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Hence, a cluster random sample is merely a simple random sample of these boxes. The operational definition is very important for collecting data because it can impact the control charts in many ways. With a larger sample size, there is a smaller probability for a given process shift. In phase I it is important that we calculate the preliminary control limits, which allows us to find the extent of variation in sample means and sample ranges if the process is stable. Notice, however, there is little difference in how we compute the control limits for the X chart and the X chart.
Though there is no official standards body that defines Six Sigma, there are accreditation organizations. Since classroom instruction is almost always necessary, start by looking for training classes close to you. Ensure that training should be provided by a six sigma certified trainer. After completing classroom training sessions, the real challenge is with the certification exam preparation. For making it simpler, you can take help of online practice exam.
Like I recommend processexam. Online guidance is necessary for the candidates to know the advanced nature of certification exam after which exam preparation becomes more fruitful.
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Wonderful assist! The online practice exam is a very good reflection of the important thing course ideas presented. If the population is in random order, a systematic random sample provides results comparable with those of a simple random sample. Moreover, a systematic sample usually covers the whole sampled population uniformly, which may provide more information about the population than that provided by a simple random sample. However, if the sampling units of a population are in some kind of periodic or cyclical order, the systematic sample may not be a representative sample.
Therefore, a systematic sample in this case will not be a representative sample. Similarly, suppose we want to estimate a workers productivity and we decide to take samples of his or her productivity every Friday afternoon.
We might be underestimating the true productivity. On the other hand, if we pick the same day in the middle of the week, we might be overestimating the true productivity. So, to have better results using systematic sampling, we must keep in mind that it includes the days when the productivity tends to be higher as well as lower.
If there is a linear trend, the systematic sample mean provides a more precise estimate of the population mean than that of a simple random sample but less precision than the stratified random sample. We denote the mean of the ith sample described earlier by y and the mean of a randomly selected systematic sample by y , where the i sy subscript sy means that the systematic sample was used. Note that the mean of a systematic sample is more precise than the mean of a simple random sample if, and only if, the variance within the systematic samples is greater than the population variance as a whole, that is,.
Sometimes the manufacturing scenario may warrant first stratifying the population and then taking a systematic sample within each stratum. For example, if there are several plants and we want to collect samples on line from each of these plants, then clearly we are first stratifying the population and then taking a systematic sample within each stratum.
In such cases, the formulas for estimating the population mean and total are the same as given in the previous section except that y i is replaced by y sy. The sample size needed to estimate the population total with margin of error E with probability 1 can be found from equation 2. However, before closing the deal, the company is interested in determining the mean timber volume per lot of one-fourth an acre.
To achieve its goal, the company conducted an every th systematic random sample and obtained the data given in the following table. Estimate the mean timber volume per lot and the total timber volume T in one thousand acres. Determine the 95 percent margin of error of estimation for estimating the mean and the total T and then find 95 percent confidence intervals for the mean and the total T.
To find the margin of error with 95 percent probability for the estimation of the mean and the total T, we need to determine the estimated variances of We first need to find the sample variance. The estimated variances of and T are given by ssy2 26, Note that in Example 2.
In order to attain the margin of error of 25 cubic feet, we will have to take a sample of at least size In systematic sampling it is normally not possible to take only a supplemental sample to achieve a sample of full size, which in Example 2.
Furthermore, we must keep the first randomly selected unit the same. Our new sample will then consist of lot numbers 7, 32, 57, 82, , , , and so on. Obviously, the original sample is part of the new sample, which means in this case we can take only a supplemental sample of size instead of taking a full sample of size That is, to prepare a good frame listing is very costly, or there are not enough funds to meet that kind of cost, or all sampling units are not easily accessible.
Another possible scenario is that all the sampling units are scattered, so obtaining observations on each sampling unit is not only very costly, but also very time consuming. In such cases we prepare the sampling frame consisting of larger sampling units, called clusters, such that each cluster consists of several original sampling units subunits of the sampled population.
Then we take a simple random sample of the clusters and make observations on all the subunits in the.
