Fig. 17 Second revision control charts for Table 2 data with two more sample deletions (samples 1 and 11, both of which exceed UCL in Fig. 16 ) resulting from workpieces produced prior to machine being properly warmed up. (a) -chart. (b) R-chart. Data now have k = 16, n = 5. Importance of Using Both and R Control Charts. This example points to the importance of maintaining both and R control charts and the significance of first focusing attention on the R-chart and establishing its stability. Initially, no points fell outside the -chart control limits and one could be led to believe that this indicates that the process mean exhibits good statistical control. However, the fact that the R-chart was initially not in control caused the limits on the - chart to be somewhat wider because of two inordinately large R values. Once these special causes of variability were removed, the limits on the -chart became narrower, and two values now fall outside these new limits. Special causes were present in the data, but initially were not recognizable because of the excess variability as seen in the R-chart. This example also points strongly to the need to have 25 or more samples before initiating control charts. In this case, once special causes were removed, only 16 subgroups remained to construct the charts. This is simply not enough data. Importance of Rational Sampling Perhaps the most crucial issue to the successful use of the Shewhart control chart concept is the definition and collection of the samples or subgroups. This section will discuss the concept of rational sampling, sample size, sampling frequency, and sample collection methods and will review some classic misapplications of rational sampling. Also, a number of practical examples of subgroup definition and selection will be presented to aid the reader in understanding and implementing this central aspect of the control chart concept. Concept of Rational Sampling. Rational subgroups or samples are collections of individual measurements whose variation is attributable only to one unique constant system of common causes. In the development and continuing use of control charts, subgroups or samples should be chosen in a way that provides the maximum opportunity for the measurements within each subgroup to be alike and the maximum chance for the subgroups to differ from one another if special causes arise between subgroups. Figure 18 illustrates the notion of a rational sample. Within the sample or subgroup, only common cause variation should be present. Special causes/sporadic problems should arise between the selection of one rational sample and another. Fig. 18 Graphical depiction of a rational subgroup illustrating effect of special causes on mean. (a) Unshifted. (b) Shifted Sample Size and Sampling Frequency Considerations. The size of the rational sample is governed by the following considerations: • Subgroups should be subject to common cause variation. The sample size should be small to minimize the chance of mixing data within one sample from a controlled process and one that is out of control. This generally means that consecutive sample selection should be used rather than distributing the sample selection over a period of time. There are, however, certain situations where distributed sampling may be preferred • Subgroups should ensure the presence of a normal distribution for the sample means. In general, the larger the sample size, the better the distribution is represented by the normal curve. In practice, sample sizes of four or more ensure a good approximation to normality • Subgroups should ensure good sensitivity to the detection of assignable causes. The larger the sample size, the more likely that a shift of a given magnitude will be detected When the above factors are taken into consideration, a sample/subgroup size of four to six is likely to emerge. Five is the most commonly used number because of the relative ease of further computation. Sampling Frequency. The question of how frequently samples should be collected is one that requires careful thought. In many applications of and R control charts, samples are selected too infrequently to be of much use in identifying and solving problems. Some considerations in sample frequency determination are the following: • If the process under study has not been charted before and appears to exhibit somewhat erratic behavior, samples should be taken quite frequently to increase the opportunity to quickly identify improvement opportunities. As the process exhibits less and less erratic behavior, the sample interval can be lengthened • It is important to identify and consider the frequency with which occurrences are taking place in the process. This might include, for example, ambient condition fluctuations, raw material changes, and process adjustments such as tool changes o r wheel dressings. If the opportunity for special causes to occur over a 15-min period is good, sampling twice a shift is likely to be of little value • Although it is dangerous to overemphasize the cost of sampling in the short term, clearly it cannot be neglected Common Pitfalls in Subgroup Selection. In many situations, it is inviting to combine the output of several parallel and assumed-to-be-identical machines into a single sample to be used in maintaining a single