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Attribute Control Charts

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2 Attribute Control Chart Learning Objectives Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability

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3 Attribute Control Chart Shewhart Control Charts - Overview

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4 Attribute Control Chart Defective and Defect Defective A unit of product that does not meet customer ’ s requirement or specification. Also known as a non-conforming unit. Example A base casting that fails porosity specification is a defective. A disc clamp that does not meet the parallelism specification is a defective.

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5 Attribute Control Chart Defect A flaw or a single quality characteristic that does not meet customer ’ s requirement or specification. Also known as a non-conformity. There can be one or more defects in a defective. Example A dent on a VCM pole that fails customer ’ s specification is a defect. A stain on a cover that fails customer ’ s specification is a defect. Defective and Defect

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6 Attribute Control Chart Shewhart Control Charts for Attribute Data There are 4 types of Attribute Control Charts: 43 np c p u Constant Lot Size Variable Lot Size Defects (Poisson Distribution) Defectives (Binomial Distribution)

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7 Attribute Control Chart Learning Objectives Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability Mean defective rate Mean defect rate

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8 Attribute Control Chart Types of Data and Distributions Discrete Data (Attribute) Binomial Poisson Continuous Data (Variable) Normal Exponential Weibull Lognormal t 2 F Discrete Distributions Continuous Distributions

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9 Attribute Control Chart Types of Distributions Normal Distribution Exponential Distribution Uniform Distribution Binomial Distribution Discrete Distributions Continuous Distributions

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10 Attribute Control Chart Binomial Distribution Useful for attribute data (or binary data) Result from inspection criteria which are binary in nature, e.g. pass/fail, go/nogo, accept/reject, etc. Data generated from counting of defectives. Discrete Distributions Plot is known as Probability Mass Function of X

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11 Attribute Control Chart Binomial Distribution If a process typically gives 10% reject rate (p = 0.10), what is the chance of finding 0, 1, 2 or 3 defectives within a sample of 20 units (n = 20)? Discrete Distributions Commonly used in Acceptance Sampling

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12 Attribute Control Chart Binomial Distribution Commonly used in Acceptance Sampling, where p is the probability of success (defective rate), n is the number of trials (sample size), and x is the number of successes (defectives found).

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13 Attribute Control Chart Binomial Distribution Properties: each trial has only 2 possible outcomes - success or failure probability of success p remains constant throughout the n trials the trials are statistically independent the mean and variance of a Binomial Distribution are and

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14 Attribute Control Chart The location, dispersion and shape of a binomial distribution are affected by the sample size (n) and defective rate (p). Binomial Distribution Discrete Distributions

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15 Attribute Control Chart James Bernoulli Binomial Distribution Discrete Distributions

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16 Attribute Control Chart Poisson Distribution Useful for discrete data involving error rate, defect rate (dpu, dpmo), particle count rate, etc. Data generated from counting of defects. Discrete Distributions

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17 Attribute Control Chart Poisson Distribution If a process typically gives 4.0 defect rate ( = 4 dpu), what is the chance of finding 0, 1, 2 or 3 defects per unit? Discrete Distributions Commonly used as an approximation of the binomial distribution when: p < 0.1 (10%) n is large

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18 Attribute Control Chart Poisson Distribution This distribution have been found to be relevant for applications involving error rates, particle count, chemical concentration, etc, where is the mean number of events (or defect rate) within a given unit of time or space.

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19 Attribute Control Chart Poisson Distribution Properties: number of outcomes in a time interval (or space region) is independent of the outcomes in another time interval (or space region) probability of an occurrence within a very short time interval (or space region) is proportional to the time interval (or space region) probability of more than 1 outcome occurring within a short time interval (or space region) is negligible the mean and variance for a Poisson Distribution are and

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20 Attribute Control Chart The location, dispersion and shape of a Poisson distribution are affected by the mean ( ). Poisson Distribution Discrete Distributions

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21 Attribute Control Chart Simeon D Poisson Poisson Distribution Discrete Distributions

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22 Attribute Control Chart Summary of Approximation Binomial p < 0.1 The smaller p and larger n the better 15 The larger the better np > 5, > 10 p = 0.5, < 0.5 Poisson Normal

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23 Attribute Control Chart Learning Objectives Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability

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24 Attribute Control Chart p Chart Fraction Non-Conforming Reject Rate / Defective Rate Percent Fallout 20100 0.5 0.4 0.3 0.2 0.1 0.0 Sample Number P r o p o r t i o n p Chart 1 P=0.2140 3.0SL=0.3880 -3.0SL=0.04000

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25 Attribute Control Chart p Chart Fraction non-conforming ( ) Ratio of number of defectives (or non-conforming items) in a population to the number of items in that population. Sample fraction non-conforming (p) Ratio of number of defectives (d) in a sample to the sample size (n), i.e. Is “ p ” a sample statistic?

