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乘數原理 (multiplication principle)

乘數原理 (multiplication principle). Fundamentals of Probability and Statistics. 假設一試驗 E1 有 n1 種可能結果，對此 n1 個可能結果進行另一試驗 E2 ，又有 n2 可能結果，則此組合試驗 E1-E2 ，共有 n1  n2 可能結果。. 抽出不放回. 抽出放回. 考慮順序. 不考慮順序. n 個事物中取 r 個之四種情況. Fundamentals of Probability and Statistics. (permutation). (combination). 問題 1.

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乘數原理 (multiplication principle)

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  1. 乘數原理(multiplication principle) Fundamentals of Probability and Statistics • 假設一試驗E1有n1種可能結果，對此n1個可能結果進行另一試驗E2，又有n2可能結果，則此組合試驗E1-E2，共有n1n2可能結果。 Quality Control

  2. 抽出不放回 抽出放回 考慮順序 不考慮順序 n個事物中取r個之四種情況 Fundamentals of Probability and Statistics (permutation) (combination) Quality Control

  3. 問題1 Fundamentals of Probability and Statistics • 某西餐廳推出系列套餐組合，第一道湯品共有5種，第二道沙拉共有4種，第三道主菜共有10種，最後附餐則有5種。試問此系列套餐共有幾種搭配方式? • 某公司推出兒童電子琴玩具，其中有兩個可撥0~9的數字鍵，此兩鍵任意撥一號碼即發一種特殊音效，試問此玩具共可發出多少種音效? Quality Control

  4. 問題2 Fundamentals of Probability and Statistics • 汽車牌照號碼前兩碼可為英文字母或數字，後四碼可為數字。試問此設計共可容納多少車輛? • 六個學生要坐一排有10張椅子的座位上，有多少種不同坐法? • 自生產線上抽出10個產品檢驗，其中檢查出三個不良品。然而此三個不良品可能是第1、2、3個，也有可能是第2、5、8個…。試問共有多少種可能情況? Quality Control

  5. Probability Distributions Fundamentals of Probability and Statistics • 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. Quality Control

  6. Two Types of Probability Distributions Fundamentals of Probability and Statistics • 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. Quality Control

  7. Important Discrete Distributions Fundamentals of Probability and Statistics • The Hypergeometric Distribution • The Binomial Distribution • The Poisson Distribution • The Pascal and Related Distributions Quality Control

  8. Hypergeometric Distribution Fundamentals of Probability and Statistics The Hypergeometric distributionis N：母體數； D：某特定類別之物品數； n：樣本數 x：抽出n個物品中正好有x個是屬於D類之個數 Quality Control

  9. Example Fundamentals of Probability and Statistics • 有100支保險絲，隨機抽5支，若在某電壓下均燒斷，則接受。若已知該批有20支不良保險絲，則允收機率為何? Quality Control

  10. Bernoulli trial Fundamentals of Probability and Statistics An experiment with two, and only two, possible outcomes. A random variable X has a Bernoulli(p) distribution if Quality Control

  11. Fundamentals of Probability and Statistics The Binomial Distribution1 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. Quality Control

  12. Fundamentals of Probability and Statistics The Binomial Distribution2 The binomial distribution with parameters n 0 and 0 < p < 1, is The mean and variance of the binomial distribution are Quality Control

  13. Example Fundamentals of Probability and Statistics • 經長時間觀察，在一項製程中所生產的產品，平均10項產品中會有1個不良品。今自生產線上抽出5項檢驗，則最多有一項不良品之機率為何? Quality Control

  14. Fundamentals of Probability and Statistics Poisson Distribution1 • 在一定連續區間內計算變化之次數 • 期本假設 • 在極短區(h)內正好發生一次的變化機率接近λh • 發生和其他不重疊的區間之變化次數無關 • 基本上在極短時間內發生兩次或兩次以上變化的機率幾乎為零 Quality Control

  15. Fundamentals of Probability and Statistics Poisson Distribution2 The Poisson distribution is Where the parameter  > 0. The mean and variance of the Poisson distribution are Quality Control

  16. Fundamentals of Probability and Statistics The Poisson Distribution3 • 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. Quality Control

  17. Example Fundamentals of Probability and Statistics • 某品牌錄音帶之瑕疵數平均1200呎有一個，則4800呎裏有0個瑕疵之機率為何? Quality Control

  18. Fundamentals of Probability and Statistics Important Continuous Distributions • The Normal Distribution • The Exponential Distribution • The Gamma Distribution • The Weibull Distribution Quality Control

  19. Fundamentals of Probability and Statistics The Normal Distribution1 The normal distribution is an important continuous distribution. • Symmetric, bell-shaped • Mean,  • Standard deviation,  Quality Control

  20. Fundamentals of Probability and Statistics The Normal Distribution2 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. Quality Control

  21. Fundamentals of Probability and Statistics Standard normal distribution1 • 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: Quality Control

  22. Fundamentals of Probability and Statistics Standard normal distribution2 • 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. Quality Control

  23. Fundamentals of Probability and Statistics 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? Quality Control

  24. Fundamentals of Probability and Statistics Central Limit Theorem • 若 是由具有平均數μ及變異數σ2的分配所抽出之一組樣本數的平均數，則當n時 Quality Control

