You mostly know: the Standard Normal Distribution is the special case of Normal Distribution, given that:

*Mean: Mu=0.0*

*Variance: Sigma^2=1.0*

Cool, as shown below: the family of normal distributions mainly vary in their mean value and/or their variance.

The standard one plays the role of being the reference distribution. Well, we can convert any normal random variable to corresponding interpretations in the standard form. Hence, we simplify different computations only using standard normal distribution.

OK, let's assume

*X*is a normal distribution with mean*Mu*and variance*Sigma^2.*We can convert to a standard normally distributed random variable by following:*Z=(X-Mu)/Sigma*

Here, we got

*Z*as the standard normal distribution. OK, but what does this mean?- Any point in
*X*(with*Mu, Sigma*) can be dealt exactly as the converted point in*Z*with mean*Mu=0.0 and Sigma=1.0*. - The numerator means: how much is the distance or difference between
*X*and the mean*Mu*. The division by*Sigma*means: how many*Sigmas*is that difference? Totally:*Z*means how many*Sigmas*the distance between*X*and*Mu*is. This information is sufficient in further computations using only the standard normal distribution.

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