This technique of sampling is known as the cluster random sampling design, or simply the cluster sampling design. Cluster sampling is not only cost effective, but also a time saver because collecting data from adjoining units is cheaper, easier, and quicker than if the sampling units are far apart from each other. In manufacturing, for example, it may be much easier and cheaper to randomly select boxes that contain several parts rather than randomly selecting individual parts.
However, while conducting a cluster sampling may be cost effective, it also may be less efficient in terms of precision than a simple random sampling when the sample sizes are the same.
Also, the efficiency of cluster sampling may further decrease if we let the cluster sizes increase. Cluster samplings are of two kinds: In one-stage cluster sampling, we examine all the sampling subunits within the selected clusters. In two-stage cluster sampling, we examine only a portion of sampling subunits, which are chosen from each selected cluster using simple random sampling. Furthermore, in cluster sampling, the cluster sizes may or may not be of the same size.
Normally in field sampling it is not feasible to have clusters of equal sizes. For example, in a sample survey of a large metropolitan area, city blocks may be considered as clusters.
If the sampling subunits are households or persons, then obviously it will be almost impossible to have the same number of households or persons in every block. However, in industrial sampling, one can always have clusters of equal sizes; for example, boxes containing the same number of parts may be considered as clusters. We take a simple random sample of n clusters. We have the following: Then, an estimator of the population mean average value of the characteristic of interest per subunit is given by n.
An estimator of the population total T total value of the characteristic of interest for all subunits in the population is given by n. The margins of error with probability 1 for estimating the population mean and population total are given by. The pump model being investigated is typically installed in six applications, which include food service operations for example, pressurized drink dispensers , dairy operations, soft-drink bottling operations, brewery operations, wastewater treatment, and light commercial sump water removal.
The quality manager cannot determine the exact warranty repair cost for each pump; however, through company warranty claims data, she can determine the total repair cost and the number of pumps used by each industry in the six applications. In this case, the quality manager decides to use cluster sampling, using each industry as a cluster.
The data on total repair cost per industry and the number of pumps owned by each industry are provided in the following table. Estimate the average repair cost per pump and the total cost incurred by the industries over a period of one year. Determine a 95 percent confidence interval for the population mean and population total. To estimate the population mean , we proceed as follows: An estimate of the total number of pumps owned by the industries is Note that in order to determine the , for which we need to have a sample and sample size, we needed s2 and m thus the sample size, which in fact we want to determine.
The solution for this problem, as discussed in details in Applied Statistics for the Six Sigma Green Belt, is to evaluate these quantities either by using some similar data if already available or by taking a smaller sample, as we did in this example. However, the concept of SQC is less than a century old.
SPC consists of several tools that are useful for process monitoring. The quality control chart, which is the center of our discussion in this chapter, is one of these tools. Walter A. Shewhart, from Bell Telephone Laboratories, was the first person to apply statistical methods to quality improvement when, in , he presented the first quality control chart. Van Nostrand Co.
The publication of this book set the notion of SQC in full swing. In the initial stages, the acceptance of the notion of SQC in the United States was almost negligible.
It was only during World War II, when the armed services adopted the statistically designed sampling inspection schemes, that the concept of SQC started to gain acceptance in American industry. The complete adoption of modern SQC was limited from the s to the late s, however, because American manufacturers believed that quality and productivity couldnt go hand in hand.
American manufacturers argued that producing goods of high quality would cost more because high quality would mean buying raw materials of higher grade, hiring more qualified personnel, and providing more training to workers, all of which would translate into higher costs. American manufacturers also believed that producing higher-quality goods would slow down productivity because better-qualified and better-trained workers would not compromise quality to meet their daily quotas, which would further add to the cost of their product.
As a consequence, they believed they would not be able to compete in the world market and would therefore lose their market share.
The industrial revolution in Japan, however, proved that American manufacturers were wrong. This revolution followed the visit of a famous American statistician, W. Edwards Deming, in Deming , 2 wrote, Improvement of quality transfers waste of manhours and of machine-time into the manufacture of good product and better service.
