Normal

The normal cumulative distribution function is defined for all real numbers μ and for positive σ by the integral

$\displaystyle \limfunc$NormalDist$\displaystyle \left(\vphantom{ x;\mu ,\sigma }\right.$x;μ, σ$\displaystyle \left.\vphantom{ x;\mu ,\sigma }\right)$ = $\displaystyle {\frac{{1}}{{\sigma \sqrt{2\pi
}}}}$$\displaystyle \int_{{-\infty }}^{{x}}$e-$\scriptstyle {\frac{{\left( u-\mu \right) ^{2}}}{{2\sigma ^{2}}}}$du

of the normal probability density function

$\displaystyle \limfunc$NormalDen$\displaystyle \left(\vphantom{ u;\mu ,\sigma }\right.$u;μ, σ$\displaystyle \left.\vphantom{ u;\mu ,\sigma }\right)$ = $\displaystyle {\frac{{1}}{{\sigma \sqrt{2\pi }%
}}}$e-$\scriptstyle {\frac{{\left( u-\mu \right) ^{2}}}{{2\sigma ^{2}}}}$

The inverse of the normal cumulative distribution function, $\limfunc$NormalInv, is also available. All three of these functions can be typed in mathematics, and they will automatically turn gray as you type the final letter.

The parameters μ and σ are optional parameters for mean and standard deviation, with the default values 0 and 1 defining the standard normal distribution

$\displaystyle \limfunc$NormalDist$\displaystyle \left(\vphantom{ x}\right.$x$\displaystyle \left.\vphantom{ x}\right)$ = $\displaystyle {\frac{{1}}{{\sqrt{2\pi }}}}$$\displaystyle \int_{{-\infty }}^{{x}}$e-$\scriptstyle {\frac{{u^{2}}}{{2}}}$du

A normal distribution table, as found in the back of a typical statistics book, lists some values of the standard normal cumulative distribution function. Certain versions of the table list the values 1 - $\limfunc$NormalDist$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$.

Note that the function $\limfunc$NormalDist can be evaluated as a function of one variable (with default parameters (0, 1)) or as a function of one variable and two parameters.

$\blacktriangleright$ Evaluate Numerically

Graphs of the normal density functions are the familiar bell-shaped curves. The following plots show the density functions $\func$NormalDen$\left(\vphantom{ x;\mu
,\sigma }\right.$x;μ, σ$\left.\vphantom{ x;\mu
,\sigma }\right)$ and distribution functions $\limfunc$NormalDist$\left(\vphantom{ x;\mu
,\sigma }\right.$x;μ, σ$\left.\vphantom{ x;\mu
,\sigma }\right)$ for the parameters $\left(\vphantom{ \mu ,\sigma }\right.$μ, σ$\left.\vphantom{ \mu ,\sigma }\right)$ = $\left(\vphantom{ 0,1}\right.$0, 1$\left.\vphantom{ 0,1}\right)$,$\left(\vphantom{ 0,5}\right.$0, 5$\left.\vphantom{ 0,5}\right)$,$\left(\vphantom{ 0,.5}\right.$0,.5$\left.\vphantom{ 0,.5}\right)$,$\left(\vphantom{ 1,1}\right.$1, 1$\left.\vphantom{ 1,1}\right)$.

itbpFU2.1577in1.7573in0inNormal density functions itbpFU2.1577in1.7573in0inNormal distribution functions