# OpenStax_CNX

You are here: Home » Content » Normal Distribution: Standard Normal Distribution

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

# Normal Distribution: Standard Normal Distribution

Summary: Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

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:
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.

## Content actions

### Give feedback:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks