ABSTRACT
This chapter is a continuation of Chapter 4 and includes some additional material on process control and related topics� The three types of control charts-X-and R-charts, the P-chart, and the C-chart-that were discussed in Chapter 4 would cover the majority of situations requiring process control in the real world, and the few minor variations of these charts included in the last chapter would cover a few additional, special situations� Many other situations, however, call for even more variations of the basic charts because of practical necessities� In this chapter, we will address some of these special situations and see how special charts are constructed to handle them�
Before we look at these special control charts, we want to discuss how the formulas for the control limits we used for the three major control charts are derived� We also want to study the operating characteristics of these charts, which reveal their strengths and weaknesses in discovering changes in the processes� These topics in the theory of control charts enable a user to understand how the control method works-which in turn helps a user to obtain the maximum benefit out of them-when using them on real processes� Furthermore, many real-world problems do not resemble the simplified textbook versions that can be solved by direct application of the methods� Some do not satisfy the assumptions needed, and some do not yield data in the format needed� It then becomes necessary to modify the basic methods to fit the situation at hand� Such modification of the methods is possible only by a user with a good understanding of the principles behind, and the assumptions that are made, in the derivation of the methods� An understanding of the fundamental theory of the charts is also necessary if one wants to read more advanced technical literature on these topics�
This chapter also includes some additional topics in process capability and experimental design beyond their coverage in earlier chapters� Specifically, in this chapter we will discuss:
• Derivation of the formulas for control limits of X-and R-charts • Derivation of limits for P-and C-charts • Operating characteristic curves of X-and R-charts • Operating characteristic curves of P-and C-charts • Control charts when the standards for μ and/or σ are given
• Control charts for slow processes • The chart for individuals • The moving average and moving range charts
• The exponentially weighted moving average chart • Control charts for short runs • Additional topics in process capability • Additional topics in design of experiments
5.1.1 Limits for the X-Chart
Derivation of the limits for the X-chart is based on the assumption that the process being controlled follows a normal distribution� We saw in Chapter 2 that if a process is normally distributed, the sample average from that process is also normally distributed according to the following law:
If then X N X N
n n~ , , ~ ,µ σ µ σ2
2( )
Furthermore, according to the central limit theorem, even if the process distribution is not normal, the sample averages will tend to be normal as sample sizes become large� That is, if X has any distribution f(x) with mean μ and variance σ2,
then X N
n n n→∞ →
µ
σ ,
Suppose we have a process that is normally distributed with a certain mean and standard deviation as shown in Figure 5�1� When we say that we want to
control this process, it means that we want to make certain the distribution of this process remains the same, with the same mean and the same standard deviation, throughout the time that is of interest to us� To verify this is so, we monitor the X values from samples taken periodically from this process�
If the process distribution remains the same, then the distribution of X will also remain the same and the observed values of X will “all” (99�73% of them) fall within three standard deviations (3σ X) from the mean μ� Conversely, if “all” the observed values of X fall within three standard deviations from the mean, we can conclude the process distribution remains the same� Because we know the X values have the normal distribution, with μX = μ and σ σX n= , the limits at three standard deviations, or the 3-sigma limits for X, are given by (see Figure 5�2):
UCL
CL
LCL
X n
X
X n
( ) = + = + ( ) = ( ) = − = −
µ σ µ σ
µ
µ σ µ σ
3 3
3 3
Because the values of μ and σ are not usually known, we can estimate them by using X and R/d2, respectively, where X is the average of about 25 sample averages and R is the average of about 25 sample ranges from an in-control process� The factor d2 is a correction factor that makes R, or R, an unbiased estimator of σ� In other words, the factor d2 is such the average of (R/d2) denoted as E (R/d2) = σ�
