What is Statistical Process Control? Chart, Process Capabilities

Statistical Process Control (SPC) is a methodology used in quality control and manufacturing to monitor, control, and improve processes. It involves using statistical methods to analyze and understand the variation within a process, with the aim of ensuring that the process operates efficiently and consistently, producing products or services that meet predefined quality standards.

Process control is achieved by taking periodic samples from the process and plotting these sample points on a chart, to see if the process is within statistical control limits. A sample can be a single item or a group of items.

If a sample point is outside the limits, the process may be out of control, and the cause is sought so that the problem can be corrected. If the sample is within the control limits, the process continues without interference but with continued monitoring. In this way, SPC prevents quality problems by correcting the process before it starts producing defects.

No production process produces identical items, one after the other. All processes contain a certain amount of variability that makes some variation between units inevitable. There are two reasons why a process might vary. The first is the inherent random variability of the process, which depends on the equipment and machinery, engineering, the operator, and the system used for measurement.

This kind of variability is a result of natural occurrences. The other reason for variability is unique or special causes that are identifiable and can be corrected. These causes tend to be non-random and, if left unattended, will cause poor quality. These might include equipment that is out of adjustment, defective materials, changes in parts or materials, broken machinery or equipment, operator fatigue poor work methods, or errors due to lack of training.

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The Basics of Statistical Process Control

SPC in Quality Management

Companies use SPC to see if their processes are in control—working properly. This requires that companies provide SPC training continuously which stresses that SPC is a tool individuals can use to monitor production or service processes to make improvements. Through the use of statistical process control, employees can be made responsible for quality in their area: to identify problems and either correct them or seek help in correcting them. By continually monitoring the production process and making improvements, the employee contributes to the goal of continuous improvement and few or no defects.

The first step in correcting the problem is identifying the causes. We have several quality-control tools used for identifying causes of problems, including brainstorming, Pareto charts, histograms, checksheets, quality circles, and fishbone (cause-and-effect) diagrams.

When an employee is unable to correct a problem, management typically initiates problem-solving. This problem-solving activity may be within a group like a quality circle, or it may be less formal, including other employees, engineers, quality experts, and management. This group will brainstorm the problem to seek out possible causes. Alternatively, quality problems can be corrected through Six Sigma projects.

Quality Measures: Attributes and Variables

The quality of a product or service can be evaluated using either an attribute of the product or service or a variable measure. An attribute is a product characteristic such as color, surface texture, cleanliness, or perhaps smell or taste. Attributes can be evaluated quickly with a discrete response such as good or bad, acceptable or not, or yes or no. Even if quality specifications are complex and extensive, a simple attribute test might be used to determine whether or not a product or service is defective. For example, an operator might test a light bulb by simply turning it on and seeing if it lights. If it does not, it can be examined to find out the exact technical cause for failure, but for SPC purposes, the fact that it is defective has been determined.

A variable measure is a product characteristic that is measured on a continuous scale such as length, weight, temperature, or time. For example, the amount of liquid detergent in a plastic container can be measured to see if it conforms to the company’s product specifications. The time it takes to serve a customer at McDonald’s can be measured to see if it is quick enough.

Since a variable evaluation is the result of some form of measurement, it is sometimes referred to as a quantitative classification method. An attribute evaluation is sometimes referred to as a qualitative classification since the response is not measured. Because it is a measurement, a variable classification typically provides more information about the product—the weight of a product is more informative than simply saying the product is good or bad.

SPC Applied to Services

Control charts have historically been used to monitor the quality of manufacturing processes. However, SPC is just as useful for monitoring the quality of services as shown in the photo. The difference is the nature of the “defect” being measured and monitored. Using Motorola’s definition—a failure to meet customer requirements in any product or service—a defect can be an empty soap dispenser in a restroom or an error with a phone catalog order, as well as a blemish on a piece of cloth or a faulty tray on a DVD player. Control charts for service processes tend to use quality characteristics and measurements such as time and customer satisfaction (determined by surveys, questionnaires, or inspections). Following is a list of several different services and the quality characteristics for each that can be measured and monitored with control charts:

• Hospitals: Timeliness and quickness of care, staff responses to requests, the accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts.
  