The result is a chain reactionlower costs, better competitive position, Deming describes this chain reaction as follows: Improve quality cost decreases because of less work, fewer mistakes, fewer delays or snags, better use of machine time and materials productivity improves capture the market with better quality and lower price stay in business provide jobs and more jobs.
Once the management in Japan adopted the philosophy of this chain reaction, everyone had one common goal of achieving quality. We define it as Deming did: The customer may be internal or external, an individual or a corporation. If a product meets the needs of the customer, the customer is bound to buy that product again and again.
On the contrary, if a product does not meet the needs of a customer, then he or she will not buy that product even if he or she is an internal customer. Consequently, that product will be deemed of bad quality, and eventually it is bound to go out of the market. Other components of a products quality are its reliability, how much maintenance it demands, and, when the need arises, how easily and how fast one can get it serviced.
In evaluating the quality of a product, its attractability and rate of depreciation also play an important role.
As described by Deming in his telecast conferences, the benefits of better quality are numerous. First and foremost is that it enhances the overall image and reputation of the company by meeting the needs of its customers and thus making them happy.
A happy customer is bound to buy the product again and again. Also, a happy customer is likely to share a good experience with the product with friends, relatives, and neighbors.
Therefore, the company gets publicity without spending a dime, and this results in more sales and higher profits. Higher profits lead to higher stock prices, which means higher net worth of the company.
Better quality provides workers satisfaction and pride in their workmanship. A satisfied worker goes home happy, which makes his or her family happy. A happy family boosts the morale of a worker, which means greater dedication and loyalty to the company. Another benefit of better quality is decreased cost. This is due to the need for less rework, thus less scrap, fewer raw materials used, and fewer workforce hours and machine hours wasted. Ultimately, this means increased productivity, a better competitive position, increased sales, and a higher market share.
On the other hand, losses due to poor quality are enormous. Poor quality not only affects sales and the competitive position, but it also carries with it high hidden costs that are usually not calculated and therefore not known with precision.
These costs include unusable product, product sold at a discounted. In most companies, the accounting departments provide only minimum information to quantify the actual losses incurred due to poor quality. Lack of awareness concerning the cost of poor quality could lead company managers to fail to take appropriate actions to improve quality. A process may be defined as a series of actions or operations performed in producing manufactured or nonmanufactured products.
A process may also be defined as a combination of workforce, equipment, raw materials, methods, and environment that works together to produce output. The flowchart in Figure 3. The quality of the final product depends on how the process is designed and executed.
However, no matter how perfectly a process is designed and how well it is executed, no two items produced by the process are identical. The difference between two items is called the variation.
Such variation occurs because of two causes: Common causes or random causes 2. Special causes or assignable causes As mentioned earlier, the first attempt to understand and remove variation in any scientific way was made by Walter A.
Shewhart recognized that special or assignable causes are due to identifiable sources, which can be systematically identified and eliminated, whereas common or random causes are not due to identifiable sources and therefore cannot be eliminated without very expensive measures. These measures include redesigning the process, replacing old machines with new machines, and renovating part of or the whole system.
A process is considered in statistical control when only common causes are present. Deming , 5 states, But a state of statistical control is not a natural state for a manufacturing process.
It is instead an achievement, arrived at by elimination, one by one, by determined efforts, of special causes of excessive variation.
Because the process variation cannot be fully eliminated, controlling it is key. If process variation is controlled, the process becomes predictable. Otherwise the process is unpredictable. To achieve this goal, Shewhart introduced the quality control chart.
Shewhart control charts are normally used in phase I implementation of SPC, when a process is particularly susceptible to the influence of special causes and, consequently, is experiencing excessive variation or large shifts. Quality control charts can be divided into two major categories: In this chapter we will study control charts for variables.
Definition 3. Examples of a variable include the length of a tie-rod, the tread depth of a tire, the compressive strength of a concrete block, the tensile strength of a wire, the shearing strength of paper, the concentration of a chemical, the diameter of a ball bearing, and the amount of liquid in a ounce can, and so on.
In general, in any process there are some quality characteristics that define the quality of the process. If all such characteristics are behaving in a desired manner, then the process is considered stable and it will produce products of good quality.
