Online Student Guide Types of Control Charts
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1 Online Student Guide Types of Control Charts OpusWorks 2016, All Rights Reserved 1
2 Table of Contents LEARNING OBJECTIVES... 4 INTRODUCTION... 4 DETECTION VS. PREVENTION... 5 CONTROL CHART UTILIZATION... 5 PROCESS CONTROL... 5 PROCESS CONTROL AND VARIATION... 6 TYPES OF VARIATION IN CONTROL... 6 TYPES OF VARIATION OUT OF CONTROL... 7 FINDING VARIATION... 7 IMPROVING AN UNSTABLE PROCESS... 8 IMPROVING A STABLE PROCESS... 8 USE OF CONTROL CHARTS... 9 CONTROL CHART... 9 CHOOSING THE RIGHT CONTROL CHART... 9 VARIABLES CONTROL CHARTS X & MR INTRODUCTION X & MR CHARTS X & MR CONTROL LIMITS X & MR SUMMARY X-BAR AND R INTRODUCTION SUBGROUP SIZES X-BAR & R CHARTS X-BAR AND R LIMITS X-BAR AND S CHARTS, AND X-BAR AND R CHARTS WHAT DOES OUT OF CONTROL MEAN? CONTROL CHART: INTERPRETATION RULES ATTRIBUTE CHART CLAIMS PROCESS P CHART P CHART LIMITS WEIGHTED AVERAGE P CHART SUMMARY NP CHART NP SUMMARY U CHART U CHART LIMITS U CHART SUMMARY C CHART C CHART LIMITS C CHART SUMMARY
3 2016 by OpusWorks. All rights reserved. Version 5.5 August, 2016 Terms of Use This guide can only be used by those with a paid license to the corresponding course in the e-learning curriculum produced and distributed by OpusWorks. No part of this Student Guide may be altered, reproduced, stored, or transmitted in any form by any means without the prior written permission of OpusWorks. Trademarks All terms mentioned in this guide that are known to be trademarks or service marks have been appropriately capitalized. Comments Please address any questions or comments to your distributor or to OpusWorks at info@opusworks.com. 3
4 Learning Objectives Upon completion of this course, student will be able to: Define Control Charts and discuss their purpose Explain how to determine whether to use an Attribute or a Variables Control Chart Describe the steps for setting up a Control Chart Discuss the basic rules for using Control Charts Explain how to identify which Control Chart type is most appropriate for monitoring a given process parameter Introduction The Control Chart, also known as a Shewhart Chart or Process-Behavior Chart, was invented by Dr. Walter A. Shewhart while he was working for Bell Laboratories in the 1920s. Bell engineers realized the importance of reducing variation in a manufacturing process. They also realized that continuously adjusting a process, in reaction to non-conformance, actually increased variation and degraded the quality of their products and service. Shewhart framed the variation issue by separating its cause into two categories, Common Cause and Special Cause. In May, 1924 he introduced the Control Chart as a tool for distinguishing between the two types of variation causes. He emphasized that in order to manage a process and predict output, the process must be maintained in a state of statistical control, where only Common Cause variation is present. 4
5 Detection vs. Prevention Before we begin our discussion of Control Charts, it will be helpful to clarify two fundamental elements of statistical process control (SPC): prevention and detection. Not so long ago, a manufacturing organization would rely on its Production department to manufacture a product, and on its Quality Control department to 100 percent inspect or screen out defective product. Likewise, in an office or administrative environment, leadership would require work to be checked and rechecked to avoid errors and catch mistakes. Both scenarios use and rely on a strategy of detection. Detection is waste, because it requires valuable resources such as time, labor, and materials to be invested in products or services without Value Add or return on investment. Today, business improvement practitioners realize it is much more effective to avoid waste altogether by not producing it in the first place. We refer to this strategy or philosophy as prevention. A prevention strategy is often summarized by a simple slogan or mission statement like "Do it right the first time. However, effectively implementing a true prevention strategy is much more complex; it requires that Value Add and quality start with design and be built into the actual product or service. Control Chart Utilization Control Charts are an SPC tool used to determine whether a manufacturing or business process is in a state of statistical control and is predictable. Typically used by data-driven businesses, control charts are an essential tool for project data analysis. Control Charts can be used in all phases of a DMAIC project. Process Control Once an organization adopts and implements a true prevention strategy, it must obtain control of the process in order to maintain the Value Add built into the product or service. Process control is a fundamental element of SPC; it is the reason Control Charts are so widely used today, in production, service, and even healthcare organizations. 5
