Defects are not free. Somebody makes them, and gets paid for making them.
—W. Edwards Deming
The previous two chapters discussed control charts for variable data, which are continuous measurement information such as height, weight, length, concentration, and pressure. This chapter discusses control charts for attribute data that are go/no-go or count information. Examples include the number of defective units, the number of defects in a unit, the number of complaints received from dissatisfied customers, and the number of patients whose meals were delivered more than 15 minutes late. The theory supporting control charts for attributes is the same as has been discussed in Chapter 3. However, while the control chart for individuals used to illustrate that discussion is based on the normal distribution, control charts for attributes are based on different distributions. Signals that indicate an out-of-control condition are the same for both variables and attributes control charts.
All of the control charts in this chapter are for count data, but we must be sure to clearly define what we are counting in order to select the appropriate chart. The first two charts discussed in this chapter are for counting nonconforming or defective products. A defective or nonconforming product or service is one that cannot be sold or delivered as is because it does not meet the required specifications. Under this definition, a product exists in one of only two states: conforming or nonconforming (acceptable or defective).
The last two charts are for counting nonconformities or defects. A nonconformity is defined as the “nonfulfillment of a specified requirement,” and similarly, a defect is defined as a “product’s or service’s nonfulfillment of an intended requirement…”1 A product may have a theoretically infinite number of defects. Whether the presence of defects renders the product nonconforming is determined by the specifications. For example, there may be a limit to the number of minor defects that are allowed in a product before it is rendered as nonconforming. The distinction between a defective (nonconforming) product and a defect (a nonconformity) is significant and is key to selecting the appropriate control chart for the job.
Proportion Defective Chart
The proportion defective control chart is also referred to as a percent chart, a fraction nonconforming chart, a fraction defective chart, or simply as a p-chart. ASQ defines a p-chart as a “control chart for evaluating the stability of a process in terms of the percentage (or proportion) of the total number of units in a sample in which an event of a given classification occurs.”2 Often the “event of a given classification” is whether the unit being examined is conforming (acceptable) or nonconforming (defective). The binomial distribution is the basis for the p-chart.
The p-chart is often used when large quantities of product are produced relatively quickly. For example, an injection molding process, which produces small parts with short cycle times in multiple cavity molds, would be a good candidate for a p-chart. One advantage of the use of p-charts is that multiple key quality characteristics (KQCs) can be combined using just one chart. When using control charts for variables, each KQC must have its own chart or pair of charts. The disadvantages of using p-charts compared to control charts for variables include the need for sample sizes significantly larger than with control charts for variables, and information about specific KQCs is not preserved on the p-chart as it would be on a control chart tracking a specific variable.
Example 6.1
p-charts in Action
An injection molding operation was producing a plastic boot that covers the end of an electronics cable. The parts were produced automatically on a large molding machine using a 50-cavity mold with a short molding cycle. Many thousands of these parts were produced during each hour of production. There are a number of attributes that must be inspected for. A problem with any one attribute renders the part nonconforming. Typically, the company considered each shift’s production to be a lot and evaluated each lot using an acceptance sampling plan. The major disadvantage of this plan is that problems that occur early in a shift are not detected until well into the next shift. Since a rejected lot must be inspected 100 percent (which was considered to be rework), considerable extra cost can be incurred as a result of failing to identify a problem as early as possible.
This process was chosen as the organization’s pilot study for the implementation of statistical process control (SPC). The implementation team decided to sample the process four times per shift using a sample size of 200 with immediate inspection of the sample units. The results of the inspection would be recorded on a p-chart. The expectations for this SPC implementation were:
• Quicker recognition of process problems so that corrective action can be taken more quickly with the result of fewer defective parts produced.
• A savings in inspection costs since the SPC sampling plan would require the inspection of 800 units per shift compared with the 1250 units per shift under the acceptance sampling plan.
• A savings in rework costs since the number of units subject to 100 percent inspection if a process problem is found would be about one quarter as many as with the acceptance sampling plan. This is because in this case we sample the output four times as often using SPC as with acceptance sampling so the number of items produced before a problem is detected is one-fourth as many as with acceptance sampling.
• Reduced product variation, which should manifest as better quality as perceived by the customer.
The implementation team made sure that the process was set up as designed and took 25 samples over the course of three shifts of production. They used the data from these samples to construct the trial control chart shown in Figure 6.1. The process was shown to be in control and SPC was instituted as the standard operating procedure (SOP) for this process using the control limits established during the trial period. About two months later, the implementation team revisited the process and found that SPC was working as designed and that all expectations were being achieved.
Figure 6.1 Trial p-chart
Source: Created using NWA Quality Analyst 6.3.
