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Chapter 6 Control Charts for Attributes Introduction to Statistical Quality Control, 4th Edition 6-1. Introduction • Data that can be classified into one of several categories or classifications is known as attribute data. • Classifications such as conforming and nonconforming are commonly used in quality control. • Another example of attributes data is the count of defects. Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming • Fraction nonconforming is the ratio of the number of nonconforming items in a population to the total number of items in that population. • Control charts for fraction nonconforming are based on the binomial distribution. Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Recall: A quality characteristic follows a binomial distribution if: 1. All trials are independent. 2. Each outcome is either a “success” or “failure”. 3. The probability of success on any trial is given as p. The probability of a failure is 1-p. 4. The probability of a success is constant. Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming • The binomial distribution with parameters n 0 and 0 < p < 1, is given by n x nx p(x) p (1 p) x • The mean and variance of the binomial distribution are np np(1 p) 2 Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Development of the Fraction Nonconforming Control Chart Assume • n = number of units of product selected at random. • D = number of nonconforming units from the sample • p= probability of selecting a nonconforming unit from the sample. • Then: n x P(D x) p (1 p) n x x Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Development of the Fraction Nonconforming Control Chart • The sample fraction nonconforming is given as D pˆ n ˆ is a random variable with mean and where p variance p(1 p) 2 p n Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Standard Given • If a standard value of p is given, then the control limits for the fraction nonconforming are p(1 p) UCL p 3 n CL p p(1 p) LCL p 3 n Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming No Standard Given • If no standard value of p is given, then the control limits for the fraction nonconforming are p (1 p ) n UCL p 3 CL p where LCL p 3 m p Di i 1 mn p (1 p ) n m pˆ i i 1 m Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Trial Control Limits • Control limits that are based on a preliminary set of data can often be referred to as trial control limits. • The quality characteristic is plotted against the trial limits, if any points plot out of control, assignable causes should be investigated and points removed. • With removal of the points, the limits are then recalculated. Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Example • A process that produces bearing housings is investigated. Ten samples of size 100 are selected. Sample # # Nonconf. 1 5 2 2 3 3 4 8 5 4 6 1 7 2 8 6 9 3 • Is this process operating in statistical control? Introduction to Statistical Quality Control, 4th Edition 10 4 6-2. Control Charts for Fraction Nonconforming Example n = 100, m = 10 Sample # # Nonconf. Fraction Nonconf. 1 5 2 2 3 3 4 8 5 4 6 1 7 2 8 6 9 3 10 4 0.05 0.02 0.03 0.08 0.04 0.01 0.02 0.06 0.03 0.04 m p ˆi p i 1 m 0.038 Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Example Control Limits are: 0.038(1 0.038) UCL 0.038 3 0.095 100 CL 0.038 0.038(1 0.038) LCL 0.038 3 0.02 0 100 Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Example P Chart for C1 Proportion 0.10 3.0SL=0.09536 0.05 P=0.03800 0.00 -3.0SL=0.000 0 1 2 3 4 5 6 7 8 9 10 Sampl e Number Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Design of the Fraction Nonconforming Control Chart • The sample size can be determined so that a shift of some specified amount, can be detected with a stated level of probability (50% chance of detection). If is the magnitude of a process shift, then n must satisfy: p(1 p) L n Therefore, 2 L n p(1 p) Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Positive Lower Control Limit • The sample size n, can be chosen so that the lower control limit would be nonzero: p(1 p) LCL p L 0 n and (1 p) 2 n L p Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming Interpretation of Points on the Control Chart for Fraction Nonconforming • Care must be exercised in interpreting points that plot below the lower control limit. – They often do not indicate a real improvement in process quality. – They are frequently caused by errors in the inspection process or improperly calibrated test and inspection equipment. Introduction to Statistical Quality Control, 4th Edition 6-2. Control Charts for Fraction Nonconforming The np control chart • The actual number of nonconforming can also be charted. Let n = sample size, p = proportion of nonconforming. The control limits are: UCL np 3 np(1 p) CL np LCL np 3 np(1 p) (if a standard, p, is not given, use p ) Introduction to Statistical Quality Control, 4th Edition 6-2.2 Variable Sample Size • In some applications of the control chart for the fraction nonconforming, the sample is