6 To help you understand how to use process control, we will address the following questions throughout this module: What is process control? How does variation affect process control and output? How are statistical techniques utilized for process control? When is a process in statistical control? What does it mean when a process is capable? What are Control Charts, and how are they used? What are the benefits of Control Charts? Process Control and Variation To fully grasp the role of Control Charts and how they help maintain process control, you must first understand variation. Recall that Dr. Shewhart divided variation into two categories: Common Cause and Special Cause. Common Cause and Special Cause are the two distinct origins of variation in a process or system. Common Cause variation is the usual, historical, quantifiable variation that is found within a process or system. The outcomes of a perfectly balanced roulette wheel are a good example of this type of variation. Think of Common Cause variation as the noise within the system. Common cause variation affects all units of output from a process. An example of Common Cause Variation is ambient temperature or humidity. Special Cause variation, on the other hand, is the unusual, not previously observed, non-quantifiable variation within a process or system. Special Cause variation always arrives as a surprise. It is the signal, within a process or system, that something is wrong. Special Causes of variation may affect only portions of the units of output. An example of a Special Cause would be a faulty air conditioner. Types of Variation In Control When a process is affected only by Common Cause variation, it is said to be in statistical control, or stable. The Control Chart provides a means to observe when sources of variation from other than Common Causes are affecting the process. 6
7 Two key characteristics of Common Causes are that they affect a process uniformly, and they usually come from many small sources. Processes that are affected only by Common Causes are relatively stable, and therefore, they have relatively predictable outputs. For example, suppose that we are recording, regularly over time, some measurements from a process. The measurements might be lengths of steel rods after a cutting operation, the amount of time to provide a service, or your weight as measured on the bathroom scales each morning. The data will vary due to natural variation, or Common Causes. Types of Variation Out of Control Two key characteristics of Special Causes are that they affect a process non-uniformly, and they usually come from a few large sources. Special Cause variations are large variations that can be traced to a cause and then corrected. When a process is affected by Special Cause variation, it is said to be out of statistical control, or unstable. Finding Variation Because of the inherent differences between Common Cause variation and Special Cause variation, a different improvement approach is required for each. For an improvement team to improve an unstable process, it must get timely data. Timely data enables the team to quickly signal Special Causes and employ an immediate fix to contain any damage. Once the quick fix is in place, the team searches for the cause. To do this, it looks to see what is different in the process, or in the conditions in which the process operates. It then assesses the extent of the problem. To help determine a long-term remedy, the team may wish to consider not only when and where the problem occurred, but when and where it did not occur. Improving a stable process is not quite so easy. The sources of Common Cause variation often lie deep within the system, without any obvious problems or signals. Common Causes work on the system day to day, batch to batch, shift to shift, and so on. Although quick fixes can be applied, and may help in the short term, they may also increase the variation in the process. One of the biggest revelations in dealing with variation is that most problems arise from Common Causes. 7
8 Improving an Unstable Process To begin its improvement initiative for an unstable process, the team should consider using a Special Cause strategy that includes the following tasks: 1. Obtain real time and timely data so that Special Causes are signaled quickly. 2. Establish a containment plan, and put in place an immediate fix to contain any damage. 3. Search for the cause by identifying what is different, and determine the extent the problem. 4. Consider when and where the problem did not occur. 5. Develop a long-term remedy or permanent fix. Improving a Stable Process Stable processes can also present opportunity for improvement. However, they can be much more difficult to improve than unstable processes because they are subject to only Common Causes of variation, which as you recall, often lie deep within the system, without any obvious signals in the data that indicate problems or opportunities for improvement. It is important to remember that Common Causes are always present, working on the system day-to-day, batch-to-batch, and shift-to-shift. Quick fixes usually work for Common Cause variation, but only for the short term, not permanently. Next, we will look at three strategies for improving stable processes. In order to improve a stable process, the team must change the system. There are three strategies it can use to help make Common Cause variation visible. One strategy is to stratify the data. To do this, the team sorts the data into groups or categories based on different factors, such as product, operator, or location. Within these groups, it looks for patterns or clusters. Another strategy is to experiment. In this strategy, the team makes intentional, planned changes in order to learn from the effects. A third strategy is to disaggregate. Here, the team divides the process into component pieces and compares the variation of each of the pieces. Now let s see how Control Charts fit in to these strategies. 8