Thus, substituting for μ and σ with estimates, we get:
UCL
CL
LCL
X X R
d n
X X
X X R
d n
( ) = + ( ) = ( ) = −
If we make A d n2 23= ( ), then:
UCL
CL
LCL
( )
( )
( )
X X A R
X X
X X A R
= +
=
= −
Values of A2 are computed for various values of n and provided in standard tables, such as Table A�4 in the Appendix�
From the above derivation, we learn the following:
1� The control limit calculations for the X-chart are based on the assumption that the process is normally distributed�
2� The control limit calculations are good, even for processes that are not normally distributed provided the sample sizes are large� (Even n = 4 or 5 is known to be large enough for this purpose)�
3� A2 provides 3-sigma limits for the X-chart� As mentioned in Chapter 4, Dr� Shewhart, the author of the control chart method, recommended the 3-sigma limits because they provide the economic trade-off between too-tight limits, which would produce excessive false alarms, and too-loose limits, which would allow assignable causes to go undetected� If 2-sigma limits are needed in a particular situation, as when someone wants to detect and eliminate assignable causes quickly, the factor to use would be 2A2/3�
4� Because the control limits are at a 3-sigma distance from the centerline (CL), there is a probability of a false alarm (Type I error) of 0�0027� In other words, the control chart could find a process to be
not-in-control when, in fact, it is in-control� This could happen 27 times out of 10,000 samples, or roughly 3 out of 1000 samples�
5� The control limits have been calculated using estimates for μ and σ computed from the data obtained from the process�
5.1.2 Limits for the R-Chart
The rule on computing the limits for any 3-sigma control chart can be generalized as follows�
If Θ is the statistic that is plotted to control a process parameter θ, then the CL should be at the average of the statistic, or expected value E(Θ), and the UCL and LCL should be at E(Θ) + 3σ(Θ) and E(Θ) − 3σ(Θ), respectively, where σ(Θ) represents the standard deviation of the statistic Θ� In other words, for any control chart, the centerline will be at the average value of the statistic being plotted, and upper and lower control limits will be placed at a distance of 3 × (standard deviation of the statistic) on either side of the centerline� Therefore, for the R-chart, the CL should be at E(R), and the two control limits should be at E(R) ± 3σ(R)�
The statistic w = R/σ, where σ is the process standard deviation, has been studied, and its distribution-along with its mean and standard deviationare known for samples taken from normal populations (Duncan 1974)� The statistic w is called the “relative range�” Specifically, E(w) = E(R/σ) = d2, a constant for a given sample size n� Therefore, E(R) = σd2� Similarly, st� dev� (w) = st� dev� (R/σ) = d3, a constant for a given n, which means that st� dev� (R) = σd3�
Values of d2 and d3 have been tabulated for various sample sizes drawn from normal populations, as in Table A�4 in the Appendix� Using these, the limits for the R-chart can be computed as:
UCL
CL
LCL
R d d d d
R d
R d d d d
( ) = + = +( ) ( ) = ( ) = − = −(
σ σ σ
σ
σ σ σ
3 3
3 3 ) If σ is estimated by R/d2, then:
UCL
CL
LCL
R R Rd d
R d d
R R
R R Rd d
R d
( ) = + = +
( ) =
( ) = − = −
3 1 3
3 1 3
2d
Setting D d d
2 1
3 = +
and D
d d
2 1
3 = −
, we get:
UCL
CL
LCL
R D R
R R
R D R
( ) = ( ) = ( ) =
Values of the constants D3 and D4 have been computed and tabulated for various values of n and are available in tables such as Table A�4 in the Appendix�
From this derivation, we note the following:
1� The control limits for the R-chart have been calculated on the assumption that the samples are drawn from normal populations�
2� The control limits are 3-sigma limits� If, for example, 2-sigma limits are needed, then the limits can be calculated as R(1 ± 2(d3/d2)), using d3 and d2 from Table A�4�
3� The control limits are not equidistant on both sides of the CL, because D3 will be smaller than D4 for all sample sizes�
4� These control limits are based on estimates for σ obtained from process data�
5.1.3 Limits for the P-Chart
To determine the control limits for the P-chart, we should first determine the expected value and the standard deviation of the statistic P that is being plotted�
P = D/n, where D is the number of defective units in a sample of size n drawn from a population with p fraction defectives� From Chapter 2, D is a binomial variable; that is, D ∼ Bi(n, p) and E(D) = np and V(D) = np(1-p)� Therefore,
E P E D n n
E D n np p
V P V D n n
V D n np
( ) = = ( ) = =
( ) = = ( ) =
1 1
1 1 12 2
.