If, however, any of these characteristics are not behaving in a desired manner, then the process is considered unstable and it is not capable of producing products of good quality.
A characteristic is usually determined by two parameters: In order to verify whether a characteristic is behaving in a desired manner, one needs to verify that these two parameters are in statistical control, which can be done by using quality control charts. In addition, there are several other such. These tools, including the control charts, constitute an integral part of SPC.
SPC is very useful in any process related to manufacturing, service, or retail industries. This set of tools consists of the following: Histogram 2. Stem-and-leaf diagram 3. Scatter diagram 4. Run chart also known as a line graph or a time series graph 5. Check sheet 6. Pareto chart 7. Cause-and-effect diagram also known as a fishbone or Ishikawa diagram 8. Defect concentration diagram 9. Control charts These tools of SPC form a simple but very powerful structure for quality improvement.
Once workers become fully familiar with these tools, management must get involved to implement SPC for an ongoing quality-improvement process. Management must create an environment where these tools become part of the day-to-day production or service process. The implementation of SPC without managements involvement and cooperation is bound to fail.
In addition to discussing these tools, we will also explore here some of the questions that arise while implementing SPC. Every job, whether in a manufacturing company or in a service company, involves a process.
As described earlier, each process consists of a certain number of steps. No matter how well the process is planned, designed, and executed, there is always some potential for variability.
In some cases this variability may be very little, while in other cases it may be very high.
If the variability is very little, it is usually due to some common causes that are unavoidable and cannot be controlled. If the variability is too high, we expect that in addition to the common causes there are some other causes, usually known as assignable causes, present in the process. Any process working under only common causes or chance causes is considered to be in statistical control.
If a process is working under both common and assignable causes, it is considered unstable, or not in statistical control. In this chapter and the two that follow, we will discuss the rest of these tools.
Check Sheet In order to improve the quality of a product, management must try to reduce the variation of all the quality characteristics; that is, the process must be brought to a stable condition.
In any SPC procedure used to stabilize a process, it is. The check sheet is an important tool to achieve this goal. We discuss this tool using a real-life example. Example 3. In order to identify these defects and their frequency, a study is launched.
This study is done over a period of four weeks. The data are collected daily and summarized in the following form see Table 3. The summary data not only give the total number of different types of defects but also provide a very meaningful source of trends and patterns of defects.
These trends and patterns can help find possible causes for any particular defect or defects. Note that the column totals in Table 3. It is important to remark here that these types of data become more meaningful if a logbook of all changes, such as a change in raw materials, calibration of machines, or training of workers or new workers hired, is well kept.
Pareto Chart The Pareto chart is a useful tool for learning more about attribute data quickly and visually. The Pareto chartnamed after its inventor, Vilfredo Pareto, an Italian economist who died in is simply a bar graph of attribute data in a descending order of frequency by, say, defect type.
For example, consider the data on defective rolls in Example 3. The chart allows the user to quickly identify those defects that occur more frequently and those that occur less frequently. This allows the user to priori-. For instance, the Pareto chart in Figure 3. Corrugation and blistering together are responsible for Corrugation, blistering, and streaks are responsible for 70 percent.
To reduce the overall rejection, one should first attempt to eliminate or at least reduce the defects due to corrugation, then blistering, streaks, and so on.
By eliminating these three types of defects, one would dramatically change the percentage of rejected paper and reduce the losses. It is important to note that if one can eliminate more than one defect simultaneously, then one should consider eliminating them even though some of them are occurring less frequently. Furthermore, after one or more defects are either eliminated or reduced, one should again collect the data and reconstruct the Pareto chart to determine whether the priority has changed.
If another defect is now occurring more frequently, one may divert the resources to eliminate such a defect first. Note that in this example, the category Other may include several defects such as porosity, grainy edges, wrinkles, or brightness that are not occurring very frequently.
So, if one has limited resources, one should not expend them on this category until all other defects are eliminated. Sometimes the defects are not equally important.
This is true particularly when some defects are life threatening while other defects are merely a nuisance or an inconvenience. It is quite common to allocate weights to each defect and then plot the weighted frequencies versus the defects to construct the Pareto chart. For example, suppose a product has five types of defects, which are denoted by A, B, C, D, and E, where A is life threatening, B is not life threatening but very serious, C is serious, D is somewhat serious, and E is not serious but merely a nuisance.
