CQPA Ch. 5-7
Probability
"There is always a 100% probability that a piece of toast will land buttered side down on new carpet" Murphy's Law Conditions for probability: The ____ of any event (E) lies between 0 and 1. The SUM of the probabilities of all possible events (E) in a sample space (S) = 1
Exponential Distribution
(Closely related to Poisson) -- applies to the useful life cycle of many products and is used to model items with a constant failure rate. If a random variable, x, is exponentially distributed, then the reciprocal of x, y = 1/x follows a Poisson distribution. It is used to model the mean time between occurrences, such as arrivals or failures, and the Poisson distribution is used to model occurrences per interval such as arrivals, failures or defects.
Sampling Plan Quality Indices
1. Acceptable Quality Level (AQL) - the worst quality level that is still considered satisfactory. The probability of accepting an AQL should be high. 2. Rejectable quality level (RQL) - defines unsatisfactory quality. The probability of accepting RQL lot should be low. 3. Indifference quality level (IQL) - quality level somewhere between AQL and RQL. Normally defined as the quality level having probability of acceptance of 0.5. The IQL is rarely used.
Major Sampling Plans
1. Attribute Plans - Defectives: A sample is taken from a lot with each unit classified as acceptable or defective. the # of defectives is then compared to the acceptance number, in order to make an accept or reject decision for the lot. - Defects: A sample is taken from a lot and the defects are counted. The rain of defects/100 units is derived. This value is compared to the acceptance number, in order to make an accept or reject decision for the lot. Ex. = MIL-STD-105E, ANSI/ASQ Z1.4, Dodge-Romig Tables 2. Variable Plans A sample is taken and one or more quality characteristic measurements are made on each unit. These measurements are then summarized into simples statistics (such as the sample average or standard deviation) which are compared with a critical value defined in the plan. A decision is then made to accept or reject the lot. Ex. = MIL-STD-414, ANSI/ASQ Z1.9
Designing for Reliability
1. Concept Phase - Work with customers that develop a product that meets their perceived needs in terms of ease of use, special training needs, special power requirements, complexity of design, support, etc. Up to 35% of the life cycle costs are determined for an item or product. Earliest consideration of reliability. 2. Design and Development phase - Issues such as ergonomics, maintainability, safety and other major design considerations become a "product on paper." 90% of the total life cycle costs have been determined by the time this phase is complete. 3. Full Scale Development - (for production items) or Build and Install (for one shot items) - the design is basically complete and prototype runs of final build are occurring. Changes during this phase are very costly and are not done in order to meet some other target (time or money considerations) 4. Operational Phase - Use of item in the field. The ease of use and of maintenance have a significant impact on the reliability of a product. 5. Disposal - designers are required to take product disposal into consideration. In designing for disassembly, it is important that product performance not be degraded.
Continuous Distribution
A distribution containing infinite (variable) data points that may be displayed on a continuous measurement scale. Ex. Normal, uniform, exponential, and Weibull distributions
Discrete Distribution
A distribution resulting from countable (attribute) data that has a finite number of possible values. Ex. binomial, Poisson, and hypergeometric distributions
Zero Defect Sampling
A growing interest in zero acceptance number plans for two reasons: 1. the advent of six sigma methodology 2. the litigious society that currently exists
Statistic
A numerical data value taken from a sample that may be used to make an inference about a population
Range
A range of a set of data is the difference between the largest and smallest values
Population
All possible observations of similar items from which a sample is drawn.
The distribution of time between failures (TBF)
Along with concern for high failures during the infant mortality period, customers must be concerned with the length of time that a product will run without failure. This measurement concerns the second stage of the bathtub curve known as the normal, chance, or random failure period.
Simple Events - Probability
An event that cannot be decomposed is a simple event (E). The set of all sample points for an experiment is called the sample space (S).
Permutations
An ordered arrangement of n distinct objects is called a ______. The number of ways of ordering n distinct objects taken r at a time is designated by the symbol P(n,r) 3 lottery numbers are drawn from a total of 50. How many arrangements can be expected? (50!)/(50-3)! = 50!/47! = (50)(49)(48) = 117,600
Binomial Probability Distribution
Applies when the population is large (N>50) and the sample size is small compared to the population. Generally, n is less than 10% of N. It is most appropriate to use when proportion defective is equal to or greater than (0.1). The binomial is an approximation to the hypergeometric. The normal distribution approximates the binomial when np>= 5. The poisson distribution can be used to approximate the binomial distribution when p is small (generally, less than 0.1) and n is large (generally, n>= 16). by using np as the mean of the Poisson distribution.
Hypergeometric Probability Distribution
Applies when the population is small compared to the sample size. Complex combination calculation. The number of successes (r)* in the sample follows the hypergeometric function. r can also equal the number of defectives in a sample.
