Control Charts for Variables

17.4 Control Charts for Variables

Providing an example is a relatively easy way to explain the rudiments of the ¯ X- chart for variables. Suppose that quality control charts are to be used on a process for manufacturing a certain engine part. Suppose the process mean is μ = 50 mm and the standard deviation is σ = 0.01 mm. Suppose that groups of 5 are sampled every hour and the values of the sample mean ¯ X are recorded and plotted on a chart like the one in Figure 17.2. The limits for the ¯ X-charts are based on the standard deviation of the random variable ¯

X. We know from material in Chapter

8 that for the average of independent observations in a sample of size n,

where σ is the standard deviation of an individual observation. The control limits are designed to result in a small probability that a given value of ¯ X is outside the limits given that, indeed, the process is in control (i.e., μ = 50). If we invoke the Central Limit Theorem, we have that under the condition that the process is in control,

X∼N 50, √

As a result, 100(1 − α)% of the ¯ X-values fall inside the limits when the process is in control if we use the limits

LCL = μ − z α/2 √ = 50 − z α/2 (0.0045), UCL = μ + z α/2 √ = 50 + z

α/2 (0.0045). Here LCL and UCL stand for lower control limit and upper control limit, respec-

tively. Often the ¯ X-charts are based on limits that are referred to as “three-sigma” limits, referring, of course, to z α/2 = 3 and limits that become

σ μ±3 √ . n

In our illustration, the upper and lower limits become

UCL = 50 + 3(0.0045) = 50.0135. Thus, if we view the structure of the 3σ limits from the point of view of hypothesis

LCL = 50 − 3(0.0045) = 49.9865,

testing, for a given sample point, the probability is 0.0026 that the ¯ X-value falls outside control limits, given that the process is in control. This is the probability

17.4 Control Charts for Variables 685

0 1 2 3 4 5 6 7 8 9 10 Figure 17.2: The 3σ control limits for the engine part example.

of the analyst erroneously determining that the process is out of control (see Table A.3).

The example above not only illustrates the ¯ X-chart for variables, but also should provide the reader with insight into the nature of control charts in general. The centerline generally reflects the ideal value of an important parameter. Control limits are established from knowledge of the sampling properties of the statistic that estimates the parameter in question. They very often involve a multiple of the standard deviation of the statistic. It has become general practice to use 3σ limits. In the case of the ¯ X-chart provided here, the Central Limit Theorem provides the user with a good approximation of the probability of falsely ruling that the process is out of control. In general, though, the user may not be able to rely on the normality of the statistic on the centerline. As a result, the exact probability of “type I error” may not be known. Despite this, it has become fairly standard to use the kσ limits. While use of the 3σ limits is widespread, at times the user may wish to deviate from this approach. A smaller multiple of σ may be appropriate when it is important to quickly detect an out-of-control situation. Because of economic considerations, it may prove costly to allow a process to continue to run out of control for even short periods, while the cost of the search and correction of assignable causes may be relatively small. Clearly, in this case, control limits that are tighter than 3σ limits are appropriate.

Rational Subgroups

The sample values to be used in a quality control effort are divided into subgroups, with a sample representing a subgroup. As we indicated earlier, time order of pro- duction is certainly a natural basis for selection of the subgroups. We may view the quality control effort very simply as (1) sampling, (2) detection of an out-of-control state, and (3) a search for assignable causes that may be occurring over time. The selection of the basis for these sample groups would appear to be straightforward, but the choice of these subgroups of sampling information can have an important effect on the success of the quality control program. These subgroups are often called rational subgroups. Generally, if the analyst is interested in detecting a

686 Chapter 17 Statistical Quality Control shift in location, the subgroups should be chosen so that within-subgroup variabil-

ity is small and assignable causes, if they are present, have the greatest chance of being detected. Thus, we want to choose the subgroups in such a way as to maximize the between-subgroup variability. Choosing units in a subgroup that are produced close together in time, for example, is a reasonable approach. On the other hand, control charts are often used to control variability, in which case the performance statistic is variability within the sample. Thus, it is more important to choose the rational subgroups to maximize the within-sample variability. In this case, the observations in the subgroups should behave more like a random sample and the variability within samples needs to be a depiction of the variability of the process.