Notice that the distance between the central line (CL) and upper control limit (UCL) in Figure 6.1 is greater than that between the CL and lower control limit (LCL). This is because the LCL cannot always be set at three standard deviations below the CL because that would result in a negative proportion defective, which is impossible. In this case, the LCL is set at the lowest level that is possible, which is zero.
P-charts can be used with both fixed and variable sample sizes. Figure 6.1 illustrates an example using fixed sample size. However, there are many instances where sample size will vary—particularly in the service and healthcare sectors, which will be more specifically discussed in Chapter 8. Whenever the sample consists of all the items produced during a period of time, it is likely that the sample size will vary. Even an automated manufacturing process that produces relatively constant amounts of product per day, all of which is subject to 100 percent automated inspection, is subject to variation in the sampling intervals, which creates a situation where sample size will vary. When the sample size varies, the control limits are adjusted for each sample, which explains why the UCL and LCL in Figure 6.2 are not straight lines but appear to have steps.
Example 6.2
p-chart for Varying Sample Size Leads to Improvement Project
A manufacturing company uses an automated system to apply labels to its products. Each labeled product is inspected using a pixel camera system, which detects any missing, crooked, or torn labels and removes the mislabeled product from the process flow. The inspection system automatically logs the number of units inspected and the number of units rejected. This allows the calculation of the proportion of the units inspected that are nonconforming. These data are plotted hourly on a p-chart. Since the number of units inspected per hour varies, the control limits for the control chart must be adjusted based on the sample size as shown in Figure 6.2. The control chart shows that the process is out of control because sample 25 is above the UCL.
The first action the company took was to determine the root cause for the out-of-control point. They then took appropriate corrective action, followed by taking a new sample to verify that the corrective action brought the process back into control.
The company also periodically reviews the control chart to determine how well the process is meeting expectations. The CL on this control chart is 0.05251, indicating that more than 5 percent of the labeled units are nonconforming. The organization in this case was not satisfied with this level of nonconforming product and initiated a planned improvement project with the goal of reducing the mean number of nonconforming labels produced by the process.
Figure 6.2 p-chart for labeling operation proportion defective per hour
Source: Created using Minitab 18.
Number-Defective Chart
The number-defective control chart is also referred to as an np-chart. The np-chart is an alternative to the p-chart. Adapting the definition of the p-chart, the np-chart is a control chart for evaluating the stability of a process in terms of the number of the total number of units in a sample in which an event of a given classification occurs. As with the p-chart, often the “event of a given classification” is whether the unit being examined is conforming (acceptable) or nonconforming (defective). Since the np-chart uses the number rather than the proportion of nonconforming units in each sample, sample size must remain constant. The binomial distribution is the basis for the np-chart.
As with the p-chart, the np-chart is often used when large quantities of product are produced relatively quickly and provides the same advantages and disadvantages as the p-chart. The main advantage of the np-chart over the p-chart is ease of understanding. Since the number of nonconforming units per sample is plotted on the chart, it provides direct evidence for the amount of nonconforming product being produced in units. Line operators often find units easier to understand than proportions or percentages. A disadvantage of the np-chart compared to the p-chart is the inability to handle variable sample sizes.
Figure 6.3 shows the p-chart from Figure 6.1 along with an np-chart using the same data. The patterns in the charts are identical and both have equal power to respond to out of control conditions in the process.
Both the p- and np-charts are based on defective or nonconforming units—that is, units that are judged not to be in conformance with specifications. The next set of charts, the c-chart and u-chart, are based on the number of nonconformities or defects found in a unit.
Figure 6.3 Trial p- and np-charts using the same data
Source: Created using NWA Quality Analyst 6.3.
Count Chart or Number of Nonconformities in a Fixed-Size Sample
The count chart, also known as a number of nonconformities chart, is a “control chart for evaluating the stability of a process in terms of the count of events of a given classification occurring in a sample, and is known as a c-chart.”3 The events are generally nonconformities or defects. When using the p-chart or np-chart we are unconcerned with how many defects are present in a sample, just that for whatever reason or reasons, some units in that sample are considered to be nonconforming. In the case of c-charts we are concerned only with the number of defects that are present in a sample and not with how many nonconforming units are present. When using c-charts, the sample size should be constant. The Poisson distribution is the basis for the c-chart.
Consider the final inspection of the finish of an automobile. If the specification allows a specified number of minor defects in the finish, then until that limit is reached, the automobile being inspected is considered to be conforming even though some number of nonconformities is present. As Example 6.3 shows, a p-chart or np-chart would fail to capture the information about the minor defects present in the finish. That is why the c-chart is appropriate in this case.
