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Statistical Quality control msbte-mqc

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  • 1. Statistical quality control INSTUCTIONAL OBJECTIVES: On completion of this lesson, the student shall learn: 1. Describe categories of statistical quality control (SQC). 2. Identify and describe causes of variation. 3. Explain the use of descriptive statistics such as mean, median and mode and frequency distribution 4. Explain the meaning of process capability and the process capability index. 5. Understand the meaning of statistical process control and control charts. 6. Describe the use of control charts. 7. Identify the differences between x-bar, R-, p-, and c-charts. 8. Explain the process of acceptance sampling and describe the use of operating characteristic (OC) curves. PREREQUISITES: In order to complete this section student should be familiar with 1. Quality characteristics. 2. Quality control and TQM concepts. Statistical quality control: When statistical techniques are employed to control the quality or to solve the quality related problems, then it is called statistical quality control. Statistical quality control can be divided into three broad categories: 1. Statistics is used to describe quality characteristics and relationships. It includes mean, standard deviation, the range, and a measure of the distribution of data. 2. Statistical process control (SPC) involves inspecting a random sample of the output from a process and deciding whether the process is producing products with characteristics that fall within a predetermined range. SPC answers the question of whether the process is functioning properly or not. 3. Acceptance sampling is the process of randomly inspecting a sample of goods and deciding whether to accept the entire lot based on the results. Acceptance sampling determines whether a batch of goods should be accepted or rejected. Variation: It is difficult to produce all items of products exactly alike. Variation means “ No two parts are made exactly alike”. The process involves various elements such as Man, machine and material. Each of these elements contributes to cause of variation in final product. Data: Data is the set of observations collected for the quality control purposes. Variable data and attribute data:
  • 2. Both of these statistical techniques, acceptance sampling and control charts may be applied to two kinds of data. 1. Attribute Data: when the quality characteristic being investigated is noted by either its presence or absence and then classified as Defective or Non-Defective. Example: Conforming or non-conforming, Pass or fail, Good or bad 2. Variable Data: The characteristics are actually measured and can take on a value along a continuous scale. Example: Length, Weight Thus, attribute data does not have information of “how much good or how much bad?” which the variable data would have, because it would record the exact measurements of each shaft. Frequency Distribution: It is the tabulation of frequently occurred data. Frequency distribution is used for predicting entire lots character from sample, for determining process capabilities, for comparing inspection results for two machines, for analysis of tool wear etc. The various/pictorial ways of representing data or frequency: (a) Frequency distribution table (b)Frequency histogram (c) Bar chart (d)Frequency polygon (e) Ogive curves Central Tendency: One of the salient characteristics of the distribution of the sample data is that most of the observations tend to concentrate in the centre of distribution is known as central tendency. The three ways for expressing central tendency are: (a) mean, (b) median (c) mode. Mean: The arithmetic average, or the mean, is a statistic that measures the central tendency of a set of data. Knowing the central point of a set of data is highly important. To compute the mean we simply sum all the observations and divide by the total number of observations. The equation for computing the mean is Where, Median:
  • 3. Diamete r in mm. 2.6 2.7 2.7 2.8 2.8 2.8 2.9 2.9 3.0 Table 1. “The value which divides a series of ordered observations so that the number of items above it is equal to the number below it. It is another measure of central tendency. For example the diameter of a part is shown in table 1 below: Here the number of observations is odd i.e. there are 9 observations. So median is the midpoint of the values i.e. 2.8 mm. (Leaving 4 observations above and 4 observations below i.e. number of observations above is equal to number of observations below.) Mode: The mode is the statistic showing the value that occurs most frequently. In table no 1, the value which has greatest frequency is 2.8mm, which is three times there in the table. Dispersion: the extent to which the data is scattered about the zone of central tendency is known as dispersion. The measures of dispersion are: (a) Range R, (b) Standard deviation (σ), Variance (δ) Range: Range is the difference between the largest and smallest observations in a set of data. R = xmax - xmin In quality control, range provides knowledge of total spread of data. Standard deviation: A statistic that measures the amount of data dispersion around the mean Where,
