QQPLOT Statement: CAPABILITY Procedure

Summary of Theoretical Distributions

You can use the QQPLOT statement to request Q-Q plots based on the theoretical distributions summarized in Table 5.71.

Table 5.71: QQPLOT Statement Distribution Options

     

Parameters

Distribution

Density Function $p(x)$

Range

Location

Scale

Shape

Beta

$\frac{(x-\theta )^{\alpha -1}(\theta +\sigma -x)^{\beta -1}}{B(\alpha ,\beta )\sigma ^{(\alpha +\beta -1)}}$

$\theta < x <\theta +\sigma $

$\theta $

$\sigma $

$\alpha $, $\beta $

Exponential

$\frac{1}{\sigma }\exp \left(-\frac{x-\theta }{\sigma }\right)$

$x \geq \theta $

$\theta $

$\sigma $

 

Gamma

$\frac{1}{\sigma \Gamma (\alpha )} \left(\frac{x-\theta }{\sigma }\right)^{\alpha -1} \exp \left(-\frac{x-\theta }{\sigma }\right)$

$x>\theta $

$\theta $

$\sigma $

$\alpha $

Gumbel

$\frac{e^{-(x-\mu )/\sigma }}{\sigma } \exp \left( -e^{-(x-\mu )/\sigma }\right)$

all x

$\mu $

$\sigma $

 

Lognormal

$\frac{1}{\sigma \sqrt {2\pi }(x-\theta )} \exp \left(-\frac{(\log (x-\theta )-\zeta )^{2}}{2\sigma ^{2}}\right)$

$x>\theta $

$\theta $

$\zeta $

$\sigma $

(3-parameter)

         

Normal

$\frac{1}{\sigma \sqrt {2\pi }} \exp \left(-\frac{(x-\mu )^2}{2\sigma ^{2}}\right)$

all x

$\mu $

$\sigma $

 

Generalized

$\alpha \neq 0$

$\frac{1}{\sigma }{(1 - \alpha (x-\theta )/\sigma )}^{1/\alpha -1}$

$x > \theta $

$\theta $

$\sigma $

$\alpha $

Pareto

$\alpha = 0$

$\frac{1}{\sigma }\exp (-(x-\theta )/\sigma )$

       

Power Function

$\frac{\alpha }{\sigma }\left(\frac{x-\theta }{\sigma }\right)^{\alpha -1}$

$x > \theta $

$\theta $

$\sigma $

$\alpha $

Rayleigh

$\frac{x-\theta }{\sigma ^2}\exp (-(x-\theta )^2/(2\sigma ^2))$

$x \geq \theta $

$\theta $

$\sigma $

 

Weibull

$\frac{c}{\sigma }\left(\frac{x-\theta }{\sigma }\right)^{c-1} \exp \left(-\left(\frac{x-\theta }{\sigma }\right)^{c}\right)$

$x>\theta $

$\theta $

$\sigma $

c

(3-parameter)

         

Weibull

$\frac{c}{\sigma }\left(\frac{x-\theta _0}{\sigma }\right)^{c-1} \exp \left(-\left(\frac{x-\theta _0}{\sigma }\right)^{c}\right)$

$x>\theta _0$

$\theta _0$

$\sigma $

c

(2-parameter)

   

(known)

    


You can request these distributions with the BETA , EXPONENTIAL , GAMMA , LOGNORMAL , NORMAL , WEIBULL , and WEIBULL2 options, respectively. If you do not specify a distribution option, a normal Q-Q plot is created.