The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 {\displaystyle \mu =0} and σ = 1 {\displaystyle \sigma =1}
, and it is described by this probability density function:[1]
φ ( x ) = 1 2 π e − 1 2 x 2 {\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}}
Here, the factor 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}} ensures that the total area under the curve φ ( x ) {\displaystyle \varphi (x)}
is equal to one.[note 1] The factor 1 / 2 {\displaystyle 1/2}
in the exponent ensures that the distribution has unit variance (i.e., variance being equal to one), and therefore also unit standard deviation. This function is symmetric around x = 0 {\displaystyle x=0}
, where it attains its maximum value 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}}
and has inflection points at x = + 1 {\displaystyle x=+1}
and x = − 1 {\displaystyle x=-1}
.
Authors differ on which normal distribution should be called the "standard" one. Carl Friedrich Gauss, for example, defined the standard normal as having a variance of σ 2 = 1 / 2 {\displaystyle \sigma ^{2}=1/2} . That is:
φ ( x ) = e − x 2 π {\displaystyle \varphi (x)={\frac {e^{-x^{2}}}{\sqrt {\pi }}}}
On the other hand, Stephen Stigler[7] goes even further, defining the standard normal as having a variance of σ 2 = 1 / ( 2 π ) {\displaystyle \sigma ^{2}=1/(2\pi )} :
φ ( x ) = e − π x 2 {\displaystyle \varphi (x)=e^{-\pi x^{2}}}
Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor σ {\displaystyle \sigma } (the standard deviation) and then translated by μ {\displaystyle \mu }
(the mean value):
f ( x ? μ , σ 2 ) = 1 σ φ ( x − μ σ ) {\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)}
The probability density must be scaled by 1 / σ {\displaystyle 1/\sigma } so that the integral is still 1.
If Z {\displaystyle Z} is a standard normal deviate, then X = σ Z + μ {\displaystyle X=\sigma Z+\mu }
will have a normal distribution with expected value μ {\displaystyle \mu }
and standard deviation σ {\displaystyle \sigma }
. Conversely, if X {\displaystyle X}
is a normal deviate with parameters μ {\displaystyle \mu }
and σ 2 {\displaystyle \sigma ^{2}}
, then the distribution Z = ( X − μ ) / σ {\displaystyle Z=(X-\mu )/\sigma }
will have a standard normal distribution. This variate is also called the standardized form of X {\displaystyle X}
.
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