This section considers the scenario in which regular samples on measurement data are obtained to monitor process behavior. In the quality area, samples of observations are often referred to as subgroups. For each subgroup, pertinent statistics are computed and then charted over time. For example, the subgroup means can be plotted over time to control the overall mean level of the process, while process variability might be controlled by plotting subgroup standard deviations or a more simplistic statistic known as the range statistic.

The effectiveness of a control chart depends on how the subgroups were collected. Three basic issues need to be addressed:

Once the sampling scheme has been designed in terms of establishing rational subgroups, sample size, and sampling frequency, data are collected to form preliminary subgroups. When first applying control charts to a process, the process behavior is not fully understood. By using a control chart to analyze a given set of initial data, we are looking back retrospectively on the performance of the process. In this phase, control charts are said to be used as a judgment.

If the process is found to be in control, then the control chart can be used pro-spectively to monitor future process performance in real time. In this phase, control charts are said to be used as an ongoing operation. If retrospective analysis shows the process to be not in control, then any numerical descriptions of the data used to construct the initial control chart will not serve as reliable guides for monitoring the process into the future. The priority is then to bring the process into control. This may mean uncovering and removing the unwanted effects of special causes. Or it may mean incorporating the favorable effects of the special causes and stabilizing the process at a more favorable position. Later, when the process has been operating in control for some time and you understand its usual behavior, control charts can be used prospectively.

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We begin with a quantitative variable that is an important measure of quality. The variable might be the diameter of a part or the time to respond to a customer call. Given this quality characteristic is subject to variation, there is a distribution underlying the process. As discussed earlier, an in-control process is a stable process whose underlying distribution remains the same over time. Associated with this distribution is a process mean and a process standard deviation .

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To make subgroup control charts, we begin by taking samples of size from the process at regular intervals. For instance, we might measure four or five consecutive parts or time the responses to four or five consecutive customer calls. For each subgroup, we can compute the sample mean . From Chapter 6, we learned that is a random variable with mean and standard deviation . Furthermore, the central limit theorem tells us that the sampling distribution of the sample mean is approximately Normal.

The 99.7 part of the 68-95-99.7 rule for Normal distributions says that as long as the process remains in control, 99.7% of the values of are expected to fall between:

where UCL and LCL stand for "upper control limit" and "lower control limit," respectively. These limits are called three-sigma control limits and serve as the basis for most control charts. Along with the control limits, it is standard practice to draw a center line (CL) at the mean.

With three-sigma limits, if the process remains in control and the process mean and standard deviation do not change, we will rarely observe an outside the control limits: only about three out of every 1000 sample means, given the assumption of Normality. From that perspective, if we do observe an outside the control limits, it serves as a strong signal that the underlying conditions of the process may have changed.

In practice, and are typically not known and must be estimated. One obvious estimate for the mean is the average of all the individual observations taken from all the preliminary subgroups. If the sample sizes are all equal, then the average of all the individual observations can be computed as the average of the subgroup means. In particular, if we are basing the construction of the control chart on subgroups, then the overall average (referred to as the grand mean) is computed as

We now need an estimate of σ. One possibility is to use the subgroup sample standard deviations. However, in practice, it is more common to use a more simplistic estimate of variation based on the range statistic . The range statistic is simply the sample range, which is the difference between the largest and the smallest observations in a sample. With preliminary subgroups, the average range can be computed as

Based on statistical theory, it can shown that if is multiplied by a constant () that is a function of the subgroup size , we then have a reasonable estimate for . Table 12.1 provides values for for various subgroup sizes.

The control chart for the mean, called an chart, is dedicated to detecting changes in the process level. With the mean and variability estimates, the estimated center line and control limits for the mean are given by

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It is also important to monitor the variability of the process. Because we have the subgroup ranges in hand, it is sensible to develop a control chart for ranges. Such a chart is known as an chart. The center line and control limits of the chart are given by

The control chart constants and are also provided in Table 12.1. On the surface, the chart does not appear to have the same format as the chart in the sense of establishing control limits a certain amount above and below the center line. However, underlying the development of the control chart constants and is a three-sigma structure for the range statistic. You will notice that for subgroup sizes of two to six, the factor is 0, which implies a lower control limit of 0. For small subgroup sizes, it can be shown that the theoretical control limits placed plus/minus three standard deviations of the range statistic above and below the range mean will result in a negative lower control limit. Because the range statistic can never be negative, the lower control limit is accordingly set to 0. Here is a summary of our discussion.