  
• Grocery stores: Waiting time to check out, frequency of out-of-stock items, quality of food items, cleanliness, customer complaints, checkout register errors.
  
  
• Airlines: Flight delays, lost luggage and luggage handling, waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness, and maintenance.
  
  
• Fast-food restaurants: Waiting time for service, customer complaints, cleanliness, food quality, order accuracy, employee courtesy.
  
  
• Catalog-order companies: Order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time.
  
  
• Insurance companies: Billing accuracy, timeliness of claims processing, agent availability, and response time.

Where to Use Control Charts

Most companies do not use control charts for every step in a process. Although that might be the most effective way to ensure the highest quality, it is costly and time-consuming. In most manufacturing and service processes, there are identifiable points where control charts should be used. In general, control charts are used at critical points in the process where historically the process has shown a tendency to go out of control, and at points where if the process goes out of control it is particularly harmful and costly.

For example, control charts are frequently used at the beginning of a process to check the quality of raw materials and parts, or supplies and deliveries for a service operation. If materials and parts are bad, to begin with, it is a waste of time and money to begin the production process with them. Control charts are also used before a costly or irreversible point in the process, after which the product is difficult to rework or correct; before and after assembly or painting operations that might cover defects; and before the outgoing final product or service is shipped or delivered.

Control Charts

Control charts are graphs that visually show if a sample is within statistical control limits. They have two basic purposes: to establish the control limits for a process and then to monitor the process to indicate when it is out of control. They exist for attributes and variables; within each category, there are several different types of control charts. We will present four commonly used control charts, two in each category: p-charts and c-charts for attributes and mean (x –-) and range (R) control charts for variables. Even though these control charts differ in how they measure process control, they all have certain similar characteristics. They all look alike, with a line through the center of a graph that indicates the process average and lines above and below the center line that represent the upper and lower limits of the process.

The formulas for conducting upper and lower limits in control charts are based on several standard deviations, z, from the process average (e.g., center line) according to a normal distribution. Occasionally, z is equal to 2.00 but most frequently is 3.00. A z value of 2.00 corresponds to an overall normal probability of 95%, and z = 3 . 00 corresponds to a normal probability of 99.73%.

The normal distribution in Figure 10.2 below shows the probabilities corresponding to z values equal to 2.00 and 3.00 standard deviations (σ). The smaller the value of z, the more narrow the control limits are and the more sensitive the chart is to changes in the production process. Control charts using z = 2.00 are often referred to as having 2-sigma (2σ) limits (referring to two standard deviations), whereas z = 3.00 means 3-sigma (3σ) limits.

Management usually selects z = 3.00 because if the process is in control it wants a high probability that the sample values will fall within the control limits. In other words, with wider limits management is less likely to (erroneously) conclude that the process is out of control when points outside the control limits are due to normal, random variations. Alternatively, wider limits make it harder to detect changes in the process that are not random and have an assignable cause. A process might change because of a non-random, assignable cause and be detectable with the narrower limits but not with the wider limits. However, companies traditionally use wider control limits.

Each time a sample is taken, the mathematical average of the sample is plotted as a point on the control chart. A process is generally considered to be in control if, for example,

• There are no sample points outside the control limits.
  
  
• Most points are near the process average (i.e., the center line), without too many close to the control limits.
  
  
• Approximately equal numbers of sample points occur above and below the center line.
  
  
• The points appear to be randomly distributed around the center line (i.e., no discernible pattern).

If any of these conditions are violated, the process may be out of control. The reason must be determined, and if the cause is not random, the problem must be corrected.

above the upper control limit, suggesting the process is out of control (i.e., something unusual has happened). The cause is not likely to be random since the sample points have been moving toward the upper limit, so management should attempt to find out what is wrong with the process and bring it back under control. Perhaps the employee was simply interrupted.

Although the other samples display some degree of variation from the process average, they are usually considered to be caused by normal, random variability in the process and are thus in control. However, sample observations can be within the control limits and the process to be out of control anyway if the observations display a discernible, abnormal pattern of movement. We discuss such patterns in a later section.