control chart for the process. Two variations of this approach can be particularly troublesome: stratification and mixing. Stratification of the Sample. Here each machine contributes equally to the composition of the sample. For example, one measurement each from four parallel machines yields a sample/subgroup of n = 4, as seen in Fig. 19. In this case, there will be a tremendous opportunity for special causes (true differences among the machine) to occur within subgroups. Fig. 19 Block diagram depicting a stratified sample selection When serious problems do arise, for example, for one or more of the machines, they will be very difficult to detect because of the use of stratified samples. This problem can be detected, however, because of the unusual nature of the - chart pattern (recall the previous pattern analysis) and can be rectified provided the concepts of rational sampling are understood. The R-charts developed from such data will usually show very good control. The corresponding control chart will show very wide limits relative to the plotted values, and their control will therefore appear almost too good. The wide limits result from the fact that the variability within subgroups is likely to be subject to more than merely common causes (Fig. 20). Fig. 20 Typical control charts obtained for a stratified sample selection. (a) -chart. (b) R-chart Mixing Production From Several Machines. Often it is inviting to combine the output of several parallel machines/lines into a single stream of well-mixed product that is then sampled for the purposes of maintaining control charts. This is illustrated in Fig. 21. Fig. 21 Block diagram of sampling from a mixture If every sample has exactly one data point from each machine, the result would be the same as that of stratified sampling. If the sample size is smaller than the number of machines with different means or if most samples do not include data from all machines, the within-sample variability will be too low, and the between-sample differences in the means tend to be large. Thus, the -chart would give an appearance that the values are too far away from the centerline. Statistical Quality Design and Control Richard E. DeVor, University of Illinois, Urbana-Champaign; Tsong-how Chang, University of Wisconsin, Milwaukee Zone Rules for Control Chart Analysis Special causes often produce unnatural patterns that are not as clear cut as points beyond the control limits or obvious regular patterns. Therefore, a more rigorous pattern analysis should be conducted. Several useful tests for the presence of unnatural patterns (special causes) can be performed by dividing the distance between the upper and lower control limits into zones defined by , 2 , and 3 boundaries, as shown in Fig. 22. Such zones are useful because the statistical distribution of follows a very predictable pattern the normal distribution; therefore, certain proportions of the points are expected to fall within the ± boundary, between and 2 , and so on. The following sections discuss eight tests that can be applied to the interpretation of and R control charts. Not all of these tests follow/use the zones just described, but it is useful to discuss all of these rules/tests together. These tests provide the basis for the statistical signals that indicate that the process has undergone a change in its mean level, variability level, or both. Some of the tests are based specifically on the zones defined in Fig. 22 and apply only to the interpretation of the -chart patterns. Some of the tests apply to both charts. Unless specifically identified to the contrary, the tests/rules apply to the consideration of data to one side of the centerline only. When a sequence of points on the chart violates one of the rules, the last point in the sequence is circled. This signifies that the evidence is now sufficient to suggest that a special cause has occurred. The issue of when that special cause actually occurred is another matter. A logical estimation of the time of occurrence may be the beginning of the sequence in question. This is the interpretation that will be used here. It should be noted that some judgment and latitude should be given. Figure 23 illustrates the following patterns: • Test 1(extreme points): The existence of a single point beyond zone A signals the presence of an out-of- control condition (Fig. 23a) • Test 2 (2 out of 3 points in zone A or beyond): The existence of 2 out of any 3 successive points in zone A or beyond signals the presence of an out-of-control condition (Fig. 23b) • Test 3 (4 out of 5 points in zone B or beyond): A situation in which there are 4 out of 5 successive points in zone B or beyond signals the presence of an out-of-control condition (Fig. 23c) • Test 4 (runs above or below the centerline): Long runs (7 or more successive points) either strictly above or strictly below the centerline; this rule applies to both the and R control charts (Fig. 23d) • Test 5 (trend identification): When 6 successive points on either the or the R co ntrol chart show a continuing increase or decrease, a systematic trend in the process is signaled (Fig. 23e) • Test 6 (trend identification): When 14 successive points oscillate up and down on either the or R control chart, a systematic trend in the process is signaled (Fig. 23f) • Test 7 (avoidance of zone C test): When 8 successive points, occurring on either side of the center line, avoid zone C, an out-of- control condition is signaled. This could also be the pattern due to mixed sampling (discussed earlier), or it could also be signaling the presence of an over- control situation at the process (Fig. 23g) • Test 8 (run in zone C test): When 15 successive points on the - chart fall in zone C only, to either side of the centerline, an out-of- control condition is signaled; such a condition can arise from stratified Fig. 22 Control chart zones to aid chart interpretation sampling or from a change (decrease) in process variability (Fig. 23h) The above tests are to be applied jointly in interpreting the charts. Several rules may be simultaneously broken for a given data point, and that point may therefore be circled more than once, as shown in Fig. 24 Fig. 23 Pattern analysis of -charts. Circ led points indicate last point in a sequence of points on a chart that violates a specific rule. Fig. 24 Example of simultaneous application of more than one test for out-of- control conditions. Point A is a violation of tests 3 and 4; point B is a violation of tests 2, 3, and 4; and point C is a violation of tests 1 and 3. See text for discussion. In Fig. 24, point A is circled twice because it is the end point of a run of 7 successive points above the centerline and the end point of 4 of 5 successive points in zone B or beyond. In the second grouping in Fig. 24, point B is circled three times because it is the end point of: • A run of 7 successive points below the centerline • 2 of 3 successive points in zone A or beyond • 4 of 5 successive points in zone B or beyond Point C in Fig. 24 is circled twice because it is an extreme point and the end point of a group of five successive points, four of which are in zone B or beyond. Two other points (D, E) in these groupings are circled only once because they violate only one rule. Statistical Quality Design and Control Richard E. DeVor, University of Illinois, Urbana-Champaign; Tsong-how Chang, University of Wisconsin, Milwaukee Control Charts for Individual Measurements In certain situations, the notion of taking several measurements to be formed into a rational sample of size greater than one simply does not make sense, because only a single measurement is available or meaningful at each sampling. For example, process characteristics such as oven temperature, suspended air particulates, and machine downtime may vary during a short period at sampling. Even for those processes in which multiple measurements could be taken, they would not provide valid within-sample variation for control chart construction. This is so because the variation among several such measurements would be primarily attributed to variability in the measurement system. In such a case, special control charts can be used. Commonly used control charts for individual measurements include: • x, R m control charts • Exponentially weighted moving average (EWMA) charts • Cumulative sum charts (CuSum charts) Both the EWMA (Ref 13, 14, 15, 16) and the CuSum (Ref 17, 18, 19, 20, 21) control charts can be used for charting sample means and other statistics in addition to their use for charting individual measurements. x and R m (Moving-Range) Control Charts. This is perhaps the simplest type of control chart that can be used for the study of individual measurements. The construction of x and R m control charts is similar to that of and R control charts except that x stands for the value of the individual measurements and R m for the moving range, which is the range of a group of n consecutive individual measurements artificially combined to form a subgroup of size n (Fig. 25). The moving range is usually comprised of the largest difference in two or three successive individual measurements. The moving ranges are calculated as shown in Fig. 25 for the case of three consecutive measurements used to form the artificial samples of size n = 3. Fig. 25 Examples of three successive measurements used to determine the moving range Because the moving range, R m , is calculated primarily for the purpose of estimating common cause variability of the process, the artificial samples that are formed from successive measurements must be of very small size to minimize the chance of mixing data from out-of-control conditions. It is noted that x and R m are not independent of each other and that successive sample R m values are overlapping. The following example illustrates the construction of x and R m control charts, assuming that x follows at least approximately a normal distribution. Here, R m is based on two consecutive measurements; that is, the artificial sample size is n = 2. Example 2: x and R m Control Chart Construction for the Batch Processing of White Millbase Component of a Topcoat. The operators of a paint plant were studying the batch processing of white millbase used in the manufacture of topcoats. The basic process begins by charging a sandgrinder premix tank with resin and pigment. The premix is agitated until a homogeneous slurry is obtained and then pumped through the sandgrinder. The grinder output is sampled to check for fineness and gloss. A batch may require adjustments by adding pigment or resin to achieve acceptable gloss. Through statistical modeling of the results of some ash tests, a quantitative method was developed for determining the amount of pigment or resin to be added when necessary, all based on the weight per unit volume (lb/gal.) of the batch. Therefore, it became important to monitor the weight per unit volume for each batch to achieve millbase uniformity. Table 3 lists weight per unit volume data for 27 consecutive batches. Table 3 x and R m control chart data for the batch processing of white millbase topcoat component of Example 2 Batch x, lb/gal. R m (a) 1 14.04 2 13.94 0.10 (14.04 - 13.94 = 0.10) 3 13.82 0.12 (13.94 - 13.82 = 0.12) 4 14.11 0.29 (14.11 - 13.82 = 0.29) [...]