The theory for the Central Limit Theorem was developed in the previous topic Distribution of the Sample Mean.

The central limit theorem states:

If random samples of size n are taken from ANY distribution with mean μ and variance σ 2then, for large n, (say, over 30), the distribution of the sample mean is approximately normal, with mean μ and standard deviation Y12_The_Central_Limit_Theorem_01.gif

When solving probability problems using the Central Limit Theorem, the z-score formula becomes Y12_The_Central_Limit_Theorem_02.gif


A sample of size 50 is taken from a normally distributed population with a mean of 42 and a standard deviation of 8.

Find the probability that the sample mean is greater than 40.

Distribution of sample means is normal with mean μ = 42 and σ = 8 / √50

Using Central Limit Theorem:


          = 0.5 + 0.4615

          = 0.9615