Comments:

• In statistics, the Central Limit Theorem is probably the most often used theorem in practice.
• If our random sample (data) comes from a normal distribution, then we know the sum of the data and the sample mean calculated from our data is normal. (Our distribution is exactly normal and our results are exact.) We don’t need to use the Central Limit Theorem.
• Most often we don’t know the distribution from which our data comes. The Central Limit Theorem or variations of it, allow us when we are working with large enough sample sizes to assume the distribution of the sum of the observations or the sample mean are approximately normally distributed.