The standard normal distribution has mean
and variance
It is a very convenient property of the normal distribution that any normal distribution, with any mean and any variance, can be transformed onto the standard normal distribution with a simple linear transformation.
If some quantity
is normally distributed with mean
and variance
and the quantity takes values
then the transformation
transforms each value
onto a standard normal score
Every value of
corresponds to a probability
and
We can use the transformation properties of the mean and variance
and
to show that the transformation
transforms
onto