The data collected over a period of study are shown in Table 3. In Figure 3. That is, by using weighted frequencies, the order of priority of removing the defects is C, A, B, D, and E, whereas without using the weighted frequencies, this order would have been E, C, D, B, and A. Cause-and-Effect Fishbone or Ishikawa Diagram In an SPC-implementing scheme, identifying and isolating the causes of a particular problem or problems is very important.
An effective tool for identifying such causes is the cause-and-effect diagram. This diagram is also known as a fishbone diagram because of its shape, and sometimes it is called an Ishikawa diagram after its inventor.
Japanese manufacturers have widely used this diagram to improve the quality of their products. In preparing a causeand-effect diagram, it is quite common to use a brainstorming technique. The brainstorming technique is a form of creative and collaborative thinking. This technique is used in a team setting. The team usually includes personnel from departments of production, inspection, purchasing, design, and management, along with any other members associated with the product under discussion.
A brainstorming session is set up in the following manner: Each team member makes a list of ideas. The team members sit around a table and take turns reading one idea at a time. As the ideas are read, a facilitator displays them on a board so that all team members can see them. Steps 2 and 3 continue until all ideas have been exhausted and displayed.
Cross-questioning concerning a team members idea is allowed only for clarification. When all ideas have been read and displayed, the facilitator asks each team member if he or she has any new ideas. This procedure continues until no team member can think of any new ideas. Once all the ideas are presented using the brainstorming technique, the next step is to analyze them.
The cause-and-effect diagram is a graphical technique used to analyze these ideas. Figure 3. The five spines in Figure 3. In most workplaces, whether they are manufacturing or nonmanufacturing, the causes of all problems usually fall into one or more of these categories. Using a brainstorming session, the team brings up all possible causes under each category.
For example, under the environment category, a cause could be the managements attitude. Management might not be willing to release any funds for research or to change suppliers; there might not be much cooperation among middle and top management; or something similar. Under the personnel category, a cause could be lack of proper training for workers, supervisors who are not helpful in solving problems, lack of communication between workers and supervisors, and workers who are afraid of asking their supervisors questions for fear of repercussions in their jobs, promotions, or raises.
Once all possible causes under each major category are listed in the cause-and-effect diagram, the next step is to isolate one or more common causes and eliminate them. A complete cause-and-effect diagram may appear as shown in Figure 3.
Defect Concentration Diagram A defect concentration diagram is a visual representation of the product under study that depicts all the defects. This diagram helps the workers determine if there are any patterns or particular locations where the defects occur and what kinds of defects, minor or major, are occurring.
The patterns or particuEnvironment.
Workers underpaid No funds for research Not enough workers Lack of management cooperation. It is important that the diagram show the product from different angles. For example, if the product is in the shape of a rectangular prism and defects are found on the surface, then the diagram should show all six faces, very clearly indicating the location of the defects. The defect concentration diagram was useful when the daughter of one of the authors made a claim with a transportation company.
In , the author shipped a car from Boston, Massachusetts, to his daughter in San Jose, California. After receiving the car, she found that the front bumpers paint was damaged. She filed a claim with the transportation company for the damage, but the company turned it down, simply stating that the damage was not caused by the company. Fortunately, a couple days later, she found similar damage under the back bumper, symmetrically opposite the front bumper.
She again called the company and explained that this damage had clearly been done by the belts used to hold the car during transportation. This time the company could not turn down her claim, because by using a defect concentration diagram, she could prove that the damage was caused by the transportation company.
Run Chart In any SPC procedure it is very important to detect any trends that may be present in the data. Run charts help identify such trends by plotting data over a certain period of time. For example, if the proportion of nonconforming parts produced from shift to shift is perceived to be a problem, we may plot the number of nonconforming parts against the shifts for a certain period of time to determine whether there are any trends.
Trends usually help us identify the causes of nonconformities. This chart is particularly useful when the data are collected from a production process over a certain period of time.