  • 4. In quality control, standard deviation gives more precise measure of variation and is a measure of dispersion of quality control work. Small values of the range and standard deviation mean that the observations are closely clustered around the mean. Large values of the range and standard deviation mean that the observations are spread out around the mean. Statistical process control methods: Statistical process control methods extend the use of descriptive statistics to monitor the quality of the product and process. As we have learned so far, there are common and assignable causes of variation in the production of every product. Using statistical process control we want to determine the amount of variation that is common or normal. Then we monitor the production process to make sure production stays within this normal range. That is, we want to make sure the process is in a state of control. The most commonly used tool for monitoring the production process is a control chart. Different types of control charts are used to monitor different aspects of the production process. Control charts: Definition: The control chart may be defined as ‘a graphical method for finding whether a process is in a state of control or not in a state of control’. A control chart (also called process chart or quality control chart) is a graph that shows whether a sample of data falls within the common or normal range of variation. A control chart has upper and lower control limits that separate common from assignable causes of variation. The common range of variation is defined by the use of control chart limits. We say that a process is out of control when a plot of data reveals that one or more samples fall outside the control limits. Types of Control Charts: Control charts are one of the most commonly used tools in statistical process control. They can be used to measure any characteristic of a product, such as the weight of a cereal box, the number of chocolates in a box, or the volume of bottled water. The different characteristics that can be measured by control charts can be divided into two groups: variables and attributes. A control chart for variables is used to monitor characteristics that can be measured and have a continuum of values, such as height, weight, or volume. A soft drink bottling operation is an example of a variable measure, since the amount of liquid in the bottles is measured and can take on a number of different values. Other examples are the weight of a bag of sugar, the temperature of a baking oven, or the diameter of plastic tubing. A control chart for attributes, on the other hand, is used to monitor characteristics that have discrete values and can be counted. Often they can be evaluated with a simple yes or no decision. Examples include color, taste, or smell. The monitoring of attributes usually takes less time than that of variables because a variable needs to be measured
  • 5. (e.g., the bottle of soft drink contains 15.9 ounces of liquid). An attribute requires only a single decision, such as yes or no, good or bad, acceptable or unacceptable (e.g., the apple is good or rotten, the meat is good or stale, the shoes have a defect or do not have a defect, the light bulb works or it does not work) or counting the number of defects (e.g., the number of broken cookies in the box, the number of dents in the car, the number of barnacles on the bottom of a boat). Statistical process control is used to monitor many different types of variables and attributes. Charts Used with Variable Data: X and R-Charts (mean and range charts) are commonly used in dealing with variable data to monitor the quality of a manufacturing process. The reason that both the charts have to be used together is that both the mean and the variation (spread) have to be under control. Recall that the variable data consists of actual measurements (e.g., shaft lengths, weight of bags in lbs, etc.) The Control Limits (UCL = Upper Control Limit and LCL = Lower Control Limit with the mean of the data as the central line) for X and R Charts are established as follows: X-Charts is further estimated using the range information as such the control limit calculations are much simplified. The simplified control limits are as follows: Where A2 is a factor available in tables for different sample sizes (see table below).
  • 6. R-Charts: Similarly, control limit calculations are much simplified and are: Where D3 and D4 are factors available in tables for different sample sizes (see above table.) Charts Used with Attribute Data: P-Chart, also known as the fraction or percent defective chart, is commonly used in dealing with attribute data to monitor the quality of a manufacturing process. The mean percent defective ( p ) is the central line. The upper and lower control limits are constructed as follows:
  • 7. Usually the Z value is equal to 3 (as was used in the X and R charts), since the variations within three standard deviations are considered as natural variations. However, the choice of the value of Z depends on the environment in which the chart is being used, and on managerial judgment. C-charts: These are used to monitor the number of defects per unit. Examples are the number of returned meals in a restaurant, the number of trucks that exceed their weight limit in a month, the number of discolorations on a square foot of carpet, and the number of bacteria in a milliliter of water. Note that the types of units of measurement we are considering are a period of time, a surface area, or a volume of liquid. The average number of defects is the center line of the control chart. The upper and lower control limits are computed as follows: Acceptance sampling: Acceptance sampling refers to the process of randomly inspecting a certain number of items from a lot or batch in order to decide whether to accept or reject the entire batch. 100% inspection Sampling inspection 1. It involves sorting and therefore results in fatigue of the inspector. 1. It causes less fatigue to the inspectors. 2. Since every part is to be inspected there is more damage during the inspection. 2. Less damage during the inspection. 3. It is time consuming and requires more manpower. 3. It is not time consuming. 4. It is not possible in destructive tests. 4. Use of sampling inspection is must during destructive testing. 5. Since hundred percent inspection is done there are less chances of rejecting good lots. 5. There is risk of accepting bad lots and rejecting good lots. Sampling Plans A sampling plan is a plan for acceptance sampling that precisely specifies the parameters of the sampling process and the acceptance/rejection criteria. The variables to be specified include the size of the lot (N), the size of the sample inspected from the
