See CAPBTA2 in the SAS/QC Sample LibraryYou can use a beta distribution to model the distribution of a quantity that is known to vary between lower and upper bounds. In this example, a manufacturing company uses a robotic arm to attach hinges on metal sheets. The attachment point should be offset 10.1 mm from the left edge of the sheet. The actual offset varies between 10.0 and 10.5 mm due to variation in the arm. Offsets for 50 attachment points are saved in the following data set:
title ; data Measures; input Length @@; label Length = 'Attachment Point Offset in mm'; datalines; 10.147 10.070 10.032 10.042 10.102 10.034 10.143 10.278 10.114 10.127 10.122 10.018 10.271 10.293 10.136 10.240 10.205 10.186 10.186 10.080 10.158 10.114 10.018 10.201 10.065 10.061 10.133 10.153 10.201 10.109 10.122 10.139 10.090 10.136 10.066 10.074 10.175 10.052 10.059 10.077 10.211 10.122 10.031 10.322 10.187 10.094 10.067 10.094 10.051 10.174 ;
The following statements create a histogram with a fitted beta density curve:
ods graphics off; legend2 frame cframe=ligr cborder=black position=center; title1 'Fitted Beta Distribution of Offsets'; proc capability data=Measures; specs usl=10.25 lusl=20 cusl=salmon cright=yellow pright=solid; histogram Length / beta(theta=10 scale=0.5 color=blue fill) cfill = ywh cframe = ligr href = 10 hreflabel = 'Lower Bound' lhref = 2 legend = legend2 vaxis = axis1; axis1 label=(a=90 r=0); inset n = 'Sample Size' beta(pchisq = 'PValue') / pos=ne cfill=ywh; run;
The histogram is shown in Output 5.8.1. The THETA= betaoption specifies the lower threshold. The SCALE= betaoption specifies the range between the lower threshold and the upper threshold (in this case, 0.5 mm). Note that in general, the default THETA= and SCALE= values are zero and one, respectively.
Output 5.8.1: Superimposing a Histogram with a Fitted Beta Curve
The FILL betaoption specifies that the area under the curve is to be filled with the CFILL= color. (If FILL were omitted, the CFILL= color would be used to fill the histogram bars instead.) The CRIGHT= option in the SPEC statement specifies the color under the curve to the right of the upper specification limit. If the CRIGHT= option were not specified, the entire area under the curve would be filled with the CFILL= color. When a lower specification limit is available, you can use the CLEFT= option in the SPEC statement to specify the color under the curve to the left of this limit.
The HREF= option draws a reference line at the lower bound, and the HREFLABEL= option adds the label Lower Bound. The option LHREF=2 specifies a dashed line type. The INSET statement adds an inset with the sample size and the pvalue for a chisquare goodnessoffit test.
In addition to displaying the beta curve, the BETA option summarizes the curve fit, as shown in Output 5.8.2. The output tabulates the parameters for the curve, the chisquare goodnessoffit test whose pvalue is shown in Output 5.8.1, the observed and estimated percents above the upper specification limit, and the observed and estimated quantiles. For instance, based on the beta model, the percent of offsets greater than the upper specification limit is 6.6%. For computational details, see the section Formulas for Fitted Curves.
Output 5.8.2: Summary of Fitted Beta Distribution
Fitted Beta Distribution of Offsets 
Parameters for Beta Distribution  

Parameter  Symbol  Estimate 
Threshold  Theta  10 
Scale  Sigma  0.5 
Shape  Alpha  2.06832 
Shape  Beta  6.022479 
Mean  10.12782  
Std Dev  0.072339 
GoodnessofFit Tests for Beta Distribution  

Test  Statistic  DF  p Value  
ChiSquare  ChiSq  1.02463588  3  Pr > ChiSq  0.795 
Percent Outside Specifications for Beta Distribution 


Upper Limit  
USL  10.250000 
Obs Pct > USL  8.000000 
Est Pct > USL  6.618103 
Quantiles for Beta Distribution  

Percent  Quantile  
Observed  Estimated  
1.0  10.0180  10.0124 
5.0  10.0310  10.0285 
10.0  10.0380  10.0416 
25.0  10.0670  10.0718 
50.0  10.1220  10.1174 
75.0  10.1750  10.1735 
90.0  10.2255  10.2292 
95.0  10.2780  10.2630 
99.0  10.3220  10.3237 