The quality measures used in attribute control charts are discrete values reflecting a simple decision criterion such as good or bad. A p-chartuses the proportion of defective items in a sample as the sam- ple statistic; a c-chartuses 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 nondefective items and to state the number of defectives as a percentage of the whole. In some processes, the proportion 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 propor- tion 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
zthe number of standard deviations from the process average the sample proportion defective; an estimate of the process average
pthe standard deviation of the sample proportion The sample standard deviation is computed as
where n is the sample size.
Example 3.1 demonstrates how a p-chart is constructed.
sp = A
p(1 - p) n p
LCL = p - zsp UCL = p + zsp c-chart:
uses the number of defective items in a sample.
The Western Jeans Company produces denim jeans. The company wants to establish a p-chart to monitor the production process and maintain high quality. Western believes that approxi- mately 99.74% of the variability in the production process (corresponding to 3-sigma limits, or z3.00) is random and thus should be within control limits, whereas 0.26% of the process variability is not random and suggests that the process is out of control.
The company has taken 20 samples (one per day for 20 days), each containing 100 pairs of jeans (n100), and inspected them for defects, the results of which are as follows.
Example 3.1
Construction of a p-Chart
Sample Number of Defectives Proportion Defectives
1 6 .06
2 0 .00
3 4 .04
4 10 .10
5 6 .06
6 4 .04
7 12 .12
8 10 .10
9 8 .08
10 10 .10
11 12 .12
12 10 .10
13 14 .14
14 8 .08
15 6 .06
16 16 .16
17 12 .12
18 14 .14
19 20 .20
20 18 .18
200
The proportion defective for the population is not known. The company wants to construct a p-chart to determine when the production process might be out of control.
Solution
Since p is not known, it can be estimated from the total sample:
The control limits are computed as follows:
The p-chart, including sample points, is shown in the following figure:
The process was below the lower control limits for sample 2 (i.e., during day 2). Although this could be perceived as a “good” result since it means there were very few defects, it might also suggest that something was wrong with the inspection process during that week that should be checked out. If there is no problem with the inspection process, then management
= 0.10 - 3.00 A
0.10(1 - 0.10) 100
= 0.010 LCL = p - z
A
p(1 - p) n
= 0.10 + 3.00 A
0.10(1 - 0.10) 100
= 0.190 UCL = p + z
A
p(1 - p) n
= 0.10
= 200 20(100)
p = total defectives total sample observations
(Continued)
would want to know what caused the quality of the process to improve. Perhaps “better”
denim material from a new supplier that week or a different operator was working.
The process was above the upper limit during day 19. This suggests that the process may not be in control and the cause should be investigated. The cause could be defective or malad- justed machinery, a problem with an operator, defective materials (i.e., denim cloth), or a number of other correctable problems. In fact, there is an upward trend in the number of de- fectives throughout the 20-day test period. The process was consistently moving toward an out-of-control situation. This trend represents a pattern in the observations, which suggests a nonrandom cause. If this was the actual control chart used to monitor the process (and not the initial chart), it is likely this pattern would have indicated an out-of-control situation before day 19, which would have alerted the operator to make corrections. Patterns are discussed in a separate section later in this chapter (page 124).
This initial control chart shows two out-of-control observations and a distinct pattern of in- creasing 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.
A L O N G T H E S U P P L Y C H A I N
Using Control Charts for Improving Health-Care Quality
The National Health Service (NHS) in the United Kingdom makes extensive use of SPC charts to improve service deliv- ery to patients. SPC charts are frequently developed using Excel spreadsheets. Employees throughout NHS attend a two-day training course where they learn about X- and R- charts and their application. They also learn about Deming’s 14 points and principles of quality management and how to improve processes in their own operation.
The number of possible applications of SPC charts for a health-care facility is very high, with many applications re- lating to waiting times (as is the case with many service or- ganizations); for example, the time to see a doctor, nurse, or other medical staff member, or to get medical results. One specific example in the NHS is “door to needle” times, which is the time it takes after a heart attack patient is regis- tered into the hospital to receive an appropriate injection.
The system-wide goverment-mandated target value for this process is that 75% of heart patients receive treatment within 20 minutes of being received into the hospital. Control charts are also used to monitor administrative errors.