  • 8. lot (n), the number of defects above which a lot is rejected (c), and the number of samples that will be taken. There are different types of sampling plans. The first one is the single sampling, in which a random sample is drawn from every lot. Each item in the sample is examined and is labeled as either “good” or “bad.” Depending on the number of defects or “bad” items found, the entire lot is either accepted or rejected. For example, a lot size of 50 cookies is evaluated for acceptance by randomly inspecting 10 cookies from the lot. The cookies may be inspected to make sure they are not broken or burned. If 4 or more of the 10 cookies inspected are bad, the entire lot is rejected. In this example, the lot size N _ 50, the sample size n _ 10, and the maximum number of defects at which a lot is accepted is c _ 4. The second type of acceptance sampling is called double sampling plan. This provides an opportunity to sample the lot a second time if the results of the first sample are inconclusive. In double sampling we first sample a lot of goods according to preset criteria for definite acceptance or rejection. However, if the results fall in the middle range, they are considered inconclusive and a second sample is taken. The third type of sampling plan is the multiple sampling plan. Multiple sampling plans are similar to double sampling plans except that criteria are set for more than two samples. THE O.C. CURVE: The x axis shows the percentage of items that are defective in a lot. This is called “lot quality.” The y axis shows the probability or chance of accepting a lot. You can see that if we use 100 percent inspection we are certain of accepting only lots with zero defects. However, as the proportion of defects in the lot increases, our chance of accepting the lot decreases. For example, we have a 90 percent probability of accepting a lot with 5 percent defects and an 80 percent probability of accepting a lot with 8 percent defects. Regardless of which sampling plan we have selected, the plan is not perfect. That is, there is still a chance of accepting lots that are “bad” and rejecting “good” lots. The steeper the OC curve, the better our sampling plan is for discriminating between “good” and “bad.”
  • 9. DEFINITION OF O.C.CURVE: O.C. curve is a graph that shows the probability or chance of accepting a lot given various proportions of defects in the lot. There is a small percentage of defects that consumers are willing to accept. This is called the acceptable quality level (AQL) and is generally in the order of 1–2 percent. However, sometimes the percentage of defects that passes through is higher than the AQL. Consumers will usually tolerate a few more defects, but at some point the number of defects reaches a threshold level beyond which consumers will not tolerate them. This threshold level is called the lot tolerance percent defective (LTPD). LTPD is the upper limit of the percentage of defective items consumers are willing to tolerate. Consumer’s risk is the chance or probability that a lot will be accepted that contains a greater number of defects than the LTPD limit. This is the probability of making a Type II error—that is, accepting a lot that is truly “bad.” Consumer’s risk or Type II error is generally denoted by beta (β). Producer’s risk is the chance or probability that a lot containing an acceptable quality level will be rejected. This is the probability of making a Type I error—that is, rejecting a lot that is “good.” It is generally denoted by alpha (α) producer’s risk. We can determine from an OC curve what the consumer’s and producer’s risks are. Process capability The ability of a production process to meet or exceed preset specifications. A critical aspect of statistical quality control is evaluating the ability of a production process to meet or exceed preset specifications. This is called process capability. To
  • 10. understand exactly what this means, let’s look more closely at the term specification. Product specifications, often called tolerances, are preset ranges of acceptable quality characteristics, such as product dimensions. For a product to be considered acceptable, its characteristics must fall within this preset range. Otherwise, the product is not acceptable. Product specifications, or tolerance limits, are usually established by design engineers or product design specialists. For example, the specifications for the width of a machine part may be specified as 15 inches + 3. This means that the width of the part should be 15 inches, though it is acceptable if it falls within the limits of 14.7 inches and 15.3 inches. As we have learned, any production process has a certain amount of natural variation associated with it. To be capable of producing an acceptable product, the process variation cannot exceed the preset specifications. Process capability thus involves evaluating process variability relative to preset product specifications in order to determine whether the process is capable of producing an acceptable product.