Explanation on how to use to develop a sample plan or Answer question: How many samples should I take to ensure confidence in my data?

Transcription

1 THE OPERATING CHARACTERISTIC CURVE (OC CURVE) The Operating characteristic curve is a picture of a sampling plan. Each sampling plan has a unique OC curve. The sample size and acceptance number define the OC curve and determine its shape. The acceptance number is the maximum allowable defects or defective parts in a sample for the lot to be accepted. The OC curve shows the probability of acceptance for various values of incoming quality. Probabiity of Acceptance Pa % 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Classic: OC Curve c= 1 = number defcts that is accepted n = sample size is varied 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% % Defective Plan A: Pa c=1 n =20 Plan B: Pa c=1 n =50 Explanation on how to use to develop a sample plan or Answer question: How many samples should I take to ensure confidence in my data?

2 Two sample plans A & B: Both inspection plans call for accepting the lot based on finding only one reject of defective part. If there are 2% defective parts in my population : Plan A: I have a probability of 93% of accepting the lot. Plan B: I have a probability of 73% of accepting the lot. Further:Acceptance PLAN A: 50:50 chance accepting 8.1% defective PLAN B: 50:50 chance accepting 3.3% defective Probabiity of Acceptance Pa % 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Classic: OC Curve c= 1 = number defcts that is accepted n = sample size is varied 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% % Defective Plan A: Pa c=1 n =20 Plan B: Pa c=1 n =50

3 Two sample plans A & B: Further:Acceptance PLAN A: 10% chance of accepting 20% defective PLAN B: 10% chance of accepting 7.6% defective Does not sound too good! How do I improve: Increase n more = $$. If you increase c (number of defects in sample you are willing to accept) chance of accepting increase! Probabiity of Acceptance Pa % 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Classic: OC Curve c= 1 = number defcts that is accepted n = sample size is varied 0% 5% 10% 15% 20% 25% 30% % Defective Plan A: Pa c=1 n =20 Plan B: Pa c=1 n =50

4 Producers risk: Probability or risk of rejecting the product when it is good: EXAMPLE: If process runs normally at 1% defective, probability of acceptance is 95%. Producers risk is or 5%! Consumers risk: Probability or risk of accepting the product when it is bad: EXAMPLE: Define a level at which the product consumer wants the product rejected at 6% defective, probability of acceptance is 10%. Consumers risk is 10% for the defined Percent defective! AQL : Defined so there is a high probability of acceptance RQL : Defined so there is a low probability of acceptance

5 OC Curves: Operating Characteristic Curves page 40 Generated using the Poisson Classic: OC Curve n = 10 formula n = sample size p = % defective in population or % difference you want to detect np = λ = # expected defects in sample x = c = number of defects found in the sample = acceptance Number. If P(x) or Pa = probability of acceptance Probabiity of Acceptance Pa % 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Pa c=1 Pa c=2 Pa c=3 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% % Defective

6 ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd OC Curves: Operating Characteristic Curves Classic: OC Curve n = 10 n = sample size 100% p = % defective in population or % difference you want to detect x = c = number of defects found in the sample = acceptance Number. If Probabiity of Acceptance Pa % 90% 80% 70% 60% 50% 40% 30% 20% 10% Pa c=1 Pa c=2 Pa c=3 P(x) or Pa = probability of acceptance 0% 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% % Defective

7 OC Curves: Operating Characteristic Curves page 40 Generated using the Poisson formula n = sample size p = % defective in population or % difference you want to detect x = c = acceptance number number of defects found in the sample. If P(x) or Pa = probability of acceptance Probabiity of Acceptance Pa % Classic: OC Curve c= 1 100% Pa c=1 n =10 90% Pa c=1 n =15 80% Pa c=1 n =20 70% Pa c=1 n =30 Pa c=1 n = 40 60% Pa c=1 n =75 50% Pa c=1 n =100 40% 30% 20% 10% 0% 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% % Defective

8 THE OPERATING CHARACTERISTIC CURVE (OC CURVE) An OC curve is developed by determining the probability of acceptance for several values of incoming quality. Incoming quality is denoted by p. The probability of acceptance is the probability that the number of defects or defective units in the sample is equal to or less than the acceptance number of the sampling plan. The AQL is the acceptable quality level and the RQL is rejectable quality level. If the units on the abscissa are in terms of percent defective, the RQL is called the LTPD or lot tolerance percent defective. The producer s risk (a ) is the probability of rejecting a lot of AQL quality. The consumer s risk (b ) is the probability of accepting a lot of RQL quality. There are three probability distributions that may be used to find the probability of acceptance. These distributions were covered in the Basic Probability chapter and are reviewed here. The hypergeometric distribution The binomial distribution The Poisson distribution Although the hypergeometric may be used when the lot sizes are small, the binomial and Poisson are by far the most popular distributions to use when constructing sampling plans.

9 THE OPERATING CHARACTERISTIC CURVE (OC CURVE) Hypergeometric Distribution The hypergeometric distribution is used to calculate the probability of acceptance of a sampling plan when the lot is relatively small. It can be defined as the true basic probability distribution of attribute data but the calculations could become quite cumbersome for large lot sizes. The probability of exactly x defective parts in a sample n:

10 Binomial Distribution The binomial distribution is used when the lot is very large. For large lots, the nonreplacement of the sampled product does not affect the probabilities. The hypergeometric takes into consideration that each sample taken affects the probability associated with the next sample. This is called sampling without replacement. The binomial assumes that the probabilities associated with all samples are equal. This is sometimes referred to as sampling with replacement although the parts are not physically replaced. The binomial is used extensively in the construction of sampling plans. The sampling plans in the Dodge- Romig Sampling Tables were derived from the binomial distribution. The probability of exactly x defective parts in a sample n: The symbol p represents the value of incoming quality expressed as a decimal. (1% =.01, 2% =.02, etc.)

11 THE OPERATING CHARACTERISTIC CURVE (OC CURVE) Poisson Distribution The Poisson distribution is used for sampling plans involving the number of defects or defects per unit rather than the number of defective parts. It is also used to approximate the binomial probabilities involving the number of defective parts when the sample (n) is large and p is very small. When n is large and p is small, the Poisson distribution formula may be used to approximate the binomial. Using the Poisson to calculate probabilities associated with various sampling plans is relatively simple because the Poisson tables can be used. The Thorndike chart, which will be discussed later, is a valuable aid in the construction of sampling plans using the Poisson distribution. The probability of exactly x defects or defective parts in a sample n: The letter e represents the value of the base of the natural logarithm system. It is a constant value (e = ).

12 Poisson Distribution An OC Curve Using the Poisson Distribution The Poisson distribution is used to compute the probability of acceptance for defects per unit sampling plans. It may also be used to approximate binomial probabilities and compute the probability of acceptance for fraction defective sampling plans. For the sampling plan n = 30 and c = 1, c may be in terms of number of defects or in terms of number of defective parts. If c is in terms number of defects, the AIQ or abscissa on the OC curve is in terms of defects per unit. The acceptance number for the sampling plan n = 30 and c = 1 may either be 1 defect or 1 defective part. The Poisson formula, of acceptance., is used to compute the probabilities

13 Poisson Distribution For all practical purposes, the probabilities of acceptance are the same as those obtained with the binomial formula. There are some minor differences. For this example, the differences increase slightly as the curve approaches the tail.

14 Poisson Distribution The Operating Characteristic Curve for n = 30 and c = 1 using the Poisson Distribution Classic: OC Curve c= 1 n % 90% Probabiity of Acceptance Pa % 80% 70% 60% 50% 40% 30% 20% 10% Pa c=1 n =30 0% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% % Defective