A run chart for data in Table 3. From this run chart we can easily see that the percentage of nonconforming units is the lowest in the morning shift shifts 1, 4, 7,. There are also some problems in the evening shift, but they are not as severe as those in the night shift. Because such trends or patterns are usually created by special or assignable.
Does the quality of the raw materials differ from shift to shift? Is there inadequate training of workers in the later shifts? Are evening- and late-shift workers more susceptible to fatigue? Are there environmental problems that increase in severity as the day wears on? Deming , points out that sometimes the frequency distribution of a set of data does not give a true picture of the data, whereas a run chart can bring out the real problems of the data.
The frequency distribution gives us the overall picture of the data, but it does not show us any trends or patterns that may be present in the process on a short-term basis. Control charts make up perhaps the most important part of SPC. We will study some of the basic concepts of control charts in the following section, and then in the rest of this chapter and the next two chapters we will study the various kinds of control charts. As noted by Duncan , a control chart is 1 a device for describing in concrete terms what a state of statistical control is, 2 a device for attaining control, and 3 a device for judging whether control has been attained.
Before the control charts are implemented in any process, it is important to understand the process and have a concrete plan for future actions that would be needed to improve the process. Process Evaluation Process evaluation includes not only the study of the final product, but also the study of all the intermediate steps or outputs that describes the actual operating state of the process. For example, in the paper production process, wood chips and pulp may be considered as intermediate outputs.
If the data on process evaluation are collected, analyzed, and interpreted correctly, they can show where and when a corrective action is necessary to make the whole process work more efficiently.
Action on Process Action on process is important for any process because it prevents the production of an out-of-specification product. An action on process may include changes in raw materials, operator training, equipment, design, or other measures. The effects of such actions on a process should be monitored closely, and further action should be taken if necessary.
Action on Output If the process evaluation indicates that no further action on the process is necessary, the last action is to ship the final product to its destination. Obviously, such an action on the output is futile and expensive. We are interested in correcting the output before it is produced.
This goal can be achieved through the use of control charts. Any process with a lot of variation is bad and is bound to produce products of inferior quality. Control charts can help detect variation in any process. As described earlier, in any process there are two causes of variation: Variation No process can produce two products that are exactly alike or possess exactly the same characteristics. Any process is bound to contain some sources of variation.
The difference between two products may be very large, moderate, very small, or even undetectable, depending on the source of variation, but certainly there is always some difference.
For example, the moisture content in any two rolls of paper, the opacity in any two spools of paper, and the brightness of two lots of pulp will always vary. Our aim is to trace back as far as possible the sources of such variation and eliminate them. The first step is to separate the common and special causes of such sources of variation.
Common Causes or Random Causes Common causes or random causes are the sources of variation within a process that is in statistical control. The causes behave like a constant system of chance. While individual measured values may all be different, as a group they tend to form a pattern that can be explained by a statistical distribution that can generally be characterized by: Location parameter 2.
Dispersion parameter 3. Shape the pattern of variation, whether it is symmetrical, right skewed, left skewed, and so on Special Causes or Assignable Causes Special causes or assignable causes refer to any source of variation that cannot be adequately explained by any single distribution of the process output, as otherwise would be the case if the process were in statistical control. Unless all the special causes of variation are identified and corrected, they will continue to affect the process output in an unpredictable way.
Any process with assignable causes is considered unstable and hence not in statistical control. However, any process free of assignable causes is considered stable and therefore in statistical control. Assignable causes can be corrected by local actions, while common causes or random causes can be corrected only by actions on the system. Are usually required to eliminate special causes of variation 2. Can usually be taken by people close to the process 3.
Can correct about 15 percent of process problems Deming believed that as much as 6 percent of all system variations is due to special or assignable causes, while no more than 94 percent of the variations is due to common causes.
Actions on the System 1. Are usually required to reduce the variation due to common causes 2. Almost always require management action for correction 3.
Are needed to correct about 85 percent of process problems Furthermore, Deming points out that there is an important relationship between the two types of variations and the two types of actions needed to reduce such variations. We will discuss this point in more detail.