Attribute data
Attribute data is discrete and counted. This means that the data values can only be integers, for example, 3, 48, 1029. Counted or attribute data answer questions like "how many" or "how often" or "what kind"
Data Accuracy and INtegrity
Bad data is not only costly too capture, but corrupt the decision making process. - Avoid emotional bias relative to targets or tolerances when counting, measuring, or recording digital or analog displays. - Avoid unnecessary rounding - If time sequence occurs, record it. - Record the measurement or classification as soon as possible after manufacturing process and stabilization period if characteristic changes over time. -Screen or filter data to remove data entry errors
Other ways to get std. dev
Can also be determined using probability paper. Can be estimated from control charts using R, this is tied in to the determination of process capability.
Poisson Probability Distribution
Closely related to exponential distribution. Used for defect counts and can be used as an approximation to the binomial, when p is equal to or less than 0.1 and the sample size is fairly large. If x is a Poisson distributed random variable, then 1/x is an exponential random variable. If x is an exponential random variable, then 1/x is a Poisson random variable.
Composition
Consists of two possibilities -- a union or intersection
Primary types of Control Charts
Control charts for variables - Pots specific measurements of a process characteristic (temp, size, weight, sales volume, shipments, etc) Types: Xbar - R charts, Run Charts, Mbar - MR charts, X - MR charts, X -s Charts, Median charts, Short run charts. There are other more advanced variable charts like CuSum (cumulative sum) and EWMA (exponentially weighted moving average) charts. Often times more valuable and useful Control charts for attributes - Plots general measurement of the total process (the number of complaints per order, the number of orders on time, absenteeism frequency, number of errors per letter, etc). Types: p charts, np charts, c charts, u charts, short run varieties of the previously listed four charts
Comparison of Control limits and Specifications
Control limits are determined by process average values. One can also see the process spread of the individual values. This process spread can be predicted and will indicate the range of the individuals being produced.
Designing for Reliability Factors -
Cost Factors - When the quality level increases, the costs increase. Environmental Factors - Two categories - Family environment in which the components must function as a system, and the second consideration is the environment in which the system must function Human Factors - human characteristics should be considered when designing equipment and assigning work. Simplification - Smallest number of components should be used without compromising performance Redundancy - the existence of more than one means for achieving a stated level of performance; all paths must fail before the system can fail Derating - can be applied to reduce the failure rates below the averages. Fail safe - failure to operate a product can lead to fatality or substantial financial loss, a fail-safe type design should be adopted -- system cutoff Producibility/Maintainability - must be designed not only for performance, but also so that they can be produced with quality Good design concepts - must contain inherent reliability characteristics. A good designer will select components and circuits that have been tried and proven and will avoid the use of unproven methods.
Types of Data
Data is objective information that everyone can agree on. The three types of data are attribute data, variables data, and locational data. Of these three attribute and variables data are more widely used.
Elements of a Reliability Program
Definition of the reliability Program Developing the reliability goals and requirements Designing for reliability Assessing reliability progress Measuring reliability Ensuring reliable performance
Failure Rate and MTBF
Failure rate of a product can be calculated from test data using the lambda failure rate formula = Number of items failed/total test items MTBF can be calculated from test data using the theta formula = MTBF = Total test items/ number of items failed ** there is an obvious relationship between failure rate and MTBF
Median
For an even set of data, the ____ is the average of the middle two values. Advantages: -Provides an idea where most data are located -Little calculation required - Insensitivity to extreme values Disadvantages: - The data must be sorted and arranged - Extreme values may be important -Two _____ cannot be averaged to obtain a combined median - The _____ will have more variation (between samples) than the average (X)
Compound Events
Formed by a composition of two or more events. Consist of more than one point in the sample space. Ex. If two dice are tossed what is the probability of getting an 8? A die and a coin are tossed. What is the probability of getting a 4 and a tail? The two most important probability theorems are the additive and multiplicative.
Union: of A and B
If A and B are two events in a sample space (S) the union of A and B (A U B) contains all sample points in event A or B or both. If A = E1, E2, E3 and B = E1, E3, E5 then A U B = E1, E2, E3, E5
Intersection: of A and B
If A and B re two events in a sample space (S), the intersection of A and B (A 'upside down U' B is composed of all sample points that are in both A and B. If A = E1, E2, E3 and B = E1, E3, E4, then A 'upside down U' B = E1, E3
Mutually Exclusive Events
If event A contains no sample points in common with event B, they they are said to be mutually exclusive. Ex. Obtaining a 3 and a 2 on the toss of a SINGLE die is a mutually exclusive event. The probability of observing both events simultaneously is zero. The probability of obtaining either a 3 or a 2 is: PE2 + PE3 = 1/6 + 1/6 = 1/3
Conditional Probabilities
If the event A (rain) = 0.2 and the event B (cloudiness) = 0.3, what is the probability of rain on a cloudy day? (Note, it will not rain without clouds). P (AIB) = P(A 'upside down U' B)/P(B) = 0.2/0.3 = 0.67 These events are dependent of one another. Two events A and B are independent if they can occur without the other occurring.