After a control chart is established, it is used to determine when a process goes out of control and corrections need to be made. As such, a process control chart should be based only on sample observations from when the process is in control so that the control chart reflects a true benchmark for an in-control process. However, it is not known whether the process is in control until the control chart is first constructed. Therefore, when a control chart is first developed if the process is found to be out of control, the process should be examined and corrections made.

A new center line and control limits should then be determined from a new set of sample observations. This “corrected” control chart is then used to monitor the process. It may not be possible to discover the cause(s) for the out-of-control sample observations. In this case, a new set of samples is taken, and a completely new control chart is constructed. Or it may be decided simply to use the initial control chart, assuming that it accurately reflects the process variation.

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Control Charts for Attributes

The quality measures used in attribute control charts are discrete values reflecting a simple decision criterion such as good or bad. A p-chart uses the proportion of defective items in a sample as the sample statistic; a c-chart uses the actual number of defects per item in a sample. A p-chart can be used when it is possible to distinguish between defective and non-defective items and to state the number of defectives as a percentage of the whole. In some processes, the proportion of defective cannot be determined. For example, when counting the number of blemishes on a roll of upholstery material (at periodic intervals), it is not possible to compute a proportion. In this case, a c-chart is required.

P-chart

With a p-chart, a sample of n items is taken periodically from the production or service process, and the proportion of defective items in the sample is determined to see if the proportion falls within the control limits on the chart. Although a p-chart employs a discrete attribute measure (i.e., number of defective items) and thus is not continuous, it is assumed that as the sample size (n) gets larger, the normal distribution can be used to approximate the distribution of the proportion defective. This enables us to use the following formulas based on the normal distribution to compute the upper control limit (UCL) and lower control limit (LCL) of a p-chart:

where n is the sample size.

This initial control chart shows two out-of-control observations and a distinct pattern of increasing defects. Management would probably want to discard this set of samples and develop a new center line and control limits from a different set of sample values after the process has been corrected. If the pattern had not existed and only the two out-of-control observations were present, these two observations could be discarded, and a control chart could be developed from the remaining sample values.

Once a control chart is established based solely on natural, random variation in the process, it is used to monitor the process. Samples are taken periodically, and the observations are checked on the control chart to see if the process is in control.

C-chart

A c-chart is used when it is not possible to compute a proportion defective and the actual number of defects must be used. For example, when automobiles are inspected, the number of blemishes (i.e., defects) in the paint job can be counted for each car, but a proportion cannot be computed, since the total number of possible blemishes is not known. In this case, a single car is the sample. Since the number of defects per sample is assumed to derive from some extremely large population, the probability of a single defect is very small.

As with the p-chart, the normal distribution can be used to approximate the distribution of defects. The process average for the c-chart is the mean number of defects per item, c, computed by dividing the total number of defects by the number of samples. The sample standard deviation, σ c, is c. The following formulas for the control limits are used:

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Control Charts for Variables

Variable control charts are used for continuous variables that can be measured, such as weight or volume. Two commonly used variable control charts are the range chart, or R-chart, and the mean chart, or (x –-) chart. A range (R-) chart reflects the amount of dispersion present in each sample; a mean (x –-) chart indicates how sample results relate to the process average or mean. These charts are normally used together to determine whether a process is in control.

Mean (X– ) Chart

In a mean (x –-) control chart, each time a sample of a group of items is taken from the process, the mean of the sample is computed and plotted on the chart. Each sample mean x – is a point on the control chart. The samples taken tend to be small, usually around four or five. The center line of the control chart is the overall process average—that is, the mean of the sample means. The x – x-chart is based on the normal distribution. It can be constructed in two ways depending on the information that is available about the distribution.

