Central Limit Theorem
One of the most important parts of statistics and probability theory is the Central Limit Theorem.
The central limit theorem states that if some certain conditions are satisfied, then the distribution of the arithmetic mean of a number of independent random variables approaches a normal distribution as the number of variables approaches infinity.
In simple words, If a random sample of n observations is selected from a population (any population), then when n is sufficiently large, the sampling distribution of x will be approximately normal.
(The larger the sample size, the better will be the normal approximation to the sampling distribution of x.)
Here are two main things you should remember and use when proving agreement with the Central Limit Theorem:
1) If the sample size is greater than 30, the Distribution of Sample Means will approximately follow a normal distribution Regardless of the underlying population distribution.
If the underlying population distribution is Normally Distributed, the Distribution of Sample Means will be normally distributed Regardless of the sample sizes used.
2) The mean of the Distribution of Sample Means will be the same as the underlying populations mean i.e. The mean of the sample means will be the population mean µ.
The standard deviation of the Distribution of Sample Means, however, will be the underlying population’s standard deviation divided by the square root of the sample size i.e. the standard deviation of the sample means will approach σ/√n.