9 Use of Control Charts As you recall, Control Charts are the basic tools used for studying variation, and they help organizations use statistical signals to monitor and/or improve performance. Control Charts can be used to study and improve nearly any performance area. Examples include performance characteristics of equipment; error rates of bookkeeping tasks; scrap rates from waste analysis; performance characteristics of computer systems; and transit times in material management systems. Control Chart Now that you have a basic understanding of a Control Chart s purpose, let s take a look at its basic construction. A Control Chart measures and displays variation over time. It starts with the same data as a histogram. At order of occurrence and at regular intervals, the data is plotted as a point on the chart. The point could represent the mean of a sample or a sample proportion. A Control Chart also includes a process average (or centerline) and control limits. If the data goes outside the control limits, it indicates that a Special Cause of variation is probably present. Choosing the Right Control Chart As you may recall, there two basic data types: Variables data and Attribute data. Accordingly, there are two basic types of Control Charts: Variables Control Charts and Attribute Control Charts, each of which is used to study a different data type. Variables data, sometimes referred to as Continuous data, is measured on a continuous scale. Attribute data, sometimes referred to as Discrete data, is measured in individual counts. This logic tree diagram is helpful for deciding which type chart to use. Next, we will examine some of the most commonly used Variables Control Charts. We will examine Attribute Control Charts later in this module. 9
10 Variables Control Charts Variables Control Charts come in pairs. In each pair, the first chart monitors the process average and the second chart monitors the process variation. Three of the most frequently used Variables Control Charts are the X and MR, the X-bar and R, and the X-bar and S. The first Control Chart we will examine is the X and MR chart, which is more commonly referred to as an Individuals, Moving Range chart. X & MR Introduction The X chart is commonly called an Individuals chart because each point plotted is an individual or a single measurement from the process. MR is the symbol used to identify the moving range, which is the team s estimate of process variation. All Control Charts share the following basic features: data from the process, a process average or centerline, and control limits that are calculated from the data. As you see here, the X and MR chart has all of these features. X & MR Charts Now let s look at an example of how an X and MR chart can be applied. This employee is installing new phone service for a customer. The company wants to improve the efficiency of their installation process to become more competitive and improve customer satisfaction. At the completion of each new installation, the employee records the installation time in a handheld computer, and the computer updates the Control Charts. Using the Variables data, the X and MR chart monitors the process mean and the process variability. As you see here, the X or Individuals chart is on top. It plots the individual installation times and monitors the process mean. The lower chart is the MR, or Moving Range, chart, which monitors process variability. 10
11 Take a close look, and you will notice that the first point on the MR chart is missing. Let s see why. As you may know, the range of a set of numbers is the difference between the largest and the smallest number. With only one measurement, the largest and the smallest numbers are the same, so the difference is zero. As shown here, because each subgroup size is one measurement, two successive measurements are needed to calculate the range. The first moving range is just the positive difference between the second data point and the first data point. The calculation for moving range is easy: just subtract the smaller of the two numbers from the larger. With our first observation, there is no point that immediately precedes it. Therefore, we cannot calculate a moving range for the first observation. As a result, it is missing. Let s continue with our example. Data on the MR chart is just a series of moving ranges calculated as the positive difference between each measurement and the one immediately preceding it. This tells us how the process is varying over time. As you see in this example, we subtracted 3.4 from 3.8 to arrive at our first moving range of 0.4; we then subtracted 2.6 from 3.4 to arrive at the next moving range of 0.8. We continued to calculate the moving range for each measurement and the one preceding it by subtracting the smaller of the two numbers from the larger. 11