If the standard deviation of the distribution is known from experience or historical data, then formulas using the standard deviation can be used to compute the upper and lower control limits. If the standard deviation is not known, then a table of values based on sample ranges is available to develop the upper and lower control limits. We will first look at how to construct an (x –-) chart when the standard deviation is known. The formulas for computing the upper control limit (UCL) and lower control limit (LCL) are

where x is the average of the sample means and R is the average range value. A2 is a tabular value that is used to establish the control limits. Values of A2 are included

Sample Size n	Factor for x – Chart A2	Factors for R-Chart D3	Factors for R-Chart D4
2	1.88	0	3.27
3	1.02	0	2.57
4	0.73	0	2.28
5	0.58	0	2.11
6	0.48	0	2.00
7	0.42	0.08	1.92
8	0.37	0.14	1.86
9	0.34	0.18	1.82
10	0.31	0.22	1.78
11	0.29	0.26	1.74
12	0.27	0.28	1.72
13	0.25	0.31	1.69
14	0.24	0.33	1.67
15	0.22	0.35	1.65
16	0.21	0.36	1.64
17	0.20	0.38	1.62
18	0.19	0.39	1.61
19	0.19	0.40	1.60
20	0.18	0.41	1.59
21	0.17	0.43	1.58
22	0.17	0.43	1.57
23	0.16	0.44	1.56
24	0.16	0.45	1.55
25	0.15	0.46	1.54

They were developed specifically for determining the control limits for (x –-) charts and are comparable to three-standard deviation (3σ) limits. These table values are frequently used to develop control charts.

Range (R-) Chart

In an R-chart, the range is the difference between the smallest and largest values in a sample. This range reflects the process variability instead of the tendency toward a mean value. The formulas for determining control limits are

R is the average range (and centerline) for the samples,

D3 and D4 are table values like A2 for determining control limits that have been developed based on range values rather than standard deviations. Table 10.1 also includes values for D3 and D4 for sample sizes up to 25.

Using (X-) and R-charts

The x – x-chart is used with the R-chart under the premise that both the process average and variability must be in control for the process to be in control. This is logical. The two charts measure the process differently. Samples can have very narrow ranges, suggesting little process variability, but the sample averages might be beyond the control limits.

For example, consider two samples, the first having low and high values of 4.95 and 5.05 centimeters, and the second having low and high values of 5.10 and 5.20 centimeters. The range of both is 0.10 centimeters, but x – for the first is 5.00 centimeters, and x – for the second is 5.15 centimeters. The two sample ranges might indicate the process is in control and x – = 5.00 might be okay, but x – = 5.15 could be outside the control limit.

Conversely, the sample averages can be in control, but the ranges might be very large. For example, two samples could both have x – = 5.00 centimeters, but sample 1 could have a range between 4.95 and 5.05 (R = 0.10 centimeter) and sample 2 could have a range between 4.80 and 5.20 (R = 0.40 centimeter). Sample 2 suggests the process is out of control.

It is also possible for an R-chart to exhibit a distinct downward trend in the range values, indicating that the ranges are getting narrower and there is less variation. This would be reflected on the (x –-) chart by mean values closer to the center line. Although this occurrence does not indicate that the process is out of control, it does suggest that some nonrandom cause is reducing process variation. This cause must be investigated to see if it is sustainable. If so, new control limits would need to be developed. Sometimes an (x –-) chart is used alone to see if a process is improving, perhaps toward a specific performance goal.

In other situations, a company might have studied and collected data for a process for a long time and already knows what the mean and standard deviation of the process are; all it wants to do is monitor the process average by taking periodic samples. In this case, it would be appropriate to use the mean chart where the process standard deviation is already known.

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Control Chart Patterns

Even if a control chart indicates that a process is in control, it is possible that the sample variations within the control limits are not random. If the sample values display a consistent pattern, even within the control limits, it suggests that this pattern has a nonrandom cause that might warrant investigation.

We expect the sample values to “bounce around” above and below the center line, reflecting the natural random variation in the process that will be present. However, if the sample values are consistently above (or below) the center line for an extended number of samples or if they move consistently up or down, there is probably a reason for this behavior; that is, it is not random. Examples of non-random patterns.

A pattern in a control chart is characterized by a sequence of sample observations that display the same characteristics—also called a run. One type of pattern is a sequence of observations either above or below the center line. For example, three values above the center line followed by two values below the line represent two runs of a pattern. Another type of pattern is a sequence of sample values that consistently go up or down within the control limits. Several tests are available to determine if a pattern is nonrandom or random.

