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Describing Variation: Graphical Displays and Numerical Summaries

Learn how to effectively describe variation in statistical quality control using graphical displays such as stem-and-leaf plots, histograms, box plots, and dot diagrams. Explore numerical summary statistics like mean, variance, standard deviation, and median. Understand important discrete and continuous probability distributions such as the binomial distribution and the Poisson distribution.

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Describing Variation: Graphical Displays and Numerical Summaries

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

  2. 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, 4th Edition

  3. 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 2-1.1 The Stem-and-Leaf Plot Introduction to Statistical Quality Control, 4th Edition

  4. 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, 4th Edition

  5. 2-1.2 The Frequency Distribution and Histogram • Histogram Introduction to Statistical Quality Control, 4th Edition

  6. 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, 4th Edition

  7. 2-1.3 Numerical Summary of Data Important summary statistics for a distribution of data can include: • Sample mean, • Sample variance, s2 • Sample standard deviation, s • Sample median, M Introduction to Statistical Quality Control, 4th Edition

  8. 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 StDev 40 215.50 211.00 634.5 25.19 Introduction to Statistical Quality Control, 4th Edition

  9. 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, 4th Edition

  10. 2-1.4 The Box Plot Introduction to Statistical Quality Control, 4th Edition

  11. 2-1.5 Sample Computer Output Introduction to Statistical Quality Control, 4th Edition

  12. 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, 4th Edition

  13. 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, 4th Edition

  14. 2-2 Important Discrete Distributions 2-2.1 The Hypergeometric Distribution 2-2.2 The Binomial Distribution 2-2.3 The Poisson Distribution 2-2.4 The Pascal and Related Distributions Introduction to Statistical Quality Control, 4th Edition

  15. 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”. • 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, 4th Edition

  16. 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, 4th Edition

  17. 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, 4th Edition

  18. 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, 4th Edition

  19. 2-3 Important Continuous Distributions 2-3.1 The Normal Distribution 2-3.2 The Exponential Distribution 2-3.3 The Gamma Distribution 2-3.4 The Weibull Distribution Introduction to Statistical Quality Control, 4th Edition

  20. 2-3.1 The Normal Distribution The normal distribution is an important continuous distribution. • Symmetric, bell-shaped • Mean,  • Standard deviation,  Introduction to Statistical Quality Control, 4th Edition

  21. 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, 4th Edition

  22. 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 occurring or percentages of nonconformities, etc.. A standardized normal random variable is: Introduction to Statistical Quality Control, 4th Edition

  23. 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, 4th Edition

  24. 2-3.2 The Normal Distribution Example The tensile strength of paper is modeled by a normal distribution with a mean of 35 lbs/in2 and a standard deviation of 2 lbs/in2. • What is the probability that the tensile strength of a sample is less than 40 lbs/in2? • If the specifications require the tensile strength to exceed 30 lbs/in2, what proportion of the samples is scrapped? Introduction to Statistical Quality Control, 4th Edition

  25. 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, 4th Edition

  26. 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, 4th Edition

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