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Central Limit Theorem

Central limit theorem is a statistical theory which states that when a large sample size is having a finite variance, the samples will be normally distributed and the mean of samples will be approximately equal to the mean of whole population. This theory is very important while dealing with stock index. This statistical theory is useful in simplifying analysis.

Statement
Central limit theorem states that if there is a population with mean and standard deviation as $\mu$ and $\sigma$ respectively, then a large sample size "n" taken from that population will be having same mean $\mu$ and standard deviation as $\frac{\sigma}{\sqrt{n}}$.


Properties of Central Limit Theorem
It has following properties -

  • Mean and standard deviation (or variance) of the population is given.
  • Sample size "n" has to be sufficiently large.
  • If probability distribution of population is normal, then the sample will be approximately normally distributed.
Central Limit Theorem Formula
Let us consider a random sample $X_{1}, X_{2}, X_{3}, ..., X_{n}$
of size "n" taken from a population. Each $X_{i}$ is independent and randomly distributed. It has mean as $\mu$ and variance as $\sigma^{2}$, then random normal variable is given by the following formula
$X_{norm}=\frac{s_{n}-n \mu}{\sigma \sqrt{n}}$
Where,
$S_{n}$ = Sample sum = $X_{1}+X_{2}+X_{3}+ ... + X_{n}$

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