Conversion of attributes data to variable data
It is desirable to use variable data vs attribute data whenever possible because it gives more information on the sample. aka the like or dislike of a product quality can be converted to a scale of how much do i like or dislike it.
Mode
It is possible for data to have more than one _____. Advantages: - No calculations or sorting is necessary -It is not influenced by extreme values -It is an actual value Disadvantages: -The data may not have a _____.
Locational Data
Locational data simply answers the question "where." Charts that utilize locational data are often called "measles charts" or "concentration charts" An example of a measles chart is a drawing showing locations of paint blemishes on an automobile.
Data Collection, Analysis, and Reporting
Manual data collection requires a checklist or data form, with some guidelines (pg. 46 ch. 5) Without an operational definition , most data is meaningless. Both attribute and variable specifications must be specified.
Sequential Sampling
Most discriminating of the acceptance sampling plans. Involves making one of three decisions as each sample item is obtained: accept the lot, reject the lot, or continue sampling. Often applied where sample economics are critical and a minimum sample size is required. More complex and more difficult to administer than other plans. Samples must be obtained one item at a time and operators require more training.
Weibull distribution
Most widely used distribution in reliability and statistical applications. It is commonly used to model time to fail, time to repair and material strength. There are two common versions of the ______ distribution, the two parameter______ and the three parameter _____. The difference is the three parameter ___ distribution has a location parameter when there is some non-zero time to first failure. The shape parameter is what gives the ___ distribution its flexibility. By changing this value, the distribution can model a wide variety of data. If Beta = 1 the ___ distribution is identical to the exponential distribution, if Beta =2 , the __ distribution is identical to the Rayleigh distribution; if Beta is between 3 and 4 the ___ distribution approximates the normal distribution. The ___ distribution approximates the lognormal distribution for several values of Beta. For most populations more than fifty samples are required to differentiate between the ___ distribution and the lognormal distributions. The scale parameter determines the range of distribution. The location parameter is used to define a failure-free zone. Generally the location parameter is assumed to be zero. A negative location parameter is caused by shipping failed units, etc.
Mean
Most widely used measure of central tendency. Advantages: -It is the center of gravity of the data -It uses all data - No sorting is needed Disadvantages: - Extreme data values may distort the picture -It can be time consuming - The mean may not be the actual value of any data points
Measurement Scales
Nominal - Data consists of names or categories only. NO ordering scheme is possible Ordinal (ranking) - Data is arranged in some order but differences between values cannot be determined or are meaningless Interval - Data is arranged in order and differences can be found. However, there is no inherent starting point and ratios are meaningless. Ratio - An extension of the interval level that includes an inherent zero starting point. Both differences and ratios are meaningful
Common Continuous Distributions
Normal (Gaussian) , Exponential, Weibull
Comparison of Central tendency in Normal and Skewed Distribution
Normal distribution: MEAN = MEDIAN = MODE Right skewed Distribution: MODE (1st), MEDIAN(2nd), MEAN(3rd) https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwij5t3dv43TAhXJ8CYKHVZ9CLUQjRwIBw&url=http%3A%2F%2Fisoconsultantpune.com%2Fstatistics-in-quality%2F&psig=AFQjCNFlYWF6s6cB43PNIO-50sY8bp8l1w&ust=1491487946887341
Measures of Dispersion (Spread)
Other than central tendency, the other important parameter to describe a set of data is spread or dispersion. Range, variance, and standard deviation
Common Discrete Distributions
Poisson, Binomial, Hypergeometric
The Multiplicative Law
Read as he probability of A AND B Identified as dependent or independent
The Additive Law
Read as the probability of A OR B Identified as mutually exclusive or non mutually exclusive
Sampling
Refers to the evaluation of a portion of a population (lot, batch, etc) for the purpose of obtaining useful information about it. Sampling is most useful when: - Inspection damages the product -The per unit inspection costs are high -The results of passing a defective unit are low -There are large amounts of product to be inspected
Measures of Central Tendency
Represent different ways of characterizing the central value of a collection of data. Mean , Median, Mode
Random Sampling
Requires giving every item an equal chance of being selected for the sample. Can be obtained from random number tables, computers, numbers drawn from a hat, to assist with sample selection. The sample must be representative of the lot and not just the product that use say to obtain, thus, the selection of samples requires some up front thought and planning.
