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.
• Mean of X-bar distribution
  
• Standard Deviation of X-bar distribution(also called “Standard Error”)
• Z- score (in this case, you must use standard deviation of X-Bar)

Example:

This is Parent’s Age with µ = 45.64.  Let’s imagine that this is the true population value. (Data is normal). As the population is normal, the drawn sample will have mean equal to the population mean, and a stand. dev. that decreases as the N of our sample increases

Taking samples from population:

For sample sizes with n > 30, the sample distribution (x-bar distribution) will be normally distributed with mean µ and standard deviation σ/√n.

Note: If we assume that the population is normally distributed, the sample size is not important.

Because of the central limit theorem, we can use the normal table for any random sample over 30 respondents to find the probability of sample averages!