a 100% inspection of the process output over some period of time. • Since different numbers of units could be produced in each period, the control chart would then have a variable sample size. Introduction to Statistical Quality Control, 4th Edition 6-2.2 Variable Sample Size Three Approaches for Control Charts with Variable Sample Size 1. Variable Width Control Limits 2. Control Limits Based on Average Sample Size 3. Standardized Control Chart Introduction to Statistical Quality Control, 4th Edition 6-2.2 Variable Sample Size Variable Width Control Limits • Determine control limits for each individual sample that are based on the specific sample size. • The upper and lower control limits are p(1 p) p3 ni Introduction to Statistical Quality Control, 4th Edition 6-2.2 Variable Sample Size Control Limits Based on an Average Sample Size • Control charts based on the average sample size results in an approximate set of control limits. • The average sample size is given by m n • ni i 1 m The upper and lower control limits are p(1 p) p3 n Introduction to Statistical Quality Control, 4th Edition 6-2.2 Variable Sample Size The Standardized Control Chart • The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties: – Centerline at 0 – UCL = 3 LCL = -3 – The points plotted are given by: zi pˆ i p p(1 p) ni Introduction to Statistical Quality Control, 4th Edition 6-2.4 The Operating-Characteristic Function and Average Run Length Calculations The OC Function • The number of nonconforming units, D, follows a binomial distribution. Let p be a standard value for the fraction nonconforming. The probability of committing a Type II error is P(pˆ UCL | p) P(pˆ LCL | p) P(D nUCL | p) P(D nLCL | p) Introduction to Statistical Quality Control, 4th Edition 6-2.4 The Operating-Characteristic Function and Average Run Length Calculations Example • Consider a fraction nonconforming process where samples of size 50 have been collected and the upper and lower control limits are 0.3697 and 0.0303, respectively.It is important to detect a shift in the true fraction nonconforming to 0.30. What is the probability of committing a Type II error, if the shift has occurred? Introduction to Statistical Quality Control, 4th Edition 6-2.4 The Operating-Characteristic Function and Average Run Length Calculations Example • For this example, n = 50, p = 0.30, UCL = 0.3697, and LCL = 0.0303. Therefore, from the binomial distribution, P(D nUCL | p) P(D nLCL | p) P(D 50(0.3697) | 0.30) P(D 50(0.0303| 0.30) P(D 18.48 | 0.30) P(D 1.515| 0.30) P(D 18 | 0.30) P(D 1 | 0.30) 0.8594 0 0.8594 Introduction to Statistical Quality Control, 4th Edition 6-2.4 The Operating-Characteristic Function and Average Run Length Calculations OC curve for the fraction nonconforming control chart with p = 20, LCL = 0.0303 and UCL = 0.3697. 1.0 B • 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 p Introduction to Statistical Quality Control, 4th Edition 6-2.4 The Operating-Characteristic Function and Average Run Length Calculations ARL • The average run lengths for fraction nonconforming control charts can be found as before: 1 • The in-control ARL is ARL 0 • 1 The out-of-control ARL is ARL1 1 Introduction to Statistical Quality Control, 4th Edition 6-3. Control Charts for Nonconformities (Defects) • • There are many instances where an item will contain nonconformities but the item itself is not classified as nonconforming. It is often important to construct control charts for the total number of nonconformities or the average number of nonconformities for a given “area of opportunity”. The inspection unit must be the same for each unit. Introduction to Statistical Quality Control, 4th Edition 6-3. Control Charts for Nonconformities (Defects) Poisson Distribution • • The number of nonconformities in a given area can be modeled by the Poisson distribution. Let c be the parameter for a Poisson distribution, then the mean and variance of the Poisson distribution are equal to the value c. The probability of obtaining x nonconformities on a single inspection unit, when the average number of nonconformities is some constant, c, is found using: ecc x p( x ) x! Introduction to Statistical Quality Control, 4th Edition 6-3.1 Procedures with Constant Sample Size c-chart • Standard Given: UCL c 3 c CL c LCL c 3 c UCL c 3 c • No Standard Given: CL c LCL c 3 c Introduction to Statistical Quality Control, 4th Edition 6-3.1 Procedures with Constant Sample Size Choice of Sample Size: The u Chart • • If we find c total nonconformities in a sample of n inspection units, then the average number of nonconformities per inspection unit is u = c/n. The control limits for the average number of nonconformities is u UCL u 3 n CL u LCL u 3 u n Introduction to Statistical Quality Control, 4th Edition 6-3.2 Procedures with Variable Sample Size Three Approaches for Control Charts with Variable Sample Size 1. Variable Width Control Limits 2. Control Limits Based on Average Sample Size 3. Standardized Control Chart Introduction to Statistical Quality Control, 4th