Variance
Sigma2 or S2, equal to the sum of the squared deviations from the mean, divided by the sample size. The ____ is equal to the standard deviation squared.
s
Standard deviation of the sample
Fixed Sampling
Taking a fixed sample size from a lot or batch only works if the lot or batch size remains relatively constant. The only advantage is that a fixed same size in inspection is easy to remember. They are more widely used in auditing.
The Operating Characteristic Curve
Th OC curve for a sampling plan quantifies these risks. It is a graph of the percent defective in a batch versus the probability that the sampling plan will accept that batch. **No perfect sampling plan exists
Standard Deviation (sigma, s )
The ___ is the square root of the variance. N is used for a population, and n-1 is used for a sample (to remove potential bias in relatively small samples - less than 30).
Average Outgoing Quality Limit (AOQL)
The average AOQL is equal to the maximum AOQ (Average outgoing quality).
Coefficient of Variation (COV)
The coefficient of variation equals the standard deviation divided by the mean and is expressed as a percentage
Complement of an Event
The complement of an event A is all sample points in the sample space (S), but not in A. The complement of a is 1-Pa EX. If Pa = 0.3, the complement of Pa would be 1 - Pa = 0.7 **equals 1 when added together, they complement each other
Rational Subgrouping --- Shewart
The key idea in the Shewhart control chart is the division of observations into what are called rational subgroups. Subgroups are selected in a way that makes each subgroup as homogeneous as possible and that gives the maximum opportunity for variation from one subgroup to another. It is very important to maintain the order of production.
Weibull distribution
The mean and variance of the ___ distribution are computed using the gamma distribution. The mean of the ____ distribution is equal to the characteristic life if the shape parameter is equal to one. The variance of the _____ distribution decreases as the value of the shape parameter increases. The gamma comes from a gamma function table.
Control Charts
The most powerful tools to analyze variation in most processes - either manufacturing or administrative. A process is under statistical control when it is characterized by plot points that do not exceed the upper or lower control limits
Continuous Sampling
The most widely used continuous sampling plan is the original Dodge CSP-1 plan. It is carried out on a stream of product, with production units inspected in order of production.
Combinations
The number of distinct ______ of n distinct object taken r at a time is denoted by the symbol C(n,r) n!/r!(n-r)! 3 lottery numbers are drawn from a total of 50. How many combinations can be expected? 50!/3!47! = (50)(49)(48)/6 = 19,600
The Central Limit Theorem
The sample means (x bar) will be more normally distributed around mu than individual readings of x. The distribution of sample means approaches normal regardless of the shape of the parent population. This is why Xbar - R Control charts work! The spread in sample means X bar is less than x with the standard deviation of x bar equal to the standard deviation of the population (individuals) divided by the square root of the sample size. s>bar (standard deviation of the x bar = mean) is referred to as the standard error of the mean.
Parameter
The true numeric population value, often unknown, estimated by a statistic
Bathtub model
Three general types of failures for complex products: Infant Mortality - These are not design related issues but quality issues. As corrections are made, the failures decrease for a given time interval until they reach a steady state or constant level. Thus, the infant mortality period is noted by a decreasing failure rate. (Weibull distribution determines when it is over). Constant Failure Rate - (random failure rate period or Normal chance) Once the failures due to components and workmanship for the most part are eliminated, the constant failure rate period is entered. The constant failure rate period is the most common time frame for making reliability predictions, where the exponential distribution is utilized. Wearout Period - Components begin to fatigue or wear out, one begins to observe failures at increasing rates for a specified interval. As time goes on, failures may occur more and more frequently to a point where it may no longer be practical to continue operating the system. Several distributions may be appropriate to model the wear out period. the Normal and log normal distributions are often used.
Compound Event - Event relationships
Three relationships in finding the probability of an event: Complementary, conditional, and mutually exclusive
Normal Distribution
Useful when it is equally likely that readings will fall above or below average. When a sample of several random measurements are averaged, distribution of such repeated sample averages tends to be ___ according to central limit theorem.
Variable Data
Variable data is continuous and measurable. Contains more info than counted or attribute. This means that data values can be any real number. Answers questions like "how long" "what volume" "how much time" and "how far" Generally measured with some instrument or device. ** provides more information
Variable Versus Attribute Sampling
Variable plans should be used when the measurement of a relative few items is less expensive than the counting of many items and the population approximates a normal distribution.
Stratified Sampling
sometimes more informative than homogeneous samples. One might be interested in determining the amount of pallet damage in a storage area. There might be a need to sample more row ends, row corners or bottom pallets in preference to top pallets in the middle.
Reliability
the probability a product will perform its intended function satisfactorily for a pre-determined period of time in a given environment. Product durability implies that the product will last for a long time.