One type of pattern test divides the control chart into three “zones” on each side of the center line, where each zone is one standard deviation wide. These are often referred to as 1-sigma, 2-sigma, and 3-sigma limits. The pattern of sample observations in these zones is then used to determine if any nonrandom patterns exist. Recall that the formula for computing an x – x-chart uses A2 from Table 10.1, which assumes 3 standard deviation control limits (or 3-sigma limits). Thus, to compute the dividing lines between each of the three zones for an x – x-chart, we use 1 / 3 A2. The formulas to compute these zone boundaries.

There are five general guidelines associated with the zones for identifying patterns in a control chart where none of the observations are beyond the control limits:

• Eight consecutive points on one side of the center line
  
  
• Eight consecutive points up or down
  
  
• Fourteen points alternating up or down
  
  
• Two out of three consecutive points in zone A (on one side of the center line)
  
  
• Four out of five consecutive points in zone A or B on one side of the center line

If any of these guidelines applied to the sample observations in a control chart, it would imply that a nonrandom pattern exists and the cause should be investigated if rules 1, 4, and 5 are violated.

Sample Size Determination

In our examples of control charts, sample sizes varied significantly. For p-charts, we used sample sizes in the hundreds, and for c-charts, sample sizes can be as small as one item, whereas for (x –-) and R-charts we used samples of four or five. In general, larger sample sizes are needed for attribute charts because more observations are required to develop a usable quality measure.

A population proportion defective of only 5% requires 5 defective items from a sample of 100. But, a sample of 10 does not even permit a result with 5% defective items. Variable control charts require smaller sample sizes because each sample observation provides usable information—for example, weight, length, or volume. After only a few sample observations (as few as two), it is possible to compute a range or a sample average that reflects the sample characteristics. It is desirable to take as few sample observations as possible, because they require the operator’s time to take them.

Some companies use sample sizes of just two. They inspect only the first and last items in a production lot under the premise that if neither is out of control, then the process is in control. This requires the production of small lots so that the process will not be out of control for too long before a problem is discovered.

Size may not be the only consideration in sampling. It may also be important that the samples come from a homogeneous source so that if the process is out of control, the cause can be accurately determined. If the production takes place on either one of two machines (or two sets of machines), mixing the sample observations between them makes it difficult to ascertain which operator or machine caused the problem. If the production process encompasses more than one shift, mixing the sample observation between shifts may make it more difficult to discover which shift caused the process to move out of control.

SPC With Excel and OM Tools

Computer software and spreadsheet packages are available that perform statistical quality control analysis, including the development of process control charts. It demonstrates how to develop a statistical process control chart on the computer using Excel and OM Tools.

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Process Capability

Control limits are occasionally mistaken for tolerances; however, they are quite different things. They provide a means for determining natural variation in a production process. They are statistical results based on sampling. Tolerances are design specifications reflecting customer requirements for a product. They specify a range of values above and below a designed target value (also referred to as the nominal value) within which product units must fall to be acceptable. For example, a bag of potato chips might be designed to have a net weight of 9.0 oz of chips with a tolerance of ±0.5 oz.

The design tolerances are thus between 9.5 oz (the upper specification limit) and 8.5 oz (the lower specification limit). The packaging process must be capable of performing within these design tolerances, or a certain portion of the bags will be defective—that is, underweight or overweight. Tolerances are not determined by the production process; they are externally imposed by the designers of the product or service. Control limits, on the other hand, are based on the production process, and they reflect process variability.

They are a statistical measure determined from the process. It is possible for a process in an instance to be statistically “in control” according to control charts, yet the process may not conform to the design specifications. To avoid such a situation, the process must be evaluated to see if it can meet product specifications before the process is initiated, or the product or service must be redesigned.

Process capability refers to the natural variation of a process relative to the variation allowed by the design specifications. They are used for process capability to determine if an existing process is capable of meeting design specifications.

The three main elements associated with process capability are process variability (the natural range of variation of the process), the processing center (mean), and the design specifications. Figure 10.5 shows four possible situations with different configurations of these elements that can occur when we consider process capability. The natural variation of a process, which is greater than the design specification limits. The process is not capable of meeting these specification limits.

This situation will result in a large proportion of defective parts or products. If the limits of a control chart measuring natural variation exceed the specification limits or designed tolerances of a product, the process cannot produce the product according to specifications. The variation that will occur naturally, at random, is greater than the designed variation.

