See CUSONES1 in the SAS/QC Sample LibraryAn alternative to the Vmask cusum chart is the onesided cusum chart with a decision interval, which is sometimes referred to as the “computational form of the cusum chart.” This example illustrates how you can create a onesided cusum chart for individual measurements.
A can of oil is selected every hour for fifteen hours. The cans are weighed, and their weights are saved in a SAS data set
named Cans
:^{[27]}
data Cans; length comment $16; label Hour = 'Hour'; input Hour Weight comment $16. ; datalines; 1 8.024 2 7.971 3 8.125 4 8.123 5 8.068 6 8.177 Pump Adjusted 7 8.229 Pump Adjusted 8 8.072 9 8.066 10 8.089 11 8.058 12 8.147 13 8.141 14 8.047 15 8.125 ;
Suppose the problem is to detect a positive shift in the process mean of one standard deviation () from the target of 8.100 ounces. Furthermore, suppose that
a known value is available for the process standard deviation
an incontrol average run length (ARL) of approximately 100 is required
an ARL of approximately five is appropriate for detecting the shift
Table 6.3 indicates that these ARLs can be achieved with the decision interval h = 3 and the reference value k = 0.5. The following statements use these parameters to create the chart and tabulate the cusum scheme:
options nogstyle; goptions ftext=swiss; symbol v=dot color=salmon h=1.8 pct; title "OneSided Cusum Analysis"; proc cusum data=Cans; xchart Weight*Hour / mu0 = 8.100 /* target mean for process */ sigma0 = 0.050 /* known standard deviation */ delta = 1 /* shift to be detected */ h = 3 /* cusum parameter h */ k = 0.5 /* cusum parameter k */ scheme = onesided /* onesided decision interval */ tableall /* table */ cinfill = ywh cframe = bigb cout = salmon cconnect = salmon climits = black coutfill = bilg; label Weight = 'Cusum of Weight'; run; options gstyle;
The NOGSTYLE system option causes ODS styles not to affect traditional graphics. Instead, the SYMBOL statement, GOPTIONS, and XCHART statement options control the appearance of the graph. The GSTYLE system option restores the use of ODS styles for traditional graphics produced subsequently. The chart is shown in Figure 6.6.
Figure 6.6: OneSided Cusum Chart with Decision Interval
The cusum plotted at Hour
=t is

where , and is the standardized deviation of the tth measurement from the target.

The cusum is referred to as an upper cumulative sum. A shift is signaled at the seventh hour since exceeds h. For further details, see OneSided Cusum Schemes.
The option TABLEALL requests the tables shown in Figure 6.7, Figure 6.8, and Figure 6.9. The table in Figure 6.7 summarizes the cusum scheme, and it confirms that an incontrol ARL of 117.6 and an ARL of 6.4 at are achieved with the specified h and k.
Figure 6.7: Summary Table
Cusum Parameters  

Process Variable  Weight (Cusum of Weight) 
Subgroup Variable  Hour (Hour) 
Scheme  OneSided 
Target Mean (Mu0)  8.1 
Sigma0  0.05 
Delta  1 
Nominal Sample Size  1 
h  3 
k  0.5 
Average Run Length (Delta)  6.40390895 
Average Run Length (0)  117.595692 
The table in Figure 6.8 tabulates the information displayed in Figure 6.6.
Figure 6.8: Tabulation of OneSided Chart
OneSided Cusum Analysis 
Cumulative Sum Chart Summary for Weight  

Hour  Subgroup Sample Size 
Individual Value 
Cusum  Decision Interval 
Decision Interval Exceeded 
1  1  8.0240000  0.0000000  3.0000  
2  1  7.9710000  0.0000000  3.0000  
3  1  8.1250000  0.0000000  3.0000  
4  1  8.1230000  0.0000000  3.0000  
5  1  8.0680000  0.0000000  3.0000  
6  1  8.1770000  1.0400000  3.0000  
7  1  8.2290000  3.1200000  3.0000  Upper 
8  1  8.0720000  2.0600000  3.0000  
9  1  8.0660000  0.8800000  3.0000  
10  1  8.0890000  0.1600000  3.0000  
11  1  8.0580000  0.0000000  3.0000  
12  1  8.1470000  0.4400000  3.0000  
13  1  8.1410000  0.7600000  3.0000  
14  1  8.0470000  0.0000000  3.0000  
15  1  8.1250000  0.0000000  3.0000 
The table in Figure 6.9 presents the computational form of the cusum scheme described by Lucas (1976).
Figure 6.9: Computational Form of Cusum Scheme
OneSided Cusum Analysis 
Computational Cumulative Sum for Weight  

Hour  Subgroup Sample Size 
Individual Value 
Upper Cusum 
Number of Consecutive Upper Sums > 0 
1  1  8.0240000  0.0000000  0 
2  1  7.9710000  0.0000000  0 
3  1  8.1250000  0.0000000  0 
4  1  8.1230000  0.0000000  0 
5  1  8.0680000  0.0000000  0 
6  1  8.1770000  1.0400000  1 
7  1  8.2290000  3.1200000  2 
8  1  8.0720000  2.0600000  3 
9  1  8.0660000  0.8800000  4 
10  1  8.0890000  0.1600000  5 
11  1  8.0580000  0.0000000  0 
12  1  8.1470000  0.4400000  1 
13  1  8.1410000  0.7600000  2 
14  1  8.0470000  0.0000000  0 
15  1  8.1250000  0.0000000  0 
Following the method of Lucas (1976), the process average at the outofcontrol point (Hour
=7) can be estimated as








where is the upper sum at Hour
=7, and is the number of successive positive upper sums at Hour
=7.