This example, which is a continuation of Example 5.21, demonstrates techniques for estimating the shape parameter, location and scale parameters, and theoretical percentiles for a lognormal distribution.
See CAPQQ2 in the SAS/QC Sample LibraryThe threeparameter lognormal distribution depends on a threshold parameter , a scale parameter , and a shape parameter . You can estimate from a series of lognormal QQ plots with different values of . The estimate is the value of that linearizes the point pattern. You can then estimate the threshold and scale parameters from the intercept and slope of the point pattern. The following statements create the series of plots in Output 5.22.1 through Output 5.22.3 for values of 0.2, 0.5, and 0.8:
ods graphics off; symbol v=plus; title 'Lognormal QQ Plot for Diameters'; proc capability data=Measures noprint; qqplot Diameter / lognormal(sigma=0.2 0.5 0.8) square; run;
Output 5.22.1: Lognormal QuantileQuantile Plot ( =0.2)
Output 5.22.2: Lognormal QuantileQuantile Plot ( =0.5)
Output 5.22.3: Lognormal QuantileQuantile Plot ( =0.8)
Note: You must specify a value for the shape parameter for a lognormal QQ plot with the SIGMA= option or its alias, the SHAPE= option.
The plot in Output 5.22.2 displays the most linear point pattern, indicating that the lognormal distribution with provides a reasonable fit for the data distribution.
Data with this particular lognormal distribution have the density function
The points in the plot fall on or near the line with intercept and slope . Based on Output 5.22.2, and , giving .
See CAPQQ2 in the SAS/QC Sample LibraryYou can use a QQ plot to estimate percentiles such as the 95th percentile of the lognormal distribution.^{[25]}
The point pattern in Output 5.22.2 has a slope of approximately 0.39 and an intercept of 5. The following statements reproduce this plot, adding a lognormal reference line with this slope and intercept.
ods graphics on; title 'Lognormal QQ Plot for Diameters'; proc capability data=Measures noprint; qqplot Diameter / lognormal(sigma=0.5 theta=5 slope=0.39) pctlaxis(grid) vref = 5.8 5.9 6.0; run;
The ODS GRAPHICS ON statement specified before the PROC CAPABILITY statement enables ODS Graphics, so the QQ plot is created using ODS Graphics instead of traditional graphics. The result is shown in Output 5.22.4.
Output 5.22.4: Lognormal QQ Plot Identifying Percentiles
The PCTLAXIS option labels the major percentiles, and the GRID option draws percentile axis reference lines. The 95th percentile is 5.9, since the intersection of the distribution reference line and the 95th reference line occurs at this value on the vertical axis.
Alternatively, you can compute this percentile from the estimated lognormal parameters. The th percentile of the lognormal distribution is

where is the inverse cumulative standard normal distribution. Consequently,

See CAPQQ2 in the SAS/QC Sample LibraryIf a known threshold parameter is available, you can construct a twoparameter lognormal QQ plot by subtracting the threshold
from the data and requesting a normal QQ plot. The following statements create this plot for Diameter
, assuming a known threshold of five:
data Measures; set Measures; Logdiam=log(Diameter5); label Logdiam='log(Diameter5)'; run;
ods graphics off; symbol v=plus; title 'TwoParameter Lognormal QQ Plot for Diameters'; proc capability data=Measures noprint; qqplot Logdiam / normal(mu=est sigma=est) square; run;
Output 5.22.5: TwoParameter Lognormal QQ Plot for Diameters
Because the point pattern in Output 5.22.5 is linear, you can estimate the lognormal parameters and as the normal plot estimates of and , which are –0.99 and 0.51. These values correspond to the previous estimates of –0.92 for and 0.5 for .