Parts that are within the control limits but outside the design specification must be scrapped or reworked. This can be very costly and wasteful. Alternatives include improving the process or redesigning the product. However, these solutions can also be costly. As such, process capability studies must be done during product design and before contracts for new parts or products are entered into.

The situation in which the natural control limits and specification limits are the same. This will result in a small number of defective items, the few that will fall outside the natural control limits due to random causes. For many companies, this is a reasonable quality goal. If the process distribution is normally distributed and the natural control limits are three standard deviations from the process mean—that is, they are 3-sigma limits— then the probability between the limits is 0.9973. This is the probability of a good item. This means the area, or probability, outside the limits is 0.0027, which translates to 2.7 defects per thousand or 2700 defects out of one million items.

However, according to strict quality philosophy, this is not an appropriate quality goal. As Evans and Lindsay point out in the book The Management and Control of Quality, this level of quality corresponding to 3-sigma limits is comparable to “at least 20,000 wrong drug prescriptions each year, more than 15,000 babies accidentally dropped by nurses and doctors each year, 500 incorrect surgical operations each week, and 2000 lost pieces of mail each hour.”

As a result, several companies have adopted “Six Sigma” quality. This represents product-design specifications that are twice as large as the natural variations reflected in 3-sigma control limits. This type of situation, where the design specifications exceed the natural control limits, is shown graphically in Figure 10.5c. The company would expect that almost all products will conform to design specifications—as long as the process mean is centered on the design target. Statistically, Six Sigma corresponds to only 0.0000002 percent defects or 0.002 defective parts per million (PPM), which is only two defects per billion! However, when Motorola announced in 1989 that it would achieve Six Sigma quality in five years, it translated this to 3.4 defects per million.

How did they get 3.4 defects per million from 2 defects per billion? Motorola took into account that the process mean will not always exactly correspond to the design target; it might vary from the nominal design target by as much as 1.5 sigma, which translates to a Six Sigma defect rate of 3.4 defects per million. This value has since become the standard for Six Sigma quality in industry and business. Applying this same scenario of a 1.5-sigma deviation from the process mean to the more typical 3-sigma level used by most companies, the defect rate is not 2700 defects per million, but 66,810 defects per million.

As indicated, Figure 10.5d shows the situation in which the design specifications are greater than the process range of variation; however, the process is off-center. The process is capable of meeting specifications, but it is not because the process is not in control. In this case, a percentage of the output that falls outside the upper design specification limit will be defective. If the process is adjusted so that the process center coincides with the design target (i.e., it is centered), then almost all of the output will meet design specifications.

Determining process capability is important because it helps a company understand process variation. If it can be determined how well a process is meeting design specifications, and thus what the actual level of quality is, then steps can be taken to improve quality. Two measures used to quantify the capability of a process—that is, how well the process is capable of producing according to design specifications—are the capability ratio (Cp ) and the capability index (Cpk).

Process Capability Measures

One measure of the capability of a process to meet design specifications is the process capability ratio (Cp ). It is defined as the ratio of the range of the design specifications (the tolerance range) to the range of the process variation, which for most firms is typically ±3σ or 6σ.

If Cp is less than 1.0, the process range is greater than the tolerance range, and the process is not capable of producing within the design specifications all the time. This is the situation depicted in Figure 10.5a. If Cp equals 1.0, the tolerance range and the process range are virtually the same—the situation shown in Figure 10.5b. If Cp is greater than 1.0, the tolerance range is greater than the process range—the situation depicted in Figure 10.5c. Thus, companies would logically desire a Cp equal to 1.0 or greater, since this would indicate that the process is capable of meeting specifications.

A second measure of process capability is the process capability index (Cpk). The Cpk differs from the Cp in that it indicates if the process mean has shifted away from the design target, and in which direction it has shifted—that is, if it is off-center. This is the situation depicted in Figure 10.5d. The process capability index specifically measures the capability of the process relative to the upper and lower specifications.

If the Cpk index is greater than 1.00, then the process is capable of meeting design specifications. If Cpk is less than 1.00, then the process mean has moved closer to one of the upper or lower design specifications, and it will generate defects. When Cpk equals Cp, this indicates that the process mean is centered on the design (nominal) target.