Distribution Analyses |

A *quantile-quantile plot* (QQ plot) compares ordered values of a variable with quantiles of a specific theoretical distribution. If the data are from the theoretical distribution, the points on the QQ plot lie approximately on a straight line. The normal, lognormal, exponential, and Weibull distributions can be used in the plot.

You can specify the type of QQ plot from the **QQ Plot** dialog after choosing **Graphs:QQ Plot** from the menu.

**Figure 38.20:** QQ Plot Dialog

In the dialog, you must specify a shape parameter for the lognormal or Weibull distribution. The normal QQ plot can also be generated with the graphs options dialog. As described later in this chapter, you can also add a reference line to the QQ plot from the **Curves** menu.

The following expression is used in the discussion that follows:

*v*_{i}= [(*i*-0.375)/(*n*+0.25)] for i = 1,2, ... ,n

where

For the normal distribution, the *i*th ordered observation is plotted against the normal quantile ,where is the inverse standard cumulative normal distribution. If the data are normally distributed with mean and standard deviation , the points on the plot should lie approximately on a straight line with intercept and slope .The normal quantiles are stored in variables named **N**_* name* for each variable, where

For the lognormal distribution, the *i*th ordered observation is plotted against the lognormal quantile for a given shape parameter .If the data are lognormally distributed with parameters , , and , the points on the plot should lie approximately on a straight line with intercept and slope .The lognormal quantiles are stored in variables named **L_*** name* for each variable, where

For the exponential distribution, the *i*th ordered observation is plotted against the exponential quantile -log(1- *v*_{i}). If the data are exponentially distributed with parameters and , the points on the plot should lie approximately on a straight line with intercept and slope .The exponential quantiles are stored in variables named **E_*** name* for each variable, where

For the Weibull distribution, the *i*th ordered observation is plotted against the Weibull quantile (-log(1- *v*_{i}))^{[1/c]} for a given shape parameter *c*. If the data are from a Weibull distribution with parameters , , and *c*, the points on the plot should lie approximately on a straight line with intercept and slope .The Weibull quantiles are stored in variables named **W_*** name* for each variable, where

A normal QQ plot is shown in Figure 38.21. You can also add a reference line to the QQ plot from the **Curves** menu. You specify the intercept and slope for the reference line from the **Curves** menu.

**Figure 38.21:** Normal QQ Plot

Further information on interpreting quantile-quantile plots can be found in Chambers et al. (1983).

If you select a **Weight** variable, a weighted normal QQ plot can be generated. Lognormal, exponential, and Weibull QQ plots are not computed.

For a weighted normal QQ plot, the *i*th ordered observation is plotted against the normal quantile , where

When each observation has an identical weight, *w*_{(j)}= *w _{0}*, the formulation reduces to the usual expression in the unweighted normal probability plot

*v*_{i}= [(*i*-0.375)/(*n*+0.25)]

If the data are normally distributed with mean and standard deviation and if each observation has approximately the same weight ( *w _{0}*), then, as in the unweighted normal QQ plot, the points on the plot should lie approximately on a straight line with intercept and slope for vardef=WDF/WGT and with slope for vardef=DF/N.

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