... Process Control, Int J Prod Res., Vol 10 (No 4), 1972, p 39 3-4 00 17 A.F Bissell, An Introduction to CuSum Charts, The Institute of Statisticians, 1984 18 "Guide To Data Analysis and Quality Control Using CuSum Techniques," BS5703 (4 parts), British Standards Institution, 198 0-1 982 19 J.M Lucas, The Design and Use of V-Mask Control Scheme, J Qual Technol., Vol 8 (No 1), 1976, p 1-1 2 20 J Murdoch, Control. .. 3, there would be only 27 - 2 = 25 moving averages Once the and m values are calculated, they are used as centerline values of x and Rm control charts, respectively The calculation of upper and lower control limits for the Rm control chart is also the same as in , R control charts, using the artificial sample size n to determine D3 and D4 values However, the upper and lower control limits for the x... statistical control Statistical Quality Design and Control Richard E DeVor, University of Illinois, Urbana-Champaign; Tsong-how Chang, University of Wisconsin, Milwaukee Design of Experiments: Factorial Designs The process of product design and its associated manufacturing processes and tolerance designs often involve many experiments to better understand the various cause-effect relationships for quality. .. Fig 32 u control chart obtained for the evaluation of leather handbag lot data in Table 6 Data are for k = 25, n = 10 Datum for sample 9 is an extreme point because it exceeds value of UCL Reference cited in this section 22 I Burr, Statistical Quality Control Methods, Marcel Dekker, 1976 Statistical Quality Design and Control Richard E DeVor, University of Illinois, Urbana-Champaign; Tsong-how Chang,... of Tolerances and Control Limits It is important to clearly differentiate between specification limits and control limits The specification limits or tolerances of a part are: • • • • Characteristic of the part/ item in question Based on functional considerations Related to/compared with an individual part measurement Used to establish the conformability of a part The control limits on a control chart... Macmillan, 1979 21 J.S Oakland, Statistical Process Control, William Heinemann, 1986 Statistical Quality Design and Control Richard E DeVor, University of Illinois, Urbana-Champaign; Tsong-how Chang, University of Wisconsin, Milwaukee Shewhart Control Charts for Attribute Data Many quality assessment criteria for manufactured goods are not of the variable measurement type Rather, some quality characteristics... molded part at a steady pace Suppose the measure of quality conformance of interest is the occurrence of flash and splay on the molded part If a part has so much as one occurrence of either flash or splay, it is considered to be nonconforming, that is, a defective part To establish the control chart, rational samples of size n = 50 parts are drawn from production periodically (perhaps, each shift), and. .. the control limits for the p-chart are then given by: UCLp = + 3 LCPp = - 3 (Eq 4a) (Eq 4b) Thus, only has to be calculated for at least 25 samples of size n to set up a p-chart The binomial distribution is generally not symmetric in quality control applications and has a lower bound of p = 0 Sometimes the calculation for the lower control limit may yield a value of less than 0 In this case, a lower control. .. The notion of random assembly, that is, random part selection from these part process distributions when more than one part is being considered in an assembly The additive law of variances as a means to determine the relationship between the variability in individual parts and that for the assembly To assume that the parts can be represented by a statistical distribution of measurements (and for the assumption... 0.01 17 3 0.03 18 2 0.02 19 4 0.04 20 2 0.02 21 1 0.01 22 2 0.02 23 0 0.00 24 2 0.02 25 3 0.03 26 4 0.04 27 1 0.01 28 0 0.00 29 0 0.00 30 0 0.00 31 0 0.00 32 1 0.01 33 2 0.02 34 3 0.03 35 3 0.03 (a) n = 100 Using this sample data to establish the p-chart: Therefore: UCLp = 0. 0208 6 + 0.04287 = 0.06373 LCLp = 0. 0208 6 - 0.04287 = -0 .0 2201 That is: LCLp = 0 The plot of the data on the corresponding p-chart . Data Analysis and Quality Control Using CuSum Techniques," BS5703 (4 parts), British Standards Institution, 198 0-1 982 19. J.M. Lucas, The Design and Use of V-Mask Control Scheme, J 1), 1976, p 1-1 2 20. J. Murdoch, Control Charts, Macmillan, 1979 21. J.S. Oakland, Statistical Process Control, William Heinemann, 1986 Statistical Quality Design and Control Richard. up. (a) -chart. (b) R-chart. Data now have k = 16, n = 5. Importance of Using Both and R Control Charts. This example points to the importance of maintaining both and R control charts and the