gone through the formula of limit theorem. Now lets discuss on central limit theorem proof. Combining various statistical properties and logical entities, the limit theorems can be proved easily and all the formulas can be derived.
Solve central limit theorem examples using above formulas:
The central limit theorem is important in statistics because it not only hods true for sample mean but also sample proportion and other sample statistics. Lets go through some central limit theorem examples.
(i) The weights of women population follow normal distribution. The mean is 60 kg while standard deviation is 16 kg. What would be the sample mean and standard deviation for a sample size of 64?
Applying the central limit theorem formula µ_{x }= µ, µ_{x }= 60 i.e. the mean of a sample would be 60 kg. Also σ_{x} = σ/√n. Substituting the values, we have σ_{x} = 16/√64 = 16/8 = 2 i.e. sample standard deviation would be 2 kg. Thus we have founded out sample mean and standard deviation for above central limit theorem example.
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