The UNIVARIATE Procedure
Distributions for Probability and Q-Q Plots
You can use the PROBPLOT and QQPLOT statements to request probability and Q-Q plots that are based on the theoretical distributions summarized in Table 33.
Table 33: Distributions and Parameters
| Parameters | ||||||
|---|---|---|---|---|---|---|
| Distribution | Density Function  | Range | Location | Scale | Shape | |
| Beta |  |  |  |  |  ,  | |
| Exponential |  |  | | | ||
| Gamma |  |  | | | | |
| Gumbel |  | all x |  | | ||
| Lognormal |  | | |  | | |
| (3-parameter) | ||||||
| Normal |  | all x | | | ||
| Generalized |  |  |  | | | |
| Pareto |  |  | ||||
| Power Function |  | | | | | |
| Rayleigh |  | | | | ||
| Weibull |  | | | | c | |
| (3-parameter) | ||||||
| Weibull |  |  |  | | c | |
| (2-parameter) | (known) | |||||
You can request these distributions with the BETA, EXPONENTIAL, GAMMA, PARETO, GUMBEL, LOGNORMAL, NORMAL, POWER, RAYLEIGH, WEIBULL, and WEIBULL2 options, respectively. If you do not specify a distribution option, a normal probability plot or a normal Q-Q plot is created.
The following sections provide details for constructing Q-Q plots that are based on these distributions. Probability plots are constructed similarly except that the horizontal axis is scaled in percentile units.
Beta Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where 
is the inverse normalized incomplete beta function, n is the number of nonmissing observations, and and are the shape parameters of the beta distribution. In a probability plot, the horizontal axis is scaled in percentile units.
The pattern on the plot for ALPHA= and BETA= tends to be linear with intercept and slope if the data are beta distributed with the specific density function
where 
and
lower threshold parameter 
scale parameter 
first shape parameter 
second shape parameter 
Exponential Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where n is the number of nonmissing observations. In a probability plot, the horizontal axis is scaled in percentile units.
The pattern on the plot tends to be linear with intercept and slope if the data are exponentially distributed with the specific density function
where is a threshold parameter, and is a positive scale parameter.
Gamma Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where 
is the inverse normalized incomplete gamma function, n is the number of nonmissing observations, and is the shape parameter of the gamma distribution. In a probability plot, the horizontal axis is scaled in percentile units.
The pattern on the plot for ALPHA= tends to be linear with intercept and slope if the data are gamma distributed with the specific density function
where
- threshold parameter
- scale parameter
- shape parameter
Gumbel Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where n is the number of nonmissing observations. In a probability plot, the horizontal axis is scaled in percentile units.
The pattern on the plot tends to be linear with intercept and slope if the data are Gumbel distributed with the specific density function
location parameter - scale parameter
Lognormal Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where 
is the inverse cumulative standard normal distribution, n is the number of nonmissing observations, and is the shape parameter of the lognormal distribution. In a probability plot, the horizontal axis is scaled in percentile units.
The pattern on the plot for SIGMA= tends to be linear with intercept and slope 
if the data are lognormally distributed with the specific density function
where
- threshold parameter
scale parameter - shape parameter
See Example 3.26 and Example 3.33.
Normal Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where is the inverse cumulative standard normal distribution and n is the number of nonmissing observations. In a probability plot, the horizontal axis is scaled in percentile units.
The point pattern on the plot tends to be linear with intercept and slope if the data are normally distributed with the specific density function
where is the mean and is the standard deviation (
).
Generalized Pareto Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
() or 
(), where n is the number of nonmissing observations and is the shape parameter of the generalized Pareto distribution. The horizontal axis is scaled in percentile units.
The point pattern on the plot for ALPHA= tends to be linear with intercept and slope if the data are generalized Pareto distributed with the specific density function
where threshold parameter scale parameter shape parameter
Power Function Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where 
is the inverse normalized incomplete beta function, n is the number of nonmissing observations, is one shape parameter of the beta distribution, and the second shape parameter, 
. The horizontal axis is scaled in percentile units.
The point pattern on the plot for ALPHA= tends to be linear with intercept and slope if the data are power function distributed with the specific density function
where
- threshold parameter
- shape parameter
- shape parameter
Rayleigh Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where n is the number of nonmissing observations. The horizontal axis is scaled in percentile units.
The point pattern on the plot tends to be linear with intercept and slope if the data are Rayleigh distributed with the specific density function
where is a threshold parameter, and is a positive scale parameter.
Three-Parameter Weibull Distribution
To create the plot, the observations are ordered from smallest to largest, and the ith ordered observation is plotted against the quantile 
, where n is the number of nonmissing observations, and c is the Weibull distribution shape parameter. In a probability plot, the horizontal axis is scaled in percentile units.
The pattern on the plot for C=c tends to be linear with intercept and slope if the data are Weibull distributed with the specific density function
where
- threshold parameter
- scale parameter
shape parameter 
See Example 3.34.
Two-Parameter Weibull Distribution
To create the plot, the observations are ordered from smallest to largest, and the log of the shifted ith ordered observation 
, denoted by 
, is plotted against the quantile 
, where n is the number of nonmissing observations. In a probability plot, the horizontal axis is scaled in percentile units.
Unlike the three-parameter Weibull quantile, the preceding expression is free of distribution parameters. Consequently, the C= shape parameter is not mandatory with the WEIBULL2 distribution option.
The pattern on the plot for THETA= tends to be linear with intercept 
and slope 
if the data are Weibull distributed with the specific density function
where
known lower threshold - scale parameter
- shape parameter
See Example 3.34.