  25. The Exponential Distribution1 Fundamentals of Probability and Statistics • The exponential distribution is widely used in the field of reliability engineering. • The exponential distribution is The mean and variance are Quality Control

  26. The Exponential Distribution2 Fundamentals of Probability and Statistics • The relationship between the exponential distribution and Poisson distributionx=0 implies that there are no occurrences of the event in (0,t], and • P(0) is the probability that the interval to the first occurrence is greater than t  Quality Control

  27. Fundamentals of Probability and Statistics 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 Quality Control

  28. 生產前後 之檢驗 生產期間之 矯正措施 將品質設計 於製程中 允收抽樣 製程管制 連續改進 進步最少 進步最多 品質管制的各種層面 Quality Control

  29. 投入 轉換 產出 允收抽樣 允收抽樣 製程管制 檢驗 • 檢驗多少量？檢驗多少次 • 要在什麼地點進行製程檢驗 • 應進行集中檢驗或現場檢驗 Quality Control

  30. 檢驗成本 成本 總成本 檢驗成本 不良品通 過的成本 最佳檢驗數 Quality Control

  31. 在製程中何處進行檢驗 • 進料 • 最終產品 • 在昂貴作業前 • 在不可變更的製程之前 • 在包裝製程之前 Quality Control

  32. 統計的製程管制 • 管制程序 • 定義 • 衡量 • 與標準比較 • 評估 • 採取矯正措施 • 評估矯正措施 Quality Control

  33. Chance and Assignable Causes of Quality Variation • A process that is operating with only chance causes of variation present is said to be in statistical control. • A process that is operating in the presence of assignable causesis said to be out of control. • The eventual goal of SPC is reduction or elimination of variability in the process by identification of assignable causes. Quality Control

  34. 管制圖的目的 • 發現製程有無引起變異的非機遇原因，進而針對原因予以消除，維持製程的穩定，防止異常原因之再次發生。 Quality Control

  35. 管制圖的種類 —數據的性質分類1 • 計量值管制圖 • These charts are applied to data that follow a continuous distribution (measurement data). • 計數值管制圖 • These charts are applied to data that follow a discrete distribution. Quality Control

  36. 計量值質管制圖 平均數與全距管制圖 中位數與全距管制圖 平均數與標準差管制圖 個別值平均數與全距管制圖 複式管制圖 機率管制圖 趨勢管制圖… 計數值管制圖 不良率管制圖(p管制圖) 不良數管制圖(np管制圖) 缺點數管制圖(c管制圖) 平均缺點數管圖(u管制圖) 管制圖的種類 —依數據的性質分類2 Quality Control

  37. 計量值與計數值管圖之優缺點 Quality Control

  38. 製程分析用管制圖 為決定方針用 為工程解析用 為工程能力研究用 為製程管制之準備用 管制用管制圖 用以控制製程的品質，於繪製完成後將管制界限延長，就每日管制特性的數據計算統計量數，並予以點繪於圖上，以管制製程，如不在管制狀態即採取下列措施： 追查異常原因 迅速消除此種原因 研究此種原因使其不再發生 管制圖的種類 —依管制圖的用途分類 Quality Control

  39. 編製管制圖的步驟 1. 決定品質特性 2. 決定合理之樣組數 3. 抽驗並收集樣本觀察值 4. 計算試驗用之管制界限與中心線 5. 建立修訂後之管制界限與中心線 Quality Control

  40. 計量值管制圖—X-R管制圖(管制界限為±3個標準差)計量值管制圖—X-R管制圖(管制界限為±3個標準差) UCL：管制上限(upper control limit) CL：管制中心線(center line) LCL：管制下限(lower control limit) Quality Control

  41. Control Charts for and R (1) Notation for variables control charts • n - size of the sample (sometimes called a subgroup) chosen at a point in time • m - number of samples selected • = average of the observations in the ith sample (where i = 1, 2, ..., m) • = grand average or “average of the averages (this value is used as the center line of the control chart) Quality Control

  42. Control Charts for and R(2) Notation and values • Ri = range of the values in the ith sample Ri = xmax - xmin • = average range for all m samples •  is the trueprocess mean • is the trueprocess standard deviation Quality Control

  43. Control Charts for and R(3) Statistical Basis of the Charts • Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, . • If x1, x2, …, xn is a sample of size n, then he average of this sample is • is normally distributed with mean, , and standard deviation, Quality Control

  44. Control Charts for and R (4) Statistical Basis of the Charts • The probability is 1 -  that any sample mean will fall between • The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known. Quality Control

  45. Control Charts for and R (5) Estimating the Process Standard Deviation • The process standard deviation can be estimated using a function of the sample average range. • This is an unbiased estimator of  Relative range： W=R/σ E(W)=E(R/σ)=d2 Quality Control

  46. Control Charts for and R (6) Control Limits for the chart Quality Control

  47. ^ σR=d3(R/d2) Control Charts for and R (7) R=Wσ Var(W)=d32 σR=d3σ Quality Control

  48. Control Charts for and R (8) Control Limits for the R chart • D3 and D4 are found in Appendix VI for various values of n. Quality Control

  49. Interpretation of and R Charts • In interpreting patterns on the X bar chart, we must first determinewhether or not theR chartis in control. • Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability. • Cyclic Patterns • Mixture • Shift in process level • Trend • Stratification Quality Control

  50. Quality Control

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