It is important to note that control charts on variability should be established before the development of charts on center of location (say, ¯ X-charts). Any control chart on center of location will certainly depend on variability. For example, we have seen an illustration of the central tendency chart and it depends on σ. In the sections that follow, an estimate of σ from the data will be discussed.

X-Chart with Estimated Parameters ¯

In the foregoing, we have illustrated notions of the ¯ X-chart that make use of the Central Limit Theorem and employ known values of the process mean and standard deviation. As we indicated earlier, the control limits

LCL = μ − z α/2 √ ,

UCL = μ + z α/2 √

n are used, and an ¯ X-value falling outside these limits is viewed as evidence that the

mean μ has changed and thus the process may be out of control. In many practical situations, it is unreasonable to assume that we know μ and σ. As a result, estimates must be supplied from data taken when the process is in control. Typically, the estimates are determined during a period in which background information or start-up information is gathered. A basis for rational subgroups is chosen, and data are gathered with samples of size n in each subgroup. The sample sizes are usually small, say 4, 5, or 6, and k samples are taken, with k being at least 20. During this period in which it is assumed that the process is in control, the user establishes estimates of μ and σ on which the control chart is based. The important information gathered during this period includes the sample means in the subgroup, the overall mean, and the sample range in each subgroup. In the following paragraphs, we outline how this information is used to develop the control chart.

A portion of the sample information from these k samples takes the form

X ¯ 1 ,¯ X 2 ,...,¯ X k , where the random variable ¯ X i is the average of the values in the ith sample. Obviously, the overall average is the random variable

X= ¯¯

k i=1

This is the appropriate estimator of the process mean and, as a result, is the cen- terline in the ¯ X control chart. In quality control applications, it is often convenient

17.4 Control Charts for Variables 687 to estimate σ from the information related to the ranges in the samples rather than

sample standard deviations. Let us define

R i =X max,i −X min,i

as the range for the data in the ith sample. Here X max,i and X min,i are the largest and smallest observations, respectively, in the sample. The appropriate estimate of σ is a function of the average range

R= ¯ 1 R i .

k i=1

An estimate of σ, say ˆ σ, is obtained by

where d 2 is a constant depending on the sample size. Values of d 2 are shown in Table A.22. Use of the range in producing an estimate of σ has roots in quality-control-type applications, particularly since the range was so easy to compute, compared to other variability estimates, in the era when efficient computation was still an issue. The assumption of normality of the individual observations is implicit in the ¯ X- chart. Of course, the existence of the Central Limit Theorem is certainly helpful in this regard. Under the assumption of normality, we make use of a random variable called the relative range, given by

R W= . σ

It turns out that the moments of W are simple functions of the sample size n (see the reference to Montgomery, 2000b, in the Bibliography). The expected value of

W is often referred to as d 2 . Thus, by taking the expected value of W above, we have

E(R) =d 2 . σ

As a result, the rationale for the estimate ˆ σ=¯ R/d 2 is readily understood. It is well known that the range method produces an efficient estimator of σ in relatively small samples. This makes the estimator particularly attractive in quality control applications, since the sample sizes in the subgroups are generally small. Using the range method for estimation of σ results in control charts with the following parameters:

3¯ R UCL = ¯¯ X+ √ ,

3¯ R

centerline = ¯¯ X,

LCL = ¯¯ X− √ .

d 2 n Defining the quantity

688 Chapter 17 Statistical Quality Control we have that

UCL = ¯ X+A ¯ 2 R, ¯

LCL = ¯¯ X−A 2 R. ¯

To simplify the structure, the user of ¯ X-charts often finds values of A 2 tabulated. Values of A 2 are given for various sample sizes in Table A.22.