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26 Attribute Control Chart The underlying principles of the p chart are based on the binomial distribution. This means that if a process has a typical fraction non-conforming, p, the mean and variance of the distribution for p ’ s are computed from the binomial equation, giving: p Chart k = number of subgroup, should be between 20 to 25 before constructing control limits. X k = number of defective unit in subgroup k which has a total sample size of n k units

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27 Attribute Control Chart The p chart also assumes a symmetrical bell- shape distribution, with symmetrical control limits on each side of the center line. This implies that the binomial distribution is approximately close to the shape of the normal distribution, which can happen under certain conditions of p and n: p 1/2 and n > 10 implying np > 5 For other values of p, the general guideline is to have np > 10 to get a satisfactory approximation of the normal to the binomial. p Chart

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28 Attribute Control Chart p Chart Following Shewhart ’ s principle, the Center Line and Control Limits of a p chart are:

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29 Attribute Control Chart If the sample size is not constant, then the Control Limits of a p chart may be computed by either method: a) Variable Control Limits where n i is the actual sample size of each sampling i b) Control Limits Based on Average Sample Size where n is the average (or typical) sample size of all the samples p Chart

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30 Attribute Control Chart When to Use Control Limits Based on Average Sample Size instead of Variable Control Limits Smallest subgroup size, n min, is at least 30% of the largest subgroup size, n max. Future sample sizes will not differ greatly from those previously observed. When using Control Limits Based on Average Sample Size, the exact control limits of a point should be determined and examined relative to that value if: There is an unusually large variation in the size of a particular sample There is a point which is near the control limits. p Chart - Average Sample Size

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31 Attribute Control Chart Example 1: p Chart S/NSampledRejects 1 50 12 2 50 15 3 50 8 4 50 10 5 50 4 6 50 7 7 50 16 8 50 9 9 50 14 10 50 10 11 50 5 12 50 6 13 50 17 14 50 12 15 50 22 16 50 8 17 50 10 18 50 5 19 50 13 20 50 11 Frozen orange juice concentrate is packed in 6-oz cardboard cans. A metal bottom panel is attached to the cardboard body. The cans are inspected for possible leak. 20 samplings of different sampling size were obtained. Verify if the process is in control. The data are found in AttributeSPC.MTW.

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32 Attribute Control Chart MiniTab: Stat Control Charts P Example 1: p Chart

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33 Attribute Control Chart Example 1: p Chart

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34 Attribute Control Chart Example 1: p Chart Minitab allows different set of control charts to be plotted on one chart MiniTab: Stat Control Charts P

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35 Attribute Control Chart Example 1: p Chart

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36 Attribute Control Chart Establish Trial Control Limits When to use it? New process, modified process, no historical data available to calculate p How to do it? Calculate p based on the preliminary 20 to 25 subgroups. Calculate the trial control limits using the formula mentioned in slide 21 or 22. Sample values of p from the preliminary subgroups to be plotted against the trial control limits. Any points exceed the trial control limits should be investigated. If assignable causes for these points are discovered, they should be discarded and new trial control limits to be determined.

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37 Attribute Control Chart np Chart If the sample size is constant, it is possible to base a control chart on the number nonconforming (np), rather than the fraction nonconforming (p). The Center Line and Control Limits of an np chart are:

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38 Attribute Control Chart Example 2: np Chart S/NSampledRejects 1 50 12 2 50 15 3 50 8 4 50 10 5 50 4 6 50 7 7 50 16 8 50 9 9 50 14 10 50 10 11 50 5 12 50 6 13 50 17 14 50 12 15 50 22 16 50 8 17 50 10 18 50 5 19 50 13 20 50 11 Frozen orange juice concentrate is packed in 6-oz cardboard cans. A metal bottom panel is attached to the cardboard body. The cans are inspected for possible leak. 20 samplings of 50 cans/sampling were obtained. Verify if the process is in control. The data are found in AttributeSPC.MTW.

