Central Limit Theorem

Suppose a researcher selects 100 samples of a specific size from a large population and computes the mean of the sample variable for each of the 100 samples. These sample means, constitute a sampling distribution of sample means.

A sampling distribution of sample mean is a distribution obtained by using the means computed from random samples of a specific size taken from a population.

If the samples are randomly selected with replacement, the sample mean, will somewhat be different from the population mean (µ). These differences are caused by sampling error

Sampling error is the difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.

Properties of the Distribution of sample Means.
1. The mean of the sample means will be the same as the population mean
2. The standard deviation (σ) of the sample means will be smaller than the standard deviation of the population, and it will be equal to the population standard deviation  by the square root of the sample size
3. Central Limit Theorem - as the sample size n without limit, the shape or the distribution of the sample means taken from a population with mean µ and standard deviation σ will approach a normal distribution.


The Central Limit Theorem is valid for any FINITE population when n > 30


Important notes:
1. When the original variable is normally distributed, the distribution of the sample means will be normally distributed, for any sample size n
2. When the distribution of the original variable departs from normality, a sample size of 30 or more is needed to use the normal distribution to approximate the distribution of the sample means. The larger the sample, the better approximation will be.


Formula: X-µ
                σ
              √n


              
~ D. Bertiz