12 X & MR Control Limits With Variables data, calculating the control limits is more complicated than calculating the moving range because there are two charts, one for variation and one for the process mean. Before we can calculate the limits for the process mean, we must calculate the limits for the MR chart and ensure the process is stable. Although the upper and lower control limits can be calculated by hand, the most common approach is to use one of the many Control Chart programs available. Let s briefly review the control limit calculation process. First, find the value of the centerline, or MR-bar. Using the MR-bar, calculate and plot the value of the control limits. Next, examine the MR chart for stability. If a point goes outside the control limits, it indicates that Special Cause variation is most likely present. After verifying the MR chart for stability, the next step is to calculate the centerline and the control limits for the Individuals, or X, chart. The centerline for the Individuals chart, called X-bar, is calculated as the simple average of the individual measurements. Using the X-bar value, the final step is to calculate and draw the control limits for the X chart. X & MR Summary There are many business applications where X and MR charts may be applied. They are most often used when only one measurement is available at each point in time. Examples of areas where X and MR charts may apply include monitoring accounting or financial data, efficiency measurements, maintenance or repair costs, and productivity data. X-bar and R Introduction Now that you understand the basic features, calculations, and applications of the X and MR chart, let s examine another commonly used Variables Control Chart, the X-bar and R chart. To illustrate the basic construction and application of the X-bar and R chart, let s look at an example. Here, an operator is working in a call center. To better understand the call center process and help to improve its efficiency, her team decided to begin tracking the time it takes to service each customer. 12
13 With the help of the organization s Lean Six Sigma practitioner, the team decides to select five calls at random each hour, and to measure the time for each call. To monitor the process for Special Causes of variation, the team will plot the data it collects on Control Charts. Before it can begin, however, the team must decide what type Control Chart to use. Subgroup Sizes In this example, the team s first task is to decide which one of the following three Variables Control Charts it should use: the X and MR, the X-bar and R, or the X-bar and S. Recall that the team decided to track five random calls per hour. Because of this subgroup size of five, the team can immediately eliminate the MR chart as a choice for the upper half of its chart, since it only utilizes a subgroup size of one (or one data point). By eliminating the MR chart as an option, the team can now narrow down its Control Chart choices to either the X-bar and R, or the X-bar and S. In both cases, the X-bar, upper half, of its chart is used to monitor central tendency. With the upper half of the chart determined, the team must now decide which measurement will be used in the lower half, to monitor variation: the range, R, or the standard deviation, S. Next, let s see how the team will construct and apply its X-bar and R chart. X-bar & R Charts The team s upper section X-bar chart is a plot of the subgroup averages, or X-bars. Each point plotted is the average of the subgroup s five measurements. The R chart, or range chart, is a plot of the subgroup ranges. For each subgroup, the range is computed and plotted on the R chart. Remember, the range is just the difference between the largest and smallest numbers within a subgroup. Some basic guidelines exist to aid in the selection criteria. As you can see in the Control Chart type selection guide shown here, the R chart works fine with smaller subgroup sizes, from about two to nine. However, with larger subgroup sizes, of about 10 or more, the S chart is preferable. Because the subgroup size in this example is five, the team decides to use X-bar and R chart. 13
14 X-bar and R Limits Once the team has collected several sample data subgroups, its next step is to calculate the average for each subgroup sample of five. It then plots the points on the chart and records each value in the appropriate chart data recording box. Now the team must calculate the process mean, and draw that line on the upper, X- bar chart, section. This value is called X- double bar. An estimate of the process mean, it is the average of the X-bars collected so far. Next, the team calculates the range within each subgroup of five. It records that number in the appropriate chart data box, and then plots that same number on the lower R chart section. Once again, after the team has plotted several data sets, it calculates the average of the range numbers. It then draws that number as a line on the R chart section. This is referred to as the R-bar value; it is an estimate of the process range mean variation. Finally, the team calculates and plots the upper and lower control limits for the X-bar chart and the R chart sections. We will review how to calculate the control limits later in this module. Take a look at the completed chart. As you see, the team has labeled the chart s UCL, LCL, X-double bar and R-bar lines. This is an important step to remember. Now that the team has completed the chart and started using it, it analyzes the process for stability. The team s review of the X-bar and R chart plot results indicate the process appears to be stable, with no data points outside the control limits. Later in this module we will discuss how to further analyze Control Charts for variation trends. X-bar and S Charts, and X-bar and R Charts As you learned earlier, deciding whether to use an X-bar and R or an X-bar and S chart depends upon the sample size selected. It also depends on whether the range or the standard deviation will be used to monitor variation in the lower half of the chart. Except for the difference in the calculations, there is no real difference in how we analyze or interpret these charts. Calculating the standard deviation is much harder than calculating the range. For this reason, many applications use X-bar and R charts, rather than X-bar and S charts. Today, many of these charting calculations can be done automatically, using computers and custom or specialty software. 14
