Transformation to Standard Normal
Given a normally distributed variable X with a population mean of and a population standard deviation of σ
Suppose a normally distributed population has μ=20, σ=5, and we want to know what percentage of the distribution is above X = 30.
This is equivalent to asking how much of the distribution is more than 2 standard deviations above the mean, or what is the probability that X is more than 2 standard deviations above the mean.
From the Z table, we can see that 2.28% of the distribution lies above Z = 2.00. Thus, 2.28% of the population which has a normal distribution with a μ of 20 and a σ of 5 lies above X = 30.
We can write this as
P(X > 30) = P(Z > 2) = 0.0228, or 2.28%