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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。

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n個事物中取r個之四種情況

Fundamentals of Probability and Statistics

(permutation)

(combination)

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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?

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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?

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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.

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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.

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Important Discrete Distributions

Fundamentals of Probability and Statistics

• The Hypergeometric Distribution

• The Binomial Distribution

• The Poisson Distribution

• The Pascal and Related Distributions

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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

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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?

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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

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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.

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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

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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?

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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

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Fundamentals of Probability and Statistics

Poisson Distribution2

The Poisson distribution is

Where the parameter  > 0. The mean and variance of the Poisson distribution are

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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.

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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?

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Fundamentals of Probability and Statistics

Important Continuous Distributions

• The Normal Distribution

• The Exponential Distribution

• The Gamma Distribution

• The Weibull Distribution

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Fundamentals of Probability and Statistics

The Normal Distribution1

The normal distribution

is an important

continuous distribution.

• Symmetric, bell-shaped

• Mean, 

• Standard deviation, 

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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.

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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:

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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.

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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?

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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

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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

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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 

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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

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• 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

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• 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

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• 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

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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.

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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。

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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.

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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

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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)

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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)

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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

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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,

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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.

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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

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Control Charts for and R (6)

Control Limits for the chart

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^

σR=d3(R/d2)

R=Wσ

Var(W)=d32

σR=d3σ

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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.

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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

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The Shewhart Control Chart for Individual Measurements(1)

• What if you could not get a sample size greater than 1 (n =1)? Examples include

• Automated inspection and measurement technology is used, and every unit manufactured is analyzed. There is no basis for rational subgrouping.

• The production rate is very slow, and it is inconvenient to allow samples sizes of N > 1 to accumulate before analysis

• Repeat measurements on the process differ only because of laboratory or analysis error, as in many chemical processes.

• The X and MR charts are useful for samples of sizes

n = 1.

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個別值與移動全距管制圖(2)

• 使用理用

• 產品需經很久的時間才能製造完成

• 分析或測定一件產品的品質較為麻煩且費時

• 產品係非常貴重的物品

• 產品品質頗為均勻(自動化生產)

• 屬破壞性試驗

• 在於管制製造條件如溫度、壓力、濕度…等

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The Shewhart Control Chart for Individual Measurements(3)

Moving Range Chart

• The moving range (MR) is defined as the absolute difference between two successive observations:

MRi = |xi - xi-1|

which will indicate possible shifts or changes in the process from one observation to the next.

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The Shewhart Control Chart for Individual Measurements(4)

X and Moving Range Charts

• The X chart is the plot of the individual observations. The control limits are

where

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The Shewhart Control Chart for Individual Measurements(5)

X and Moving Range Charts

• The control limits on the moving range chart are:

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The Shewhart Control Chart for Individual Measurements(5)

Example

Ten successive heats of a steel alloy are tested for hardness. The resulting data are

Heat HardnessHeat Hardness

152 6 52

251 750

354 851

455 958

550 1051

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Example

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The Shewhart Control Chart for Individual Measurements(7)

Interpretation of the Charts

• X Charts can be interpreted similar to charts. MR chartscannotbe interpreted the same as or R charts.

• Since the MR chart plots data that are “correlated” with one another, then looking for patterns on the chart does not make sense.

• MR chartcannotreally supply useful information about process variability.

• More emphasis should be placed on interpretation of the X chart.

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The Shewhart Control Chart for Individual Measurements(8)

Average Run Lengths (ARL=1/(1-β))

• The ability of the individuals control chart to detect small shifts is very poor.

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The Shewhart Control Chart for Individual Measurements(9)

• The normality assumption is often taken for granted.

• When using the individuals chart, the normality assumption is very important to chart performance.

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• 計數值管制圖1

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管制圖的判讀1

1) One point plots outside the 3-sigma control limits.

2) Two out of three consecutive points plot beyond the 2-sigma warning limits.

