Adding a Normal Curve to the Histogram

HISTOGRAM Statement

Adding a Normal Curve to the Histogram

See CAPHST1 in the SAS/QC Sample Library

This example is a continuation of the preceding example.

The following statements fit a normal distribution using the thickness measurements and superimpose the fitted density curve on the histogram:

   title 'Process Capability Analysis of Plating Thickness';
   legend1 frame cframe=ligr cborder=black position=center;
   proc capability data=trans noprint;
      spec lsl = 3.45 llsl = 2 clsl = black
           usl = 3.55 lusl = 2 cusl = black;
      histogram / normal(color=yellow w=3)
                  cfill  = blue
                  frame = ligr
                  legend = legend1;
   run;

The NORMAL option summarizes the fitted distribution in the printed output shown in Figure 4.3, and it specifies that the normal curve be displayed on the histogram shown in Figure 4.4.

The CAPABILITY Procedure

Fitted Normal Distribution for thick

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	3.49533
Std Dev	Sigma	0.032117

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		DF	p Value
Kolmogorov-Smirnov	D	0.05563823		Pr > D	>0.150
Cramer-von Mises	W-Sq	0.04307548		Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.27840748		Pr > A-Sq	>0.250
Chi-Square	Chi-Sq	6.96953022	5	Pr > Chi-Sq	0.223

Percent Outside Specifications for Normal Distribution
Lower Limit		Upper Limit
LSL	3.450000	USL	3.550000
Obs Pct < LSL	8.000000	Obs Pct > USL	5.000000
Est Pct < LSL	7.906248	Est Pct > USL	4.435722

Quantiles for Normal Distribution
Percent	Quantile
Percent	Observed	Estimated
1.0	3.42950	3.42061
5.0	3.44300	3.44250
10.0	3.45750	3.45417
25.0	3.46950	3.47367
50.0	3.49600	3.49533
75.0	3.51650	3.51699
90.0	3.53550	3.53649
95.0	3.55300	3.54816
99.0	3.57200	3.57005

Figure 4.3: Summary for Fitted Normal Distribution

Figure 4.4: Histogram Superimposed with Normal Curve

The printed output includes the following:

parameters for the normal curve. The normal parameters $\mu$ and $\sigma$ are estimated by the sample mean ( $\hat{\mu}=3.49533$ )and the sample standard deviation ( $\hat{\sigma}=0.03211691$ ).
a chi-square goodness-of-fit test. Compared to the usual cutoff values of 0.05 and 0.10, the p-value of 0.2229 for this test indicates that the thicknesses are normally distributed.
goodness-of-fit tests based on the empirical distribution function (EDF): the Anderson-Darling, Cramer-von Mises, and Kolmogorov-Smirnov tests. The p-values for these tests are smaller than the usual cutoff values of 0.05 and 0.10, indicating that the thicknesses are normally distributed.
a chi-square goodness-of-fit test. The p-value of 0.2229 for this test indicates that the thicknesses are normally distributed. In general EDF tests (when available) are preferable to chi-square tests. See the "EDF Goodness-of-Fit Tests" section for details.
observed and estimated percentages outside the specification limits
observed and estimated quantiles

For details, including formulas for the goodness-of-fit tests, see "Printed Output". Note that the NOPRINT option in the PROC CAPABILITY statement suppresses only the printed output with summary statistics for the variable THICK. To suppress the printed output in Figure 4.3, specify the NOPRINT option enclosed in parentheses after the NORMAL option.

The NORMAL option is one of many options that you can specify in the HISTOGRAM statement. See the "Syntax" section for a complete list of options or the "Dictionary of Options" section for detailed descriptions of options.

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