Special causes of variation can be detected by simple statistical techniques. These causes of variation are not the same in all the operations involved. Detecting special causes of variation and removing them is usually the responsibility of someone directly connected to the operation, although management is sometimes in a better position to correct them. Resolving a special cause of variation usually requires local action. The extent of common causes of variation can be indicated by simple statistical techniques, but the causes themselves need more exploration in order to be isolated.
It is usually managements responsibility to correct the common causes of variation, although personnel directly involved with the operation are sometimes in a better position to identify such causes and pass them on to management for an appropriate action. Overall, resolving common causes of variation usually requires action on the system.
As we noted earlier, about 15 percent or according to Deming, 6 percent of industrial process troubles are correctable by the local action taken by people directly involved with the operation, while 85 percent are correctable only by managements action on the system. Confusion about the type of action required is very costly to the organization in terms of wasted efforts, delayed resolution of trouble, and other aggravating problems.
So, it would be wrong to take local action for example, changing an operator or calibrating a machine when, in fact, management action on the system was required for example, selecting a supplier that can provide better and consistent raw materials. All of this reasoning shows that strong statistical analysis of any operation in any industrial production is necessary.
Control charts are perhaps the best tool to separate the special causes from the common causes. Control Charts 1. Are used to describe in concrete terms what a state of statistical control is 2. Are used to judge whether control has been attained and thus detect whether assignable causes are present 3.
Are used to attain a stable process Suppose we take a sample of size n from a process at approximately regular intervals, and for each sample we compute a sample statistic, say X. This statistic may be the sample mean, a fraction of nonconforming product, or any other appropriate measure. Now, because X is a statistic, it is subject to some fluctuation or variation. If no special causes are present, the variation in X will have characteristics that can be described by some statistical distribution.
By taking enough samples, we can estimate the desired characteristics of such a distribution. For instance, we now suppose that the statistic X is distributed as normal, and we divide the vertical scale of a graph in units of X, and the horizontal scale in units of time or any other such characteristic.
Then we draw horizontal lines through the mean, called the center line CL , and the extreme values of X, called the upper control limit UCL and the lower control limit LCL. This results in the device shown in Figure 3. The main goal of using control charts is to reduce the variation in the process and bring the process target value to the desired level. In other words, the process should be brought into a state of statistical control. If we plot data pertaining to a process on a control chart and the data conform to a pattern of random variation that falls within the upper and lower control limits, then we say that the process is in statistical control.
If, however, the data fall outside these control limits and do not conform to a pattern of random variation, then the process is considered to be out of control. In the latter case, an investigation is launched to find and correct the special causes responsible for the process being out of control. If any special cause of variation is present, an effort is made to eliminate it.
In this manner, the process can eventually be brought into a state of statistical control. Shewhart strongly recommended that a process should not be judged to be in control unless the pattern of random variation has persisted for some time and for a sizable volume of output. If a control chart shows a process in statistical control, it does not mean that all special causes have been completely eliminated; rather, it simply means that for all practical purposes, it is reasonable to assume or adopt a hypothesis of common causes only.
Preparation for Use of Control Charts In order for control charts to serve their intended purpose, it is important to take the following preparatory steps prior to implementing the control charts: Establish an environment suitable for action: Any statistical method will fail unless management has prepared a responsive environment.
Define the process and determine the characteristics to be studied: The process must be understood in terms of its relationship to the other operations and users and in terms of the process elements for example, people, equipment, materials, methods, and environment that affect it at each stage.
Some of the techniques discussed earlier, such as the Pareto chart or the fishbone diagram, can help make these relationships visible. Once the process is well understood, the next step is to determine which characteristics are affecting the process, which characteristics should be studied in depth, and which characteristics should be controlled.
Determine the correlation between characteristics: For an efficient and effective study, take advantage of the relationship between characteristics. If several characteristics of an item are positively correlated, it may be sufficient to chart only one of them.
If there are some characteristics that are negatively correlated, a deeper study is required before any corrective action on such characteristics can be taken. Define the measurement system: Templates and Calculators. Project Acceleration Techniques. Search active job openings related to Six Sigma.