R-Charts to Control Variation

Up to this point, all illustrations and details have dealt with the quality control analysts’ attempts at detection of out-of-control conditions produced by a shift in the mean. The control limits are based on the distribution of the random variable

¯ X and depend on the assumption of normality of the individual observations. It is important for control to be applied to variability as well as center of location.

In fact, many experts believe that control of variability of the performance char- acteristic is more important and should be established before center of location is considered. Process variability can be controlled through the use of plots of the sample range. A plot over time of the sample ranges is called an R-chart. The same general structure can be used as in the case of the ¯ X-chart, with ¯ R being the centerline and the control limits depending on an estimate of the standard devia- tion of the random variable R. Thus, as in the case of the ¯ X-chart, 3σ limits are established where “3σ” implies 3σ R . The quantity σ R must be estimated from the

data just as σ X ¯ is estimated. The estimate of σ R , the standard deviation, is also based on the distribution of the relative range

R W= . σ

The standard deviation of W is a known function of the sample size and is generally

denoted by d 3 . As a result, σ R = σd 3 . We can now replace σ by ˆ σ=¯ R/d 2 , and thus the estimator of σ R is

Rd ¯ 3

Thus, the quantities that define the R-chart are

UCL = ¯ RD 4 ,

centerline = ¯ R,

LCL = ¯ RD 3 ,

where the constants D 4 and D 3 (depending only on n) are

D 4 =1+3 ,

D 3 =1−3 .

d 2 d 2 The constants D 4 and D 3 are tabulated in Table A.22.

17.4 Control Charts for Variables 689

¯ X- and R-Charts for Variables

A process manufacturing missile component parts is being controlled, with the performance characteristic being the tensile strength in pounds per square inch. Samples of size 5 each are taken every hour and 25 samples are reported. The data are shown in Table 17.1.

Table 17.1: Sample Information on Tensile Strength Data Sample Number

12 As we indicated earlier, it is important initially to establish “in control” condi-

tions on variability. The calculated centerline for the R-chart is

R= ¯ 1 R i = 10.72.

25 i=1

We find from Table A.22 that for n = 5, D 3 = 0 and D 4 = 2.114. As a result, the control limits for the R-chart are

LCL = ¯ RD 3 = (10.72)(0) = 0,

UCL = ¯ RD 4 = (10.72)(2.114) = 22.6621.

690 Chapter 17 Statistical Quality Control The R-chart is shown in Figure 17.3. None of the plotted ranges fall outside the

control limits. As a result, there is no indication of an out-of-control situation.

Figure 17.3: R-chart for the tensile strength example. The ¯ X-chart can now be constructed for the tensile strength readings. The

centerline is

X= ¯ ¯ 1 X ¯ i = 1507.328.

25 i=1

For samples of size 5, we find A 2 = 0.577 from Table A.22. Thus, the control limits are

UCL = ¯ X+A ¯ 2 R = 1507.328 + (0.577)(10.72) = 1513.5134, ¯ LCL = ¯ X−A ¯ 2 R = 1507.328 − (0.577)(10.72) = 1501.1426. ¯

The ¯ X-chart is shown in Figure 17.4. As the reader can observe, three values fall outside the control limits. As a result, the control limits for ¯ X should not be used for line quality control.

Further Comments about Control Charts for Variables

A process may appear to be in control and, in fact, may stay in control for a long period. Does this necessarily mean that the process is operating successfully? A process that is operating in control is merely one in which the process mean and variability are stable. Apparently, no serious changes have occurred. “In control” implies that the process remains consistent with natural variability. Quality control charts may be viewed as a method in which the inherent natural variability governs the width of the control limits. There is no implication, however, to what extent an in-control process satisfies predetermined specifications required of the process. Specifications are limits that are established by the consumer. If the current natural

17.4 Control Charts for Variables 691

Figure 17.4: ¯ X-chart for the tensile strength example.

variability of the process is larger than that dictated by the specifications, the process will not produce items that meet specifications with high frequency, even though the process is stable and in control.

