Chapter 2 Modeling Process Quality Introduction to Statistical

Chapter 2 Modeling Process Quality Introduction to Statistical Quality Control, 4 th Edition

2 -1. Describing Variation • Graphical displays of data are important tools for investigating samples and populations. • Displays can include stem and leaf plots, histograms, box plots, and dot diagrams. • Graphical displays give an indication of the overall “distribution” of the data. Introduction to Statistical Quality Control, 4 th Edition

2 -1. 1 The Stem-and-Leaf Plot • The numbers on the left are the “stems” • The values on the right are the “leaves” • The smallest number in this set of data is 175 • The median is 211 17| 558 18| 357 19| 00445589 20| 1399 21| 00238 22| 005 23| 5678 24| 1555899 25| 158 Introduction to Statistical Quality Control, 4 th Edition

2 -1. 2 The Frequency Distribution and Histogram • Frequency Distribution – Arrangement of data by magnitude – More compact than a stem-and-leaf display – Graphs of observed frequencies are called histograms. Introduction to Statistical Quality Control, 4 th Edition

2 -1. 2 The Frequency Distribution and Histogram • Histogram Introduction to Statistical Quality Control, 4 th Edition

Graphical Displays • What is the overall shape of the data? • Are there any unusual observations? • Where is the “center” or “average” of the data located? • What is the spread of the data? Is the data spread out or close to the center? Introduction to Statistical Quality Control, 4 th Edition

2 -1. 3 Numerical Summary of Data Important summary statistics for a distribution of data can include: • Sample mean, • Sample variance, s 2 • Sample standard deviation, s • Sample median, M Introduction to Statistical Quality Control, 4 th Edition

2 -1. 3 Numerical Summary of Data • For the data shown in the previous histogram and stem and leaf plot, the summary statistics are: N Mean Median Var St. Dev 40 215. 50 211. 00 634. 5 25. 19 Introduction to Statistical Quality Control, 4 th Edition

2 -1. 4 The Box Plot • The Box Plot is a graphical display that provides important quantitative information about a data set. Some of this information is – Location or central tendency – Spread or variability – Departure from symmetry – Identification of “outliers” Introduction to Statistical Quality Control, 4 th Edition

2 -1. 4 The Box Plot Introduction to Statistical Quality Control, 4 th Edition

2 -1. 5 Sample Computer Output Introduction to Statistical Quality Control, 4 th Edition

2 -1. 6 Probability Distributions • Definitions – Sample A collection of measurements selected from some larger source or population. – Probability Distribution A mathematical model that relates the value of the variable with the probability of occurrence of that value in the population. – Random Variable variable that can take on different values in the population according to some “random” mechanism. Introduction to Statistical Quality Control, 4 th Edition

2 -1. 6 Probability Distributions • Two Types of Probability Distributions – Continuous When a variable being measured is expressed on a continuous scale, its probability distribution is called a continuous distribution. The probability distribution of piston-ring diameter is continuous. – Discrete When the parameter being measured can only take on certain values, such as the integers 0, 1, 2, …, the probability distribution is called a discrete distribution. The distribution of the number of nonconformities would be a discrete distribution. Introduction to Statistical Quality Control, 4 th Edition

2 -2 Important Discrete Distributions 2 -2. 1 2 -2. 2 2 -2. 3 2 -2. 4 The Hypergeometric Distribution The Binomial Distribution The Poisson Distribution The Pascal and Related Distributions Introduction to Statistical Quality Control, 4 th Edition

2 -2. 2 The Binomial Distribution 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, 4 th Edition

2 -2. 2 The Binomial Distribution The binomial distribution with parameters n 0 and 0 < p < 1, is The mean and variance of the binomial distribution are Introduction to Statistical Quality Control, 4 th Edition

2 -2. 3 The Poisson Distribution The Poisson distribution is Where the parameter > 0. The mean and variance of the Poisson distribution are Introduction to Statistical Quality Control, 4 th Edition

2 -2. 3 The Poisson Distribution • The Poisson distribution is useful in quality engineering – Typical model of the number of defects or nonconformities that occur in a unit of product. – Any random phenomenon that occurs on a “per unit” basis is often well approximated by the Poisson distribution. Introduction to Statistical Quality Control, 4 th Edition

2 -3 Important Continuous Distributions 2 -3. 1 2 -3. 2 2 -3. 3 2 -3. 4 The Normal Distribution The Exponential Distribution The Gamma Distribution The Weibull Distribution Introduction to Statistical Quality Control, 4 th Edition

2 -3. 1 The Normal Distribution The normal distribution is an important continuous distribution. • Symmetric, bellshaped • Mean, • Standard deviation, Introduction to Statistical Quality Control, 4 th Edition

2 -3. 1 The Normal Distribution For a population that is normally distributed: • approx. 68% of the data will lie within 1 standard deviation of the mean; • approx. 95% of the data will lie within 2 standard deviations of the mean, and • approx. 99. 7% of the data will lie within 3 standard deviations of the mean. Introduction to Statistical Quality Control, 4 th Edition

2 -3. 1 The Normal Distribution • Standard normal distribution – Many situations will involve data that is normally distributed. We will often want to find probabilities of events occuring or percentages of nonconformities, etc. . A standardized normal random variable is: Introduction to Statistical Quality Control, 4 th Edition

2 -3. 1 The Normal Distribution • Standard normal distribution – Z is normally distributed with mean 0 and standard deviation, 1. – Use the standard normal distribution to find probabilities when the original population or sample of interest is normally distributed. – Tables, calculators are useful. Introduction to Statistical Quality Control, 4 th Edition

2 -3. 2 The Normal Distribution Example The tensile strength of paper is modeled by a normal distribution with a mean of 35 lbs/in 2 and a standard deviation of 2 lbs/in 2. a) What is the probability that the tensile strength of a sample is less than 40 lbs/in 2? b) If the specifications require the tensile strength to exceed 30 lbs/in 2, what proportion of the samples is scrapped? Introduction to Statistical Quality Control, 4 th Edition

2 -3. 3 The Exponential Distribution • • The exponential distribution is widely used in the field of reliability engineering. The exponential distribution is The mean and variance are Introduction to Statistical Quality Control, 4 th Edition

2 -4 Some Useful Approximations • In certain quality control problems, it is sometimes useful to approximate one probability distribution with another. This is particularly useful if the original distribution is difficult to manipulate analytically. • Some approximations: – Binomial approximation to the hypergeometric – Poisson approximation to the binomial – Normal approximation to the binomial Introduction to Statistical Quality Control, 4 th Edition