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Before we proceed to the construction of control charts for Case 12.1, we must mention a subtle implementation issue with respect to the and charts. The chart limits rely on the average_range . If process variability is not stable and is affected by special causes, then is not a reliable estimate of variability, and thus, the chart limits are less meaningful. There is a lesson here: it is difficult to interpret an chart unless the ranges are in control. When you look at and charts, always start with the chart.

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Example 12.2 shows that the lab-testing process for the ER is stable and in control. Does this mean the process is acceptable? This is not a statistical question but rather a managerial question for the ER and the hospital administration. Currently, the mean turnaround time is estimated to be 40.62 minutes. The process is stable and in control around that estimated mean. If this process performance is considered acceptable, then the control chart limits for both the chart and the chart can be projected out into the future to monitor the process so as to maintain performance. If, however, the mean time of around 41 minutes is viewed as unacceptably high, then efforts need to be initiated to find ways to improve the process. Here is where the basic tools of flowcharts, Pareto charts, and cause-and-effect diagrams can be used by quality improvement teams to better understand the lab turnaround process and to search out the underlying causes for delay.

When efforts are made to improve a process, the control chart can play an important role. Control charts can be used to judge whether an attempt to improve a process has resulted in a successful change or not. Checking for successful change is a critical aspect of the Check phase of the PDCA cycle (page 593) and the Improve phase of the DMAIC model (page 594). Figure 12.5(a) shows a case where the control chart demonstrates a successful attempt to decrease the turnaround time. However, notice that a three-sigma signal doesn't occur until subgroup 41.

So far, we have considered only the basic "one point beyond the control limits" criterion to signal that a process may have gone out of control. We would like a quick signal when the process has changed from its original in-control state, but we also want to avoid "false alarms," signals that occur just by chance when the process hasn't changed. If the shift in the process is small to moderate, it may take some time before a subgroup falls outside the control limits. We can speed the response of a control chart to process change—at the cost of also enduring more false alarms—by adding patterns other than "one-point-out" as signals. The most common step in this direction is to add a runs signal to the control chart.

There are numerous types of runs signals available with software. From Figure 12.5(a), we find that subgroups 34 and 35 have been highlighted. Both of these subgroups are beyond two standard deviations from the center line. Minitab uses a rule that signals if two out of three consecutive observations are more than two standard deviations from the center line (same side), while JMP signals if two consecutive observations are more than two standard deviations from the center line (same side). In either case, we have a signal of a possible change in the process earlier than the three-sigma signal at subgroup 41.

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In light of the evidence of changed process with Figure 12.5(a), control chart limits should be revised, and the new process should be monitored to maintain the gains, as called for in the "Control" phase of the DMAIC model.

In contrast to Figure 12.5(a), if sample means after an attempt to improve are as seen in Figure 12.5(b), then the control chart provides no evidence of an impact from the attempted process improvement; the organization should seek alternative improvement ideas.

The preliminary samples of Example 12.2 appear to come from an in-control process. Let us now consider control chart implementation issues when there is evidence of special cause effects in the preliminary samples.

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and charts

In the construction of subgroup control charts, the use of the simplistic range statistic instead of the sample standard deviation statistic is a historical artifact from when calculations were done by hand. Given the availability of computer software to do calculations, the need for computational simplicity is no longer a compelling argument. The fact that the range statistic is still in widespread use is probably due to training issues. It is much easier for corporate trainers to explain and for employees to comprehend the range statistic (largest minus smallest observation) than a statistic based on summing squared deviations, dividing the sum by , and then taking a square root!

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The primary advantage of the sample standard deviation is that it uses all the data as opposed to the range statistic, which utilizes only two observation values (largest and smallest). For small subgroup sizes (), the range statistic competes well with the sample standard deviation statistic. However, for larger subgroup sizes, it is generally advisable to utilize the more efficient sample standard deviation.⁵

When using sample standard deviations, the chart is replaced by an chart. The chart is a plot of the subgroup standard deviations with appropriate control limits superimposed. In addition, the construction of the chart is based on the subgroup standard deviation values, not the range values. There is no difference in the calculation of the grand mean (), but we need to calculate the average sample standard deviation from the preliminary samples:

Here is a summary of how to construct and charts.