We have alluded to the normality assumption on the individual observations in a variables control chart. For the ¯ X-chart, if the individual observations are normal, the statistic ¯ X is normal. As a result, the quality control analyst has control over the probability of type I error in this case. If the individual X’s are not normal, ¯ X is approximately normal and thus there is approximate control over the probability of type I error for the case in which σ is known. However, the use of the range method for estimating the standard deviation also depends on the normality assumption. Studies regarding the robustness of the ¯ X-chart to departures from normality indicate that for samples of size k ≥ 4 the ¯ X chart results in an α-risk close to that advertised (see the work by Montgomery, 2000b, and Schilling and Nelson, 1976, in the Bibliography). We indicated earlier that the ±kσ R approach to the R-chart is a matter of convenience and tradition. Even if the distribution of individual observations is normal, the distribution of R is not normal. In fact, the distribution of R is not even symmetric. The symmetric control limits of ±kσ R only give an approximation to the α-risk, and in some cases the approximation is not particularly good.

Choice of Sample Size (Operating Characteristic Function) in the Case of the ¯ X-Chart

Scientists and engineers dealing in quality control often refer to factors that affect the design of the control chart. Components that determine the design of the chart include the sample size taken in each subgroup, the width of the control limits, and the frequency of sampling. All of these factors depend to a large extent on economic and practical considerations. Frequency of sampling obviously depends on the cost of sampling and the cost incurred if the process continues out of control for a long period. These same factors affect the width of the “in-control” region. The cost that is associated with investigation and search for assignable causes has an impact

692 Chapter 17 Statistical Quality Control on the width of the region and on frequency of sampling. A considerable amount

of attention has been devoted to optimal design of control charts, and extensive details will not be given here. The reader should refer to the work by Montgomery (2000b) cited in the Bibliography for an excellent historical account of much of this research.

Choice of sample size and frequency of sampling involves balancing available resources allocated to these two efforts. In many cases, the analyst may need to make changes in the strategy until the proper balance is achieved. The analyst should always be aware that if the cost of producing nonconforming items is great,

a high sampling frequency with relatively small sample size is a proper strategy. Many factors must be taken into consideration in the choice of a sample size. In the illustrations and discussion, we have emphasized the use of n = 4, 5, or

6. These values are considered relatively small for general problems in statistical inference but perhaps proper sample sizes for quality control. One justification, of course, is that quality control is a continuing process and the results produced by one sample or set of units will be followed by results from many more. Thus, the “effective” sample size of the entire quality control effort is many times larger than that used in a subgroup. It is generally considered to be more effective to sample frequently with a small sample size.

The analyst can make use of the notion of the power of a test to gain some insight into the effectiveness of the sample size chosen. This is particularly impor- tant since small sample sizes are usually used in each subgroup. Refer to Chapters

10 and 13 for a discussion of the power of formal tests on means and the analysis of variance. Although formal tests of hypotheses are not actually being conducted in quality control, one can treat the sampling information as if the strategy at each subgroup were to test a hypothesis, either on the population mean μ or on the standard deviation σ. Of interest is the probability of detection of an out-of-control condition for a given sample and, perhaps more important, the expected number of runs required for detection. The probability of detection of a specified out-of- control condition corresponds to the power of a test. It is not our intention to show development of the power for all of the types of control charts presented here, but rather to show the development for the ¯ X-chart and present power results for the R-chart.

Consider the ¯ X-chart for σ known. Suppose that the in-control state has μ = μ 0 . A study of the role of the subgroup sample size is tantamount to investigating the β-risk, that is, the probability that an ¯ X-value remains inside the control limits given that, indeed, a shift in the mean has occurred. Suppose that the form the shift takes is

μ=μ 0 + rσ.

Again, making use of the normality of ¯

X, we have

β = P (LCL ≤ ¯ X ≤ UCL | μ = μ 0 + rσ).

