As the sample size from a population become large, the distribution of the sample approaches that of a normal distribution and so the sample distribution can be treated as a normal distribution.

When we make judgment or inference about the sample, we will use the central limit theorem to decide if whether to use a normal approximation if the sample size is large or some other distribution.

1. Know the properties of the Central Limit Theorem.

Let  be data or a random variable of a population with mean,  and variance, , then for large sample of n:

1. The mean of x, is .

2. The standard deviation of x is .

3. The random variable  is approximately normal when n is large (n). The larger the value of n, the closer the distribution of  will approach a normal distribution.
 

Central Limit Theorem: As samples size increases the sample distribution approaches the normal distribution.

2. Know the importance of the central limit theorem in making generalization about the sample distribution.

Since as the sample size increases the random variable regardless of its true distribution, approaches the normal distribution (n >= 30).

Therefore the sample mean and sample standard deviation (standard error of the mean) can be used to approximate the sample distribution.

The mean of x, is .

The standard error of  is .

So when we are working with sample data, a good estimate of the sample standard deviation is given by ..

The z=score can be used to estimate the probability distribution.