Edition 6-3.2 Procedures with Variable Sample Size Variable Width Control Limits • Determine control limits for each individual sample that are based on the specific sample size. • The upper and lower control limits are u u3 ni Introduction to Statistical Quality Control, 4th Edition 6-3.2 Procedures with Variable Sample Size Control Limits Based on an Average Sample Size • Control charts based on the average sample size results in an approximate set of control limits. • The average sample size is given by m n • ni i 1 m The upper and lower control limits are u u3 n Introduction to Statistical Quality Control, 4th Edition 6-3.2 Procedures with Variable Sample Size The Standardized Control Chart • The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties: – Centerline at 0 – UCL = 3 LCL = -3 – The points plotted are given by: ui u zi u ni Introduction to Statistical Quality Control, 4th Edition 6-3.3 Demerit Systems • When several less severe or minor defects can occur, we may need some system for classifying nonconformities or defects according to severity; or to weigh various types of defects in some reasonable manner. Introduction to Statistical Quality Control, 4th Edition 6-3.3 Demerit Systems Demerit Schemes 1. 2. 3. 4. Class A Defects - very serious Class B Defects - serious Class C Defects - Moderately serious Class D Defects - Minor • Let ciA, ciB, ciC, and ciD represent the number of units in each of the four classes. Introduction to Statistical Quality Control, 4th Edition 6-3.3 Demerit Systems Demerit Schemes • The following weights are fairly popular in practice: – Class A-100, Class B - 50, Class C – 10, Class D - 1 di = 100ciA + 50ciB + 10ciC + ciD di - the number of demerits in an inspection unit Introduction to Statistical Quality Control, 4th Edition 6-3.3 Demerit Systems Control Chart Development • Number of demerits per unit: D ui n where n = number of inspection units n D = di i 1 Introduction to Statistical Quality Control, 4th Edition 6-3.3 Demerit Systems Control Chart Development UCL u 3ˆ u CL u LCL u 3ˆ u where and u 100u A 50u B 10u C u D 100 u A 50 u B 10 u C u D ˆ u n 2 2 2 Introduction to Statistical Quality Control, 4th Edition 1/ 2 6-3.4 The OperatingCharacteristic Function • • The OC curve (and thus the P(Type II Error)) can be obtained for the c- and uchart using the Poisson distribution. For the c-chart: P( x UCL| c) P( X LCL| c) where x follows a Poisson distribution with parameter c (where c is the true mean number of defects). Introduction to Statistical Quality Control, 4th Edition 6-3.4 The OperatingCharacteristic Function • For the u-chart: P(x UCL | u) P(x LCL | u) P(c nUCL | u) P(c nLCL| u) Introduction to Statistical Quality Control, 4th Edition 6-3.5 Dealing with Low-Defect Levels • • • When defect levels or count rates in a process become very low, say under 1000 occurrences per million, then there are long periods of time between the occurrence of a nonconforming unit. Zero defects occur Control charts (u and c) with statistic consistently plotting at zero are uninformative. Introduction to Statistical Quality Control, 4th Edition 6-3.5 Dealing with Low-Defect Levels Alternative • Chart the time between successive occurrences of the counts – or time between events control charts. • If defects or counts occur according to a Poisson distribution, then the time between counts occur according to an exponential distribution. Introduction to Statistical Quality Control, 4th Edition 6-3.5 Dealing with Low-Defect Levels Consideration • Exponential distribution is skewed. • Corresponding control chart very asymmetric. • One possible solution is to transform the exponential random variable to a Weibull random variable using x = y1/3.6 (where y is an exponential random variable) – this Weibull distribution is well-approximated by a normal. • Construct a control chart on x assuming that x follows a normal distribution. • See Example 6-6, page 326. Introduction to Statistical Quality Control, 4th Edition 6-4. Choice Between Attributes and Variables Control Charts • • • • • Each has its own advantages and disadvantages Attributes data is easy to collect and several characteristics may be collected per unit. Variables data can be more informative since specific information about the process mean and variance is obtained directly. Variables control charts provide an indication of impending trouble (corrective action may be taken before any defectives are produced). Attributes control charts will not react unless the process has already changed (more nonconforming items may be produced. Introduction to Statistical Quality Control, 4th Edition 6-5. Guidelines for Implementing Control Charts 1. Determine which process characteristics to control. 2. Determine where the charts should be implemented in the process. 3. Choose the proper type of control chart. 4. Take action to improve processes as the result of SPC/control chart analysis. 5. Select data-collection systems and computer software. Introduction to Statistical Quality Control, 4th Edition