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39 Attribute Control Chart Example 2: np Chart MiniTab: Stat Control Charts NP

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40 Attribute Control Chart Example 2: np Chart

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41 Attribute Control Chart p Chart vs np Chart For ease of recording, the np chart is preferred. The p chart offers the following advantages: accommodation for variable sample size provides information about process capability X = X n Distribution of Sampling Averages X X

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42 Attribute Control Chart Sample Size for p and np Charts Sample Size is determined based on the 2 criteria: 1.Assumption to approximate Binomial Distribution to a Normal Distribution 2.To ensure that the LCL is greater than zero. For p 0.5 For p = other values

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43 Attribute Control Chart Learning Objectives Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability

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44 Attribute Control Chart c Chart Defects per Unit (DPU) Error Rate / Defect Rate Defects per Opportunity 20100 20 10 0 Sample Number S a m p l e C o u n t c Chart C=9.650 3.0SL=18.97 -3.0SL=0.3307

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45 Attribute Control Chart c Chart Each specific point at which a specification is not satisfied results in a defect or nonconformity. The c chart is a control chart for the total number of defects in an inspection unit based on the normal distribution as an approximation for the Poisson distribution, which can happen when: c or 15

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46 Attribute Control Chart c Chart Inspection Unit The area of opportunity for the occurrence of nonconformities. –e.g. a HSA, a media, a PCBA This is an entity chosen for convenience of record-keeping. It may constitute more than 1 unit of product. –e.g. a HSA, both surfaces of a media, 10 pieces of PCBA

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47 Attribute Control Chart c Chart If the number of nonconformities (defects) per inspection unit is denoted by c, then: The Center Line and Control Limits of a c chart are:

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48 Attribute Control Chart u Chart In cases where the number of inspection units is not constant, the u chart may be used instead, with: If the average number of defects per inspection unit is denoted by u, then Where c i is the count of the number of defects in number of inspection units, a i

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49 Attribute Control Chart u Chart The Center Line and Control Limits of a u chart are:

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50 Attribute Control Chart Example 3: c and u Charts S/NUnitsDefects 1 5 10 2 5 12 3 5 8 4 5 14 5 5 10 6 5 16 7 5 11 8 5 7 9 5 10 10 5 15 11 5 9 12 5 5 13 5 7 14 5 11 15 5 12 16 5 6 17 5 8 18 5 10 19 5 7 20 5 5 A personal computer manufacturer plans to establish a control chart for nonconformities at the final assembly line. The number of nonconformities in 20 samples of 5 PCs are shown here. Verify if the process is in-control.

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51 Attribute Control Chart Example 3: c and u Charts MiniTab’s Stat Control Charts C

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52 Attribute Control Chart Example 3: c and u Charts MiniTab’s Stat Control Charts U

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53 Attribute Control Chart Example 3: c and u Charts

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54 Attribute Control Chart u (or c) Chart vs p (np) Chart The u (or c) chart offers the following advantages: More informative as the type of nonconformity is noted. Facilitates Pareto analysis. Facilitates Cause & Effect Analysis.

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55 Attribute Control Chart Learning Objectives Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability

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56 Attribute Control Chart c - Chart Measures the total number of defects in a subgroup The subgroup size can be 1 unit of product if we expect to have a relatively large number of defects/unit Requires a constant subgroup size u - Chart Measures the number of defects/unit of product (dpu) The subgroup size can be constant or variable p - Chart Measures the proportion of defective units in a subgroup The subgroup size can be constant or variable np - Chart Measures the number of defective items in a subgroup Requires a constant subgroup size Selecting the Appropriate Chart

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57 Attribute Control Chart Exercise #1 Strength of 5 test pieces sampled every hour(Xbar-R) Number of defectives in 100 parts(np) Number of solder defects in a printed circuit board assembly(C) Diameter of 40 units of products sampled every day(Xbar-S) Percent defective of a lot produced in every 30-min period(p) Surface defects of surface area of varying sizes(u) In a maintenance group dealing with repair work, the number of maintenance requests that require a second call to complete the repair every week

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58 Attribute Control Chart Test for Instability Suitable for all charts Suitable only for X-Chart _

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59 Attribute Control Chart Tests for Instability CAUTION : CAUTION : Do not apply “ tests ” blindly Not every “ test ” is relevant for all charts Excessive number of “ tests ” Increased -error Nature of application

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60 Attribute Control Chart Variables vs Attributes Charts Attributes Control Charts facilitate monitoring of more than 1 quality characteristics. Variables Control Charts provide leading indicators of trouble; Attributes Control Charts react after the process has actually produced bad parts. For a specified level of protection against process drift, Variables Control Charts require a smaller sample size.

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61 Attribute Control Chart Learning Objectives Defective vs Defect Binomial and Poisson Distribution p Chart np Chart c Chart u Chart Tests for Instability

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62 Attribute Control Chart End of Topic What Question Do You Have

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63 Attribute Control Chart Reading Reference Introduction to Statistical Quality Control, Douglas C. Montgomery, John Wiley & Sons, ISBN 0-471-30353-4

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