CHAPTER 7 Attributes Control Charts 1 Introduction Data
CHAPTER (7) Attributes Control Charts 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. 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. 3
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. 4
Control Charts for Fraction Nonconforming • The binomial distribution with parameters n 0 and 0 < p < 1, is given by • The mean and variance of the binomial distribution are 5
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Design of Fraction Nonconforming Chart • Three parameters must be specified 1. The sample size 2. The frequency of sampling 3. The width of the control limits • • 20 Common to base chart on 100% inspection of all process output over a period of time Rational subgroups may also play role in determining sampling frequency
Sample size • If p is very small, we should choose n sufficiently large to find at least one nonconforming unit • Otherwise the presence of only one non-conforming in the sample would indicate out-of-control condition (example) • To avoid this, choose n such that the probability of finding at least one nonconforming per sample is at least γ (example) 21
Example • p = 0. 01 and n = 8 • If there is one nonconforming in the sample, then p =1/8=0. 125 and we conclude that the process is out of control 22
Sample size • The sample size can be determined so that the probability of finding at least one nonconforming unit per sample is at least γ • Example p = 0. 01 and γ = 0. 95 • Find n such that P(D ≥ 1) ≥ 0. 95 • Using Poisson approximation of the binomial with λ =np • From cumulative Poisson table λ must exceed 3. 00 np ≥ 3 n ≥ 300 23
Sample size • 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). • UCL = pout If is the magnitude of a process shift, then n must satisfy: Therefore, 24
Positive Lower Control Limit • The sample size n, can be chosen so that the lower control limit would be nonzero: and 25
Interpretation of Points on the Control Chart for Fraction Nonconforming 26
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Variable Sample Size Variable-Width Control Limits 28
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Variable Sample Size Control Limits Based on an Average Sample Size 32
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Variable Sample Size The Standardized Control Chart 34
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The Opening Characteristic Function and Average Run Length Calculations 37
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Control Charts for Nonconformities (Defects) Procedures with Constant Sample Size 42
• 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. 43
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Control Charts for Nonconformities (Defects) Procedures with Variable Sample Size 55
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