Charts for individual observations

Up to this point we have concentrated on the application of control charts to statistics based on samples of two or more observations. There are, however, many applications where it is not practical to gather a sample of observations at a given time. For example, in low-volume manufacturing environments, the production rate is often slow, and therefore, the time between measurements is too long to allow rational subgroups to be formed. Some processes are just naturally viewed as a series of individual measurements. Data arising once a day, once a week, or once every two weeks do not lend themselves to being grouped into subgroups. Weekly sales or inventory levels are common company performance measurements monitored as individual observations.

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A series of individual observations can be viewed as a special case of the chart with . For an in-control process with mean and standard deviation , the sample mean control chart with known parameters and (page 602) is

If the goal is to maintain the process at a target level mean of , then is used to establish the control limits. In other cases, we need to estimate . For a set of consecutive observations , the estimate of is simply given by the sample mean .

In the area of quality control, several estimators have been developed in the estimation of . Interestingly, the estimator used by many statistical software packages is not based on the sample standard deviation. Instead, a more common alternative estimate of process variability is based on the variability observed between successive observations. For a series of observations, we define moving ranges as

for . The moving range statistic is simply a special case of the general range statistic , which is the largest observation minus the smallest observation in a given sample. In this case, successive observations are paired to form samples of size 2, allowing for the range to simply be computed as the absolute value of the difference between the two observations. For a series of observations, there will be a series of moving ranges. As a result, the mean moving range is given by:

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By using the mean moving range, it can be shown that an unbiased estimate for the process standard deviation is given by , where the value of depends on the number of observations used in determining the individual ranges, which in this case is 2. From Table 12.1 (page 603), we find that . The control limits based on are given by

A control chart based on these limits is known as an individuals () chart.

In conjunction with the individuals chart, some practitioners will plot the moving ranges with limits for the detection of changes in process variability. The moving range limits are simply the earlier R chart limits with used in place of :

Referring to Table 12.1 (page 603), we find that for ranges determined from two observations and . The result limits are then:

A plot of the moving ranges with the preceding limits is known as as a moving-range (MR) chart. Here is a summary of how to construct and charts.

The individuals chart, like other control charts, is a three-sigma control chart with interpretation grounded in the Normal distribution. Unlike the mean chart based on sample sizes greater than one, the individuals chart does not have the advantage of the central limit theorem effect. Checking for Normality of the individual observations then becomes a particularly important step in the implementation of the individuals chart. Figure 12.12 shows the Normal quantile plot for the points per minute data. Aside from the one noted outlier, we see that the data are compatible with the Normal distribution.

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Don't confuse control with capability!

A process in control is stable over time. With an in-control process, we can predict the mean and the amount of variation the process will show. Control charts are, so to speak, the voice of the process telling us what state it is in. There is no guarantee that a process in control produces products of satisfactory quality. “Satisfactory quality” is measured by comparing the product or service to some standard outside the process, set by technical specifications, customer expectations, or the goals of the organization. These external standards are unrelated to the internal state of the process, which is all that statistical control pays attention to.

Capability has nothing to do with control—except for the very important point that, if a process is not in control, it is hard to tell if it is capable or not.

Managers must understand that, if a process that is in control does not have adequate capability, fundamental changes in the process are needed. The process is doing as well as it can and displays only the chance variation that is natural to its present state. Slogans to encourage the workers or disciplining the workers for poor performance will not change the state of the process. Better training for workers is a change in the process that may improve capability. New equipment or more uniform material may also help, depending on the findings of a careful investigation.

Figure 12.13 gives us a visual summary of the capability of the process, but managers often like a numerical summary of capability. One measure of capability is simply the percent of process outcomes that meet the specifications. When the variable we measure has a Normal distribution, we can use the estimated mean and standard deviation along with the Normal distribution to estimate this percent. When the variable is not Normal, we can use the actual percents of the measurements in the samples that meet the specifications.