Bellin Health System is as an integrated health-care orga- nization serving Green Bay, Wisconsin, and the surrounding region. It specializes in cardiac care and its 167-bed hospital is known as the regon’s heart center. In addition it operates a psychiatric center, 20 regional clinics, and a college of nurs- ing. Bellin’s quality management program is based on im- proving processes by identifying measures of success, setting
goals for improvement that can be measured, applying Deming’s PDCA cycle (see Chapter 2) for improving processes, and using statistical process control charts to monitor processes for stability and to see if improvement efforts are successful. Bellin has over 1,250 quality indicators reported monthly, quarterly, or annually, and over 90% are monitored with SPC charts. For example, one such quality indicator is for compliance with the Centers for Disease Control guide- lines on health-care hand hygiene, which Bellin measures across its entire system. The lack of hand hygiene is a leading cause of fatal hospital-borne infections, and thus a very important quality indicator. The quality resource department (QRD) at Bellin uses a p-chart to monitor hand hygiene for system nursing care, where the control measure is the portion of hand opportunities that meet the criteria divided by the total number of opportunities available to meet the criteria. Using SPC charts Bellin was able to improve the hand hygiene process such that a 90% target level was acheived. Throghout the Bellin Healthcare System control charts are employed to identify special cause variations outside the norm in processes that require improvement. SPC has become an essential part of the continuous quality improvement program at Bellin used by all members of the organization to improve service and health care, and reduce costs.
Identify and discuss other applications of SPC charts in a healthcare facility that you think might improve processes.
Sources: M. Owen, “From Sickness to Health,” Qualityworld 29 (8: August 2003), pp. 18–22; and C. O’Brien and S. Jennings, Quality Progress 41 (3; March 2000), pp. 36–43.
Checking cracker texture at a Nabisco plant as part of the quality-control process.
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 popula- tion, 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, , computed by dividing the total number of defects by the number of samples. The sample standard deviation,c, is . The following formulas for the control limits are used:
LCL = c - zsc UCL = c + zsc
2c c
The Ritz Hotel has 240 rooms. The hotel’s housekeeping department is responsible for maintaining the quality of the rooms’ appearance and cleanliness. Each individual house- keeper is responsible for an area encompassing 20 rooms. Every room in use is thoroughly cleaned and its supplies, toiletries, and so on are restocked each day. Any defects that the housekeeping staff notice that are not part of the normal housekeeping service are supposed to be reported to hotel maintenance. Every room is briefly inspected each day by a house- keeping supervisor. However, hotel management also conducts inspection tours at random for a detailed, thorough inspection for quality-control purposes. The management inspectors not only check for normal housekeeping service defects like clean sheets, dust, room sup- plies, room literature, or towels, but also for defects like an inoperative or missing TV re- mote, poor TV picture quality or reception, defective lamps, a malfunctioning clock, tears or stains in the bedcovers or curtains, or a malfunctioning curtain pull. An inspection sample includes 12 rooms, that is, one room selected at random from each of the twelve 20-room
Example 3.2
Construction of a c-Chart
Corbis/Bettmann
(Continued)
blocks serviced by a housekeeper. Following are the results from 15 inspection samples con- ducted at random during a one-month period:
The hotel believes that approximately 99% of the defects (corresponding to 3-sigma limits) are caused by natural, random variations in the housekeeping and room maintenance service, with 1% caused by nonrandom variability. They want to construct a c-chart to monitor the housekeeping service.
Solution
Because c, the population process average, is not known, the sample estimate, , can be used instead:
The control limits are computed using z3.00, as follows:
The resulting c-chart, with the sample points, is shown in the following figure:
All the sample observations are within the control limits, suggesting that the room quality is in control. This chart would be considered reliable to monitor the room quality in the future.
2 4 6 8 10 12
3
14 16
UCL = 23.35
Sample number 6
9 12 15 18 21 24 27
Number of defects
LCL = 1.99 c = 12.67
LCL = c - z2c = 12.67 - 3212.67 = 1.99 UCL = c + z2c = 12.67 + 3212.67 = 23.35
c = 190
15 = 12.67
c Sample Number of Defects
1 12
2 8
3 16
4 14
5 10
6 11
7 9
8 14
9 13
10 15
11 12
12 10
13 14
14 17
15 15
190
The Goliath Tool Company produces slip-ring bearings, which look like flat doughnuts or washers. They fit around shafts or rods, such as drive shafts in machinery or motors. At an early stage in the production process for a particular slip-ring bearing, the outside diameter of the bearing is measured. Employees have taken 10 samples (during a 10-day period) of 5 slip- ring bearings and measured the diameter of the bearings. The individual observations from each sample (or subgroup) are shown as follows:
Example 3.3
Constructing an -Chart
x
Range (R-) chart:
uses the amount of dispersion in a sample.
Mean ( -) chart:
uses the process average of a sample.
x
Observations (Slip-Ring Diameter, cm), n
Subgroupk 1 2 3 4 5
1 5.02 5.01 4.94 4.99 4.96 4.98
2 5.01 5.03 5.07 4.95 4.96 5.00
3 4.99 5.00 4.93 4.92 4.99 4.97
4 5.03 4.91 5.01 4.98 4.89 4.96
5 4.95 4.92 5.03 5.05 5.01 4.99
6 4.97 5.06 5.06 4.96 5.03 5.01
7 5.05 5.01 5.10 4.96 4.99 5.02
8 5.09 5.10 5.00 4.99 5.08 5.05
9 5.14 5.10 4.99 5.08 5.09 5.08
10 5.01 4.98 5.08 5.07 4.99 5.03
50.09 x