15 What Does Out of Control Mean? As you recall, all processes may have natural variability due to Common Causes; and unnatural variability due to Special Causes, often referred to as assignable causes. Business improvement practitioners use Control Charts to monitor and/or improve process variability. A Control Chart for a process operating with only Common Causes of variation will have a random pattern of X or X-bar points, mostly near and around the centerline, with only a few approaching the control limits. If there is a change to the process due to the introduction of a Special or assignable cause, or a change in the variation of the process, it can be detected with a series of tests called Run Rules. To understand how Run Rules are applied to Control Charts, take a closer look at the blank chart on the right. Now, imagine the Control Chart divided into six zones, as shown by the blue, pink, and green colors. Each zone represents one standard deviation (or Sigma), in width variation, from the centerline of our chart, in both a positive and negative direction. Keeping in mind that Control Charts are one method typically used to statistically track performance of processes and related product parameters, we can then apply the general rules related to the normal distribution. For a normal distribution, we would expect 68 percent of all collected data points to fall within plus or minus one standard deviation of the mean. Likewise, we would expect 95 percent to fall within two standard deviations, and percent within three standard deviations. Whenever a Special or assignable cause introduces variation into a process, these percentages will change. Next, we will explain how to apply Run Rules to analyze Control Chart variation and help control a process. Control Chart: Interpretation Rules Over the years, several sets of rules for Control Chart interpretation have been developed. Today, the Nelson Rules, and the Western Electric Rules which were developed in the 1920s, are the two most commonly used by Lean Six Sigma practitioners. Both of these sets of rules look for unusual patterns or trends in the data as it is plotted on a Control Chart. In this module, we will use the Nelson Rules. 15
16 The Nelson Rules are used in process control to determine if some measured variable is out of control, that is, unpredictable versus consistent. The rules are applied to a Control Chart on which the values of variable data points (or X-bar) are plotted against time. The rules are based around the mean value (or X-double bar) and the standard deviation of the samples. Attribute Chart Claims Process Four of the most frequently used Attribute Control Charts are the p chart, the np chart, the u chart, and the c chart. With Attribute data, there are two ways to look at, or count, the data: either by defectives or by defects. P and np charts are used for studying defectives, while u and c charts are used for studying defects. Before we continue, let s take a quick look at the difference between defectives and defects. To help clarify the difference between defects and defectives, consider this example: A medical insurance auditor examines Claim A and determines that it is "Correct; it has no errors. However, on Claim B, the auditor finds three errors. Claim B, therefore, is "Incorrect" and contains one or more errors. The auditor examines 40 claims in all. He finds 21 claims with errors. On those 21 Incorrect claims, there were a total of 44 errors. An improvement team could count the data in two ways. It could count the number of "Claims with Errors, which are referred to as defectives, or it could count the "Number of Errors" found, which are referred to as defects. In this example, one Incorrect claim is a defective unit, and each error is a defect. Therefore, the team could count the number defectives, which is 21, or it could count the number of defects, which is 44. The way in which a team decides to count data is a major factor in determining which type of Control Chart to use. You now understand what it means to count data as either defectives or defects. Now we will examine the four most commonly used Attributes Control Charts. To begin, let s look at the p chart. To illustrate how to set-up and use a p chart, let s continue with our previous medical claims example. The team decides to count data as defectives. It will use a p chart to track the proportion of claims that 16
17 have at least one error. Here, it is important to note that if the team had decided to count the number of good claims, instead of the bad ones, its p chart would be just as effective. The symbol p is used to represent the proportion of the sample that is defective. The p chart is a graph of these proportions over time. In our example, the team uses p to represent the proportion of Claims with Errors, or defectives. It calculates this proportion by dividing the number of claims found with errors, 21, by the total number of claims audited, 40. Therefore, the proportion of Claims with Errors for this work day is.525. Each time a sample of claims is audited, the team calculates the proportion of Claims with Errors, and plots the value on the chart. p Chart Although a computer can be used to record, analyze, and plot data on a Control Chart we will use this Attribute