There is a subtle point when it comes to estimating the standard deviation of the process. When performing process capability analysis in conjunction with subgroup charts, we could derive an estimate for based on the subgroup variability estimates or . However, these estimates are based solely on the within-subgroup variation. An alternative method of estimating the process standard deviation is to ignore the sub-grouping of the data and to simply estimate the standard deviation from the single set of all individual observations. If the process is out of control, then the single-set estimate will be larger than the subgroup-based estimates because the single-set estimate is capturing the extra variation associated with the mean shifts occurring between the subgroups. It makes little sense to summarize process capability on an out-of-control process. The summarization of process capability should occur only after the process is brought to a stable state. When the process is in control, the issue of whether to base capability analysis on a within-subgroup estimate or a single-set estimate becomes a moot point since either type will produce nearly identical results.

When performing capability analysis on a series of individual measurements in conjunction with an chart, software offers many options for estimating . One option is to simply use the sample standard deviation . It can be shown, however, that the sample standard deviation is a biased estimate of . The corrected estimate is

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For series of more than 25 observations, is well approximated⁸ by

where is the number of individual observations. In the end, for most reasonably lengthed series, the differences between and are fairly inconsequential.

Even though percent meeting specifications seems to be a reasonable measure of process capability, there are some situations that can call into question its appropriateness. Figure 12.14 shows why. This figure compares the distributions of the diameter of the same part manufactured by two processes. The target diameter and the specification limits are marked. All the parts produced by Process A meet the specifications, but about 1.5% of those from Process B fail to do so. Nonetheless, Process B is superior to Process A because it is less variable: much more of Process B's output is close to the target. Process A produces many parts close to the lower specification limit (LSL) and the upper specification limit (USL). These parts meet the specifications, but they will fit and perform more poorly than parts with diameters close to the center of the specifications. A distribution like that for Process A might result from inspecting all the parts and discarding those whose diameters fall outside the specifications. That's not an efficient way to achieve quality.

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We need a way to measure process capability that pays attention to the variability of the process (smaller is better). The standard deviation does that, but it doesn't measure capability because it takes no account of the specifications that the output must meet. Capability indices start with the idea of comparing process variation with the specifications. Process B will beat Process A by such a measure. Capability indices also allow us to measure process improvement—we can continue to drive down variation, and so improve the process, long after 100% of the output meets specifications. Continual improvement of processes is our goal, not just reaching “satisfactory” performance. The real importance of capability indices is that they give us numerical measures to describe ever-better process quality. Statistical software offers many capability indices, but we consider only the most basic ones.

Capability indices start from the fact that Normal distributions are in practice about 6 standard deviations wide. That's the 99.7 part of the 68-95-99.7 rule. Conceptually, is the specification width as a multiple of the process width 6a. When , the process output will just fit within the specifications if the center is midway between LSL and USL. Larger values of are better—the process output can fit within the specs with room to spare. But a process with high can produce poor-quality product if it is not correctly centered. As we see with the next example, remedies this deficiency by considering both the center and the variability of the measurements.

Example 12.8 shows that, as a process moves off target, remains constant while decreases in value. Off-target processes have the potential of having higher capability if the mean is adjusted to the center of the specs. For these reasons, can be viewed as a measure of process potential while attempts to measure actual process performance.

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Our discussion of capability indices has focused on capability relative to two specification limits. When there is only one specification limit involved, we can define one-sided indices as follows:

The change in denominator from 6 to 3 standard deviations reflects the focus on one specification rather than two.

We end our discussion on process capability indices on two cautionary notes. First, their interpretation is based on the assumption that the individual process measurements are Normally distributed. It is hard to interpret indices when the measurements are strongly non-Normal. It is best to apply capability indices only when a Normal quantile plot or histogram shows that the distribution is at least roughly Normal. Second, as we saw with Examples 12.9 and 12.10, process indices need to be estimated from the process data in hand. The implication is that estimated indices are statistics and thus are subject to sampling variation. A supplier under pressure from a large customer to measure often may base calculations on small samples from the process. The resulting estimate can differ greatly from the true process in either direction. As a rough rule of thumb, it is best to rely on indices computed from samples of at least 50 measurements.