Example 6.3
The np-chart versus the c-chart
Each automobile moving off the end of the assembly line is hand inspected for minor defects in the paint finish. The specification allows a maximum of eight minor defects per automobile. More than eight minor defects renders the automotive finish nonconforming or defective. The results of recent inspections are shown in Table 6.1.
Note that while there is a clear trend of increasing numbers of defects in the samples over time, none of the samples would be considered to be nonconforming until the 10th sample. When plotted on an np-chart, samples 1 through 9 would plot as zero defectives, indicating no significant variation in the process. In fact, the process would appear to be absolutely stable with zero defective paint finishes until point 10. Clearly the np-chart does not accurately depict the real state of control of the process.
Table 6.1 Inspection record for automotive finishes
Sample number |
Number of defects |
1 |
0 |
2 |
1 |
3 |
2 |
4 |
3 |
5 |
4 |
6 |
5 |
7 |
6 |
8 |
7 |
9 |
8 |
10 |
9 |
The correct control chart to use in this situation is the c-chart. The c-chart, shown in Figure 6.4 shows that the process is out of control at point 8 using the “8 points in a row on a rising trend” run rule. The c-chart clearly shows that the process is deteriorating over time. The out-of-control signal at point 8 is received before we exceed the specification limit of eight minor defects, which provides time to investigate and correct the problem before we have produced a defective finish. While this is a contrived example, it clearly illustrates the importance of selecting the correct control chart for the job. When counting nonconformities (defects) in samples of fixed size, the c-chart is the correct control chart to use.
Figure 6.4 Automotive paint finish inspection c-chart
Source: Created using NWA Quality Analyst 6.3.
Count Chart for Nonconformities per Unit Where Sample Size May Vary
The count chart per unit, called a u-chart, is an adaptation of the c-chart that evaluates the stability of a process in terms of the count of events of a given classification occurring per unit in a sample. Unlike the c-chart, which uses fixed-size samples, the u-chart allows for the use of variable size samples. As with the c-chart, the events are generally nonconformities or defects. And as with the c-chart, when using a u-chart we are concerned only with the number of defects that are present in a sample and not with how many nonconforming units are present. The Poisson distribution is the basis for the u-chart.
There are many situations where the sample size varies. Many service applications involve variable sample sizes and will be discussed in more detail in Chapter 8. In manufacturing, when the entire population is inspected rather than samples taken from the population, sample size often varies. The u-chart handles this by calculating and plotting the number of defects per unit and adjusts the UCL and LCL for the sample size.
Example 6.4
The u-Chart in Action
The manufacturer of coated polyester film uses an automatic inspection process to identify defects in the coating. The specific location of these defects is automatically recorded so that the manufacturer knows exactly where within the run the defects are located. Each hour, the number of defects is recorded along with the length of the coated product produced in linear feet. Data for 16 hours of production are contained in Table 6.2.
Table 6.2 Inspection record for coating defects
Sample number |
Number of defects |
Linear feet produced |
1 |
27 |
9,250 |
2 |
17 |
8,100 |
3 |
16 |
8,500 |
4 |
24 |
8,900 |
5 |
22 |
8,650 |
6 |
28 |
9,040 |
7 |
19 |
8,750 |
8 |
26 |
9,100 |
9 |
24 |
9,200 |
10 |
21 |
8,850 |
11 |
25 |
7,250 |
12 |
19 |
8,850 |
13 |
20 |
8,750 |
14 |
26 |
8,950 |
15 |
28 |
8,800 |
16 |
19 |
9,100 |
Since we are tracking defects and sample size varies, the correct control chart to use in this situation is the u-chart. The u-chart, shown in Figure 6.5, indicates that the process is in control. Notice that the UCL and LCL vary as a result of adjustments due to variations in sample size. When counting nonconformities (defects) in samples of variable size, the u-chart is the correct control chart to use.
The manufacturer was not happy with the level of defects in the coated rolls. The CL on the u-chart is 0.002577835, indicating that there are approximately 0.00258 defects per linear foot. Based on the information on the u-chart in Figure 6.5, the manufacturer recognized that the process was in control—operating as currently designed. So a project to improve the process was initiated with a goal of reducing the number of defects per linear foot by 25 percent within three months.
Figure 6.5 Coating defect u-chart
Source: Created using NWA Quality Analyst 6.3.
Chapter Take-Aways
Figure 6.6 provides guidance for selecting among the control charts discussed in this chapter.
Figure 6.6 Attribute control chart selection guide
Questions You Should Be Asking About Your Work Environment
• Which processes have KQCs that are measured using attribute data and are candidates for monitoring with attribute control charts?
• For the processes identified by the previous question, what would be the value of monitoring those process using SPC?
• What data that are monitored by your organization might have greater value for decision-making purposes if tracked using an attribute control chart?