Control Chart form to illustrate the basic principles. This generic form for Attribute Charts contains three basic sections. The top section contains process and chart descriptive information. In our example, this top section shows that the team is constructing a p chart for processing medical claims, and that it is monitoring the proportion of Claims with Errors. The middle section of the chart is for recording the data. Here, for each subgroup, the auditor randomly sampled claims submitted for processing; recorded the number of claims found with errors and the number of claims examined; and calculated the proportion of Claims with Errors. Remember, this value is just the number of claims found with errors divided by the number of claims audited. The bottom section is for the chart itself. Here, the proportion of Claims with Errors is plotted on the chart. In this example, a daily sample was examined. Each point represents the proportion of defective claims for a given day. This chart shows how the process was doing over the last 25 days. 17
18 If you look carefully at the chart, you will notice that the sample or subgroup sizes are not equal. This is because number of claims audited each day can vary. Having equal subgroup or sample sizes are not a requirement of a p chart. p Chart Limits As you learned during our discussion of Variables Control Charts, all Control Charts share the same basic features. These common features are data from the process, a process average or centerline, and control limits calculated from the data. The centerline or process average for this a p chart is called p-bar. This value represents the proportion one would expect from a stable process. It is important to note that p-bar is not just the average of all the p values. With unequal subgroup sizes, p-bar is a weighted average of the individual sub-group proportions. A weighted average just means the individual p s don t all carry the same weight in the calculation of p-bar. Weighted Average In basic arithmetic, we learn that the average is calculated by adding up all given values and dividing by the number of values. For example, suppose you took two tests for a course, Test One and a Final exam, and your grades were 60 and 100, respectively. You would probably say your average grade was 80. This is because you assumed that both grades carry an equal weight. However, if the grades do not carry equal weight, this average will be inaccurate. To find the accurate average you need to know how much each test counts toward your final grade. Next, let s see what happens if the tests are not equally weighted. One way to visualize the weighted average is called the Seesaw Method. Take a look at the graphic shown here. In this example, Test One and the Final exam carry equal weight, with each one worth 50 percent of your grade. Therefore, if equal weights were placed on this Seesaw at 60 and 100, the seesaw would balance if the fulcrum was placed at the average, 80. See, it balances! 18
19 Now let s consider what happens when the Final exam is worth 75 percent of your final grade. The first thing that happens is that the weights change. The Final exam now counts for 75 percent of your class grade, and since there is only one other test, it must count for 25 percent. This means the Final exam is worth three times as much as Test One, because 75 is three times 25. On the Seesaw, that would be the same as placing three weights at 100 and one weight at 60 (all weights being identical). The average will move towards the value that is carrying more weight. It's like a tug-of-war. Calculating this value is simple. Take each grade, multiply it by the value of its assigned weight, and sum up the products. For this calculation to work, the assigned weights must be expressed as fractions or decimal equivalents, rather than percentages. Take a moment to review the calculation shown for this example. As you see, the weighted average is 90. p Chart Summary Let s review what we know about p charts. The p chart is used to monitor either the proportion of bad items or the proportion of good items. With either choice, the values plotted will always be a proportion that results in a number from zero to one. Unlike Variables Control Charts, which come in pairs, only the p chart is required to monitor the process average and dispersion. With Attribute data, the average and dispersion are closely related. Therefore, only one chart is needed. The p chart can also accommodate situations where the sub-group sample sizes vary. The control limits will vary, but the centerline stays constant. The larger the sub-group size, the narrower the limits. This is because the larger sample size reduces the sampling error. 19
