Normal Distribution: Standard Normal Distribution 1.5 2008/06/06 14:18:08 GMT-5 2008/10/27 19:08:00.470 GMT-5 Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Connexions cnx@cnx.org elementary statistics 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 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 produces the distribution Z ~ N ( 0 , 1 ) The value x comes from a normal distribution with mean μ and standard deviation σ. Standard Normal Distribution A continuous random variable (RV) 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) size 12{Z "~" N \( 0,1 \) } {}. z-score Let’s consider the linear transformation of the form z=x−μσ size 12{z= { {x-μ} over {σ} } } {}. If this transformation is applied to any normal distribution X~N(μ,σ) size 12{X "~" N \( μ,σ rSup { size 8{} } \) } {} , the result is the standard normal distribution Z~N(0,1) size 12{Z "~" N \( 0,1 \) } {}. If this transformation is applied to any specific value x size 12{x} {} of a Normal RV with mean μ size 12{μ} {} and standard deviation σ size 12{σ} {} , the result is called the z-score of x size 12{x} {}. The z-score allows you to compare data that are normally distributed but scaled differently.