The second part sets the cell equal to +1 if the cumulative binomial distribution of x-1, given a sample of n and a nonconformance rate of p=0.005 is more than 1-a/2. C6-1*(C6>0) simply prevents the use of a negative number in case x=0, in which case the upper control limit is not relevant anyway.
The cell format [Green][<0]“L”;[Red][>0]“H”;”” writes a green L for “low” if the nonconformance count is below the lower control limit, a red H for “high” if it is above the upper control limit and nothing if it is within the control limit.
For example, suppose sample 7 has six (instead of two) nonconformances because of cause A. First consider the upper control limit. Calculations with the true distributions—binomial and Poisson—involve integers so it is important to identify the cutoff point correctly. CumBinomPr(x=5|n=300,p0=0.005) is the 0.9957 quantile of this distribution, so isn’t x=5 out of control if a/2=0.01 in each tail? It is important to realize, however, that a 99.57% chance of getting five or fewer nonconformances is not the same as a 0.43% chance of getting five or more. The graphic, “UCL for A Multiple Attribute Chart,” shows that it is the same thing as a 0.43% chance of getting six or more, so UCL=5.
Because six nonconformances exceed the upper control limit of 5, the entry is flagged as an out of control signal as shown in the table, “Multiple Attribute Control Chart With x>UCL.” In actual practice, of course, investigation and corrective action would be required because the point is evidence that the nonconformance rate now exceeds the historical rate.
There is, incidentally, no lower control limit (LCL) for this entry because CumBinPr(0|300,0.005)=0.2223; there is a 22.23% chance of getting no nonconformances from a sample of 300 if the nonconformance rate is 0.5%.
Now suppose that a self-directed work team makes a process improvement that it expects to reduce the rate for nonconformance B. Also assume that Sample 8 consists of 500 instead of 400 pieces and there were no nonconformances for B. CumBinPr(0|500,0.010)=0.0066 so there is a 0.66% chance (less than
a/2=1%) that getting no rework or scrap in this sample was just luck. The entry is flagged as being below the lower control limit, which shows that the process improvement probably worked. The table, “Multiple Attribute Control Chart With x
The point below the LCL helps verify that the process improvement worked. This supports the “check” part of the plan-do-check-act improvement cycle. After the work team is satisfied that the improvement has been successful, it must hold the gains by making the change permanent in the work standard. The tally sheet arrangement of this control chart supports the requirement to estimate the new nonconformance rate—now less than 1.00%—and re-set the control limits accordingly.
Control chart for defects
Defects are assumed to come from a “random arrival process” for which the Poisson distribution is the correct model. Use the spreadsheet function POISSON(c,m=nu0,1) where c is the defect count, n is the sample size and u0 is the historical defect density. n also can be an area, for example, square meters of plastic, and u0 is in defects per square meter. As with BINOMDIST, the 1 tells the spreadsheet to use the cumulative distribution.
Modern spreadsheets make it easy to use the binomial and Poisson distributions to track nonconformances and defects. The control charts are therefore not subject to the theoretical limitations of traditional attribute control charts that rely on the normal approximations to these distributions. Multiple attribute control charts can track several different problems independently on one worksheet by using the format of the simplest basic quality tool, the tally sheet or check sheet. Q
TECH tIPS
• Spreadsheets allow quality practitioners to not only use the actual binomial or Poisson distribution but also to track multiple problem sources.
• The control charts are not subject to the theoretical limitations of traditional attribute control charts that rely on the normal approximations to these distributions.
• Multiple attribute control charts can track several different problems independently on one worksheet by using the format of the simplest basic quality tool, the tally sheet or check sheet.