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The Standard Normal Distribution

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

The standard normal distribution is a normal distribution of standardized values called z-scores. A z-score is measured in units of the standard deviation. For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean. The calculation is:

x  =  μ  +  ( z ) σ  =  5  +  ( 3 ) ( 2 )  =  11 x = μ + (z)σ = 5 + (3)(2) = 11
(1)

The z-score is 3.

The mean for the standard normal distribution is 0 and the standard deviation is 1. The transformation

z = x - μ σ z= x - μ σ produces the distribution Z Z~ N ( 0 , 1 ) N(0,1) . The value xx comes from a normal distribution with mean μμ and standard deviation σσ.

Glossary

Standard Normal Distribution:
A continuous random variable (RV) X~N(0,1).X~N(0,1). size 12{X "~" N \( 0,1 \) } {}. When X follows the standard normal distribution, it is often noted as Z~N(0,1)Z~N(0,1) size 12{Z "~" N \( 0,1 \) } {}.
z-score:
The linear transformation of the form z=x−μσz=x−μσ size 12{z= { {x-μ} over {σ} } } {}. If this transformation is applied to any normal distribution X~N( μ , σ)X~N( μ , σ) , the result is the standard normal distribution Z~N(0,1)Z~N(0,1) size 12{Z "~" N \( 0,1 \) } {}. If this transformation is applied to any specific value xx size 12{x} {} of the RV with mean μμ size 12{μ} {} and standard deviation σσ size 12{σ} {} , the result is called the z-score of xx size 12{x} {}. Z-scores allow us to compare data that are normally distributed but scaled differently.

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