20 np Chart There is a specialized version of the p chart, known as the np chart. The np chart can be used when the sample size is always the same. The np chart serves the same basic function as the p chart, but is used to monitor the number, rather than the proportion, of items with a defined characteristic. Let s examine the relationship between these two types of charts. In our earlier example, the p chart plotted the proportion of Claims with Errors. The np chart, however, does not require this proportion; it only requires the number of Claims with Errors for each subgroup. A very important difference between the p and np charts is that the np chart requires subgroups of equal size. In this example, a subgroup size of 40 was chosen. To construct and use the np chart, all the subgroup sizes must be the same. The scale on the vertical axis differs from the p chart as well. Because the p chart plotted a proportion, the scale of the y-axis went from zero to one. For the np chart, however, the data plotted is a series of whole numbers, with the minimum being zero. Like the p chart, the np chart has a centerline, or process average, and upper and lower control limits. The centerline here represents the overall average number of Claims with Errors found in a subgroup size of 40. The centerline is labeled np-bar. np Summary Let s review what we know about np charts. The np chart is used to track either the actual number of defective items or the actual number of good items. With either choice, the values plotted will always be the actual number counted per subgroup. As with the p chart, the average and dispersion are closely related. Therefore, only one chart is required. The np chart requires equal subgroup sizes. It should be noted that whenever an np chart is used, a p chart could have been selected instead. But it doesn t work the other way; an np chart cannot always be used in place of a p chart. 20
21 u Chart The next chart type we will discuss is the u chart. In this example, similar to our p chart example, a dental claims auditor randomly sampled claims submitted for processing. This time, however, the number of Errors per Claim is being measured, rather than the proportion of Claims with Errors. As with the p chart, for each sample taken, the auditor recorded the number of errors found and the number of claims audited. Only this time, the error rate per claim is recorded and plotted, rather than the proportion of defective claims. The error rate is the number of errors found, divided by the number of claims audited. This value is plotted on the chart. The data shown here represents the results of the audit for the previous 25 days. Notice that the sample or subgroup sizes are not equal. Equal subgroup size is not a requirement of a u chart. u Chart Limits Once enough data has been collected, the next step is to calculate and plot the value of the process average, or centerline. This value is called u-bar. The centerline for the u chart is plotted at this value. Similar to the p-bar on the p chart, u- bar is a weighted average of the individual u values. As always, the upper and lower control limits are calculated using the data from the process. These values are then plotted, and the process is examined for stability. Here we have a completed u chart. This chart is now ready to be used to monitor the claims payable process. 21
22 u Chart Summary Let s summarize what we know about the u chart. The u chart is used to track the number of occurrences of some event per unit. In our example, the event is an error. Because an item may have more than one error, the total number of errors may be greater than the number of units sampled per subgroup. This will result in u values that are greater than one. Similar to p and np charts, the average and dispersion are closely related. Therefore, only one chart is needed. Like the p chart, the u chart can accommodate situations where the subgroup sample sizes vary. The larger the subgroup size, the narrower the limits. This is because the larger sample size reduces the sampling error. c Chart There is a specialized version of the u chart, called the c chart. Previously, you learned that the p and np charts are closely related. A similar relationship exists between the u and c charts. To illustrate, let s go back to our dental claims example. Here, the team uses a c chart to monitor the Number of Errors found. The c chart requires that the subgroup size remains constant. This requirement assures that the chance, or opportunity, for errors to occur is the same from subgroup to subgroup. In this example, the auditor collects and records data that counts the Number of Errors found, in subgroup sizes of 40. Now that subgroup sizes are the same, the team plots the Attribute data on the c chart. As you may recall, the u chart plotted the error rate, or Errors per Claim. This error rate is not needed for the c chart. The scale on the y-axis differs as well. On the c-chart, the y-axis reflects the fact that the Number of Errors per subgroup, rather than the error rate, is being plotted. 22
23 c Chart Limits Once the team has plotted the data on the C chart, it calculates and draws the centerline, or c-bar. C-bar is just the average of all the subgroup c values. Next, the team calculates and plots the upper and lower control limits. This c chart is now complete, stable, and ready to use in monitoring the process. c Chart Summary Deciding which chart to use, the u or c chart, is very similar to deciding between the p chart and the np chart. A key deciding factor is the subgroup size. Is it constant or does it change from subgroup to subgroup? If the subgroup size changes or varies, the u chart is the correct one to use. Otherwise, with a constant subgroup size, both the u and c charts are appropriate. In summary, a c chart can be used to track occurrences per unit only when the number of units, or subgroup size, remains constant. On the other hand, a u chart can be used whether or not the subgroup size varies. When sample size is constant, the choice between the u chart and c chart may depend on whether it is more meaningful to plot the actual number of errors on a c chart or the error rate on a u-chart. 23
Introduction to Control Charts
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