March 15, 2006
Demerit Control Charts
A control chart is an important tool in statistical process control. Control charts
provide a way to see whether a process is in control by being able to see if the process is
within the set upper and lower limits. Control charts were created by Walter A. Shewart
in the 1920’s and then picked up and used widely by W. Edward Demings.
The demerit control chart was created by H.F. Dodge while working at Bell
Laboratories as a means to chart products with more than one kind of possible defect.
Some defects had very serious consequences on the performance of the product and some
were not very serious. Dodge classified the various types of defects into four different
categories. Conerly, et al described these different categories as follows:
Class quot;Aquot; Defects-Very Serious.
-Will render unit totally unfit for service.
-Will surely cause operating failure of the unit in service which cannot be readily
corrected on the job.
-Liable to cause personal injury or property damage.
Class quot;Bquot; Defects-Serious.
-Will probably, but not surely cause Class quot;Aquot; operating failure of the unit in
-Will surely cause trouble of a nature less serious than Class quot;Aquot;
-Will surely cause increased maintenance or decreased life.
Class quot;Cquot; Defects-Moderately Serious.
-Will possibly cause operating failure of the unit in service. Likely to cause
trouble of a nature less serious than operating failure.
-Likely to cause increased maintenance or decreased life.
-Major defects of appearance, finish, or workmanship.
Class quot;Dquot; Defects-Not Serious.
-Will not cause operating failure of the unit in service.
-Minor defects of appearance, finish, or workmanship.
The different types of defects first need to be classified as either A, B, C, or D.
Once each defect is classified, a weight is given to each class depending on its severity.
The most common weights used today are 100 for As, 50 for Bs, 10 for Cs, and 1 for Ds.
Demerits are a defect of a product multiplied by its weight.
Steps to Constructing a Demerit Control Chart
1. Calculate the number of demerits per sample. This is done by counting the number of
A, B, C, and D defects and multiplying that number by the weight of each class. If there
were 3 Class A defects then 3 would be multiplied by the Class A weight of 100 for a
total of 300 demerits. Add up the total demerits for each class to get the number of
demerits per inspection sample represented by d. This process is shown in the following
d = 100XA + 50XB + 10XC + XD, where XA is the number of Class A defects
2. Find the average number of demerits per unit, represented by u, by taking the number
of demerits per sample, d, and dividing by the number of units in the sample, N.
3. Find the average number of each class of defect per unit. For Class A, take the total
number of Class A defects in the sample and divide by the number of units in the sample,
N. Repeat for each class.
A= XA / N
4. Calculate the standard deviation through the following formula.
σ = SqRt ((1002 A + 502 B + 102 C + D)/N)
-where A is the average number of Class A defects per unit
-N is the number of units in the sample
5. Calculate the control limits through the following formulas.
Center Line = = 100 A + 50 B + 10 C + D
Upper Control Limit = + 3σ
Lower Control Limit = - 3σ
6. Now that the center line and control limits have been calculated, a control chart can be
constructed. Information from future inspections can be charted using these limits to
monitor the performance of the process and to insure that the process is in control.
The easiest way to learn to construct a demerit control chart is by following an
example using real numbers. For this example, the following information will be used.
Sample size = 1500 Class of Defects
A B C D
Number of Defects 2 5 12 13
First, calculate the total number of demerits in the sample using the formula from step 1.
d = 100(2) + 50(5) + 10(12) + 13 = 583
Next, find the average number of demerits per unit.
u = 583/1500 = .39
Next, find the average number of each class of defect per unit.
A = 2/1500 = .0013
B = 5/1500 = .0033
C = 12/1500 = .008
D = 13/1500 = .009
Next, calculate the standard deviation.
σ = SqRt ((1002(.0013) + 502(.0033) + 102(.008) + .009)/N) = .12
Next, calculate the control limits.
Center Line = 100(.0013) + 50(.0033) + 10(.008) + .009 =.384
UCL = .384 + 3(.12) = .744
LCL = .384 – 3(.12) = .024
A large quantity of information can be found discussing control charts and the
uses of control charts. However, compared to regular control charts, not much has been
published describing the demerit control chart and the reasons for it. The best resources
found are the articles cited in the bibliography of this paper. A demerit chart is a type of
control chart, however, so a lot can be learned about them by reading what is available
about control charts and seeing how a demerit chart can relate.
Conerly, Michael D.; Jones, L Allison,; Woodall, William H. “Exact properties of
demerit control charts.” Journal of Quality Technology. Milwaukee: Apr
1999.Vol.31, Iss. 2; pg. 207, 10 pgs.
Crossley, Mark L. “Weighted Pareto Control Charts Work”
Dawson, Cree S.; McCallum, Jr ,Charles J.; Murphy, R. Bradford; Wolman, Eric.
“Operations Research at Bell Laboratories through the 1970s: Part III.”
Operations Research, Vol. 48, No. 4. (Jul. - Aug., 2000), pp. 517-526.
Dodge, H. F. (1928). quot;A Method of Rating a Manufactured Productquot;. Bell System
Technical Journal 7, pp. 350-368.