For the case of kσ limits,

kσ

kσ

LCL = μ 0 − √

and UCL = μ 0 + √ .

17.4 Control Charts for Variables 693 As a result, if we denote by Z the standard normal random variable,

= P (Z < k − r n) − P (Z < −k − r n). Notice the role of n, r, and k in the expression for the β-risk. The probability of

not detecting a specific shift clearly increases with an increase in k, as expected. β decreases with an increase in r, the magnitude of the shift, and decreases with an increase in the sample size n.

It should be emphasized that the expression above results in the β-risk (prob- ability of type II error) for the case of a single sample. For example, suppose that in the case of a sample of size 4, a shift of σ occurs in the mean. The probability of detecting the shift (power) in the first sample following the shift is (assuming 3σ limits)

1 − β = 1 − [P (Z < 1) − P (Z < −5)] = 0.1587. On the other hand, the probability of detecting a shift of 2σ is

1 − β = 1 − [P (Z < −1) − P (Z < −7)] = 0.8413. The results above illustrate a fairly modest probability of detecting a shift of mag-

nitude σ and a fairly high probability of detecting a shift of magnitude 2σ. The complete picture of how, say, 3σ control limits perform for the ¯ X-chart described here is depicted in Figure 17.5. Rather than plotting the power functions, a plot is given of β against r, where the shift in the mean is of magnitude rσ. Of course, the sample sizes of n = 4, 5, 6 result in a small probability of detecting a shift of 1.0σ or even 1.5σ on the first sample after the shift.

But if sampling is done frequently, the probability may not be as important as the average or expected number of runs required before detection of the shift. Quick detection is important and is certainly possible even though the probability of detection on the first sample is not high. It turns out that ¯ X-charts with these small samples will result in relatively rapid detection. If β is the probability of not detecting a shift on the first sample following the shift, then the probability of detecting the shift on the sth sample after the shift is (assuming independent samples)

P s = (1 − β)β s−1 .

The reader should recognize this as an application of the geometric distribution. The average or expected value of the number of samples required for detection is

Thus, the expected number of samples required to detect the shift in the mean is the reciprocal of the power (i.e., the probability of detection on the first sample following the shift).

694 Chapter 17 Statistical Quality Control

Figure 17.5: Operating characteristic curves for the ¯ X-chart with 3σ limits. Here β is the type II probability error on the first sample after a shift in the mean of rσ.

Example 17.1: In a certain quality control effort, it is important for the quality control analyst to quickly detect shifts in the mean of ±σ while using a 3σ control chart with a sample size n = 4. The expected number of samples that are required following the shift for the detection of the out-of-control state can be an aid in the assessment of the quality control procedure.

From Figure 17.5, for n = 4 and r = 1, it can be seen that β ≈ 0.84. If we allow s to denote the number of samples required to detect the shift, the mean of s is

E(s) =

Thus, on the average, seven subgroups are required before detection of a shift of ±σ.

Choice of Sample Size for the R-Chart

The OC curve for the R-chart is shown in Figure 17.6. Since the R-chart is used for control of the process standard deviation, the β-risk is plotted as a function of

the in-control standard deviation, σ 0 , and the standard deviation after the process goes out of control. The latter standard deviation will be denoted σ 1 . Let

For various sample sizes, β is plotted against λ.

17.4 Control Charts for Variables 695

Figure 17.6: Operating characteristic curve for the R-charts with 3σ limits.

