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Normal Distribution: Standard Normal Distribution

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

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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

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.


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 \( 0,1 \) } {}.
Let’s consider the linear transformation of the form z=xmsz=xms size 12{z= { {x-μ} over {σ} } } {}. If this transformation is applied to any normal distribution X~N(μ,σ2)X~N(μ,σ2) size 12{X "~" N \( μ,σ rSup { size 8{2} } \) } {}, 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 RV with mean μμ size 12{μ} {} and standard deviation σσ size 12{σ} {} , the result is called z-score of xx size 12{x} {}. z-score allows to compare data that are normally distributed but scaled differently.

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