What is the difference between normal distribution and standard normal distribution? [migrated]

Two terms normal distribution and standard normal distribution are used in statistics. Does standard term contribute to the normal distribution anything? Please give a simple-however a substantive reasoning in the back of these terms.

Solution 1:

A normal distribution is determined by two parameters the mean and the variance. Often in statistics we refer to an arbitrary normal distribution as we would in the case where we are collecting data from a normal distribution in order to estimate these parameters. Now the standard normal distribution is a specific distribution with mean $0$ and variance $1$. This is the distribution that is used to construct tables of the normal distribution.

Conveniently if $X$ has the a normal distribution with mean $m$ and variance $s^2$ then if we define

$$Z=\frac{X-m}{s}$$ then $Z$ has the standard normal distribution. So for any specific normal distribution we can calculate probabilities of the form $P[a < X < b]$ from the tables for $Z$. This is because we can write $X=sZ+m$. So the standard normal plays a special role with respect to the general family of normal distributions.