¯ X- and S-Charts for Variables

It is natural for the student of statistics to anticipate use of the sample variance in the ¯ X-chart and in a chart to control variability. The range is efficient as an estimator for σ, but this efficiency decreases as the sample size gets larger. For n as large as 10, the familiar statistic

should be used in the control chart for both the mean and the variability. The reader should recall from Chapter 9 that S 2 is an unbiased estimator for σ 2 but that S is not unbiased for σ. It has become customary to correct S for bias in control chart applications. We know, in general, that

In the case in which the X i are independent and normally distributed with mean

μ and variance σ 2 ,

2 1/2 Γ(n/2)

E(S) = c 4 σ,

where

Γ[(n − 1)/2] and Γ(·) refers to the gamma function (see Chapter 6). For example, for n = 5, √

n−1

c 4 = (3/8) 2π. In addition, the variance of the estimator S is

Var(S) = σ 2 (1 − c 2 4 ).

696 Chapter 17 Statistical Quality Control We have established the properties of S that will allow us to write control limits

for both ¯ X and S. To build a proper structure, we begin by assuming that σ is known. Later we discuss estimating σ from a preliminary set of samples.

If the statistic S is plotted, the obvious control chart parameters are

5 5 UCL = c 4 σ + 3σ 1−c 2 4 , centerline = c 4 σ, LCL = c 4 σ − 3σ 1−c 2 4 .

As usual, the control limits are defined more succinctly through use of tabulated constants. Let

B 6 =c 4 +3 1−c 2 4 , and thus we have

B 5 =c 4 −3 1−c 2 4 ,

UCL = B 6 σ,

LCL = B 5 σ. The values of B 5 and B 6 for various sample sizes are tabulated in Table A.22.

centerline = c 4 σ,

Now, of course, the control limits above serve as a basis for the development of the quality control parameters for the situation that is most often seen in practice, namely, that in which σ is unknown. We must once again assume that a set of base samples or preliminary samples is taken to produce an estimate of σ during what is

assumed to be an “in-control” period. Sample standard deviations S 1 ,S 2 ,...,S m are obtained from samples that are each of size n. An unbiased estimator of the type

c 4 m i=1

is often used for σ. Here, of course, ¯ S, the average value of the sample standard deviation in the preliminary sample, is the logical centerline in the control chart to control variability. The upper and lower control limits are unbiased estimators of the control limits that are appropriate for the case where σ is known. Since

E = σ,

the statistic ¯ S is an appropriate centerline (as an unbiased estimator of c 4 σ) and the quantities

c 4 1−c 4 are the appropriate lower and upper 3σ control limits, respectively. As a result,

the centerline and limits for the S-chart to control variability are

UCL = B 4 S, ¯ where

LCL = B 3 S, ¯

centerline = ¯ S,

B 3 =1−

1−c 2 4 ,

B 4 =1+

1−c 2 4 .

17.5 Control Charts for Attributes 697

The constants B 3 and B 4 appear in Table A.22.

We can now write the parameters of the corresponding ¯ X-chart involving the use of the sample standard deviation. Let us assume that S and ¯ X are available from the base preliminary sample. The centerline remains ¯¯ X and the 3σ limits are merely of the form ¯ ¯

X ± 3ˆσ/ n, where ˆ σ is an unbiased estimator. We simply

supply ¯ S/c 4 as an estimator for σ, and thus we have

UCL = ¯¯ X+A 3 S, ¯ where

LCL = ¯ X−A ¯ 3 S, ¯

centerline = ¯¯ X,

The constant A 3 appears for various sample sizes in Table A.22. Example 17.2: Containers are produced by a process where the volume of the containers is subject

to quality control. Twenty-five samples of size 5 each were used to establish the quality control parameters. Information from these samples is documented in Table

17.2. From Table A.22, B 3 = 0, B 4 = 2.089, and A 3 = 1.427. As a result, the control limits for ¯ X are given by

LCL = ¯ X−A ¯ 3 S = 62.2741, ¯ and the control limits for the S-chart are

UCL = ¯ X+A ¯ 3 S = 62.3771, ¯

UCL = B 4 S = 0.0754. ¯ Figures 17.7 and 17.8 show the ¯ X and S control charts, respectively, for this

LCL = B 3 S = 0, ¯

example. Sample information for all 25 samples in the preliminary data set is plotted on the charts. Control seems to have been established after the first few samples.