Normal probabilities
We showed on the previous page that all normal distributions have the same basic shape, but with their centre and spread determined by the parameters \(\mu\) and \(\sigma\). The following diagram therefore describes the pdf of any normal distribution.
Observe how the tails of the distribution fade away.
Some probabilities for normal distributions
All normal distributions have the same shape when expressed in terms of standard deviations from the mean. This allows us to give a few probabilities that hold for all normal distributions.
The diagram below shows the probabilities of being within one, two and three standard deviations of the normal distribution's mean graphically.
The next probabilities are particularly important.
In particular, there is a 95% probability that a normally distributed random variable \(X\) is within 1.96 standard deviations of the distribution's mean.
The next diagram shows a few more important normal probabilities.