3) Four out of five consecutive points plot at a distance of 1-sigma or beyond from the center line.

4) Eight consecutive points plot on one side of the center line.

5) Six points in a row steadily increasing or decreasing.

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管制圖的判讀2

6) Fifteen points in a row in zone C (both above and below the center line).

7) Fourteen points in a row steadily increasing or decreasing.

8) Eight points in a row on both sides of the center line with none in zone C.

9) An unusual or nonrandom pattern in the data.

10) One or more points near a warning or control limit.

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連串檢定方法

• 連串(run)：指擁有某種特性的一連串觀測值。

• 連串檢定方法：

• 檢查連串的向上與向下的連串數目

• 計算中位數上方與下方的連串數目

連串檢定方法之步驟

• 將數字資料轉換並計算個別的連串數。

• 轉換成一系列的U 與D（即代表「上」與「下」）

• 轉換為一系列的A 與B（即「大於」中位數與「小於」中位數）。

• 計算連串數期望值與標準差。

• 區分出連串數的觀測值與期望值不同後，計算標準差數字z。

• 將數字與完全隨機系列的期望連串數比較。

• 將z 值與±2（95.5% 的z值）或其他期望值（例如，95% 的±1.96 或98% 的±2.33）互相比較。

• 若z 值超過期望界限，則表示存在非隨機模式。

連串檢定公式彙整

N ＝總觀測數

r ＝ A 與B 或U 與D 的觀察連串數，取決於所涉及的檢定。

Choice of Control Limits

“99.73% of the Data”

• If approximately 99.73% of the data lies within 3 of the mean (i.e., 99.73% of the data should lie within the control limits), then 1 - 0.9973 = 0.0027 or 0.27% of the data can fall outside 3 (or 0.27% of the data lies outside the control limits).

• 0.0027 is the probability of a Type I error or a false alarm in this situation.

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Types of Error1

• Type I error - rejecting the null hypothesis when it is true.

• The risk of a point falling beyond the control limits, indicating an out-of-control condition when no assignable cause is present.

• Pr(Type I error) = . Sometimes called the producer’s risk.

• 可能造成不必要的預防成本與鑑定成本

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Types of Error2

• Type II error - not rejecting the null hypothesis when it is false.

• The risk of a point falling between the control limits when the process is really out of control.

• Pr(Type II error) = . Sometimes called the consumer’s risk.

• 可能造成不必要的內、外部失敗成本

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管制界限設定為±3σ的理由

• 在統計推論時，將發生型I或型II誤差。而在1930年代，所有工業型態之廣大經驗指出，±3σ界限提供此兩種錯誤型態所產生的成本之間的經濟平衡。

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規格界限與管制上下限

• 規格界限

• 乃是顧客要求，而工程設計建立的規範。指示產品必須落在此範圍內才會被接受。

• 管制界限

• 乃是統計上的界限。指示樣本平均或樣本全距因隨機變化可跨越的範圍界限。

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規格界限與管制上下限之比較

• 規格界限與管制界限並無直接相關，既使製程落在管制界限內，亦可能不合規範要求。

• 管制界限<規格界限

• 管制界限優良，製程能滿足標準。

• 管制界限>規格界限

• 離散大時，改善製程，減小變異。

• 偏心時，改善固定性變異，移動平均值至規格中心。

• 無法改善製程時亦可考慮修改標準。

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UCL

UCL

UCL

LCL

LCL

LCL

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製程能力(process capability)

• 意義

• 對於穩定之製程所持有之特定成果，能夠合理達成之能力界限。

• 製程能力的表示法

• 製程準確度(Ca值)

• 製程精密度(Cp值)製程能力比(process capability ratio，PCR)

• 製程能力指數(Cpk值)

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Ca=

=

X-μ

=

T/2

T：規格公差=規格上限-規格下限

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製程準確度之評價方法

• A級

• 小於等於12.5%理想的狀態，故維持現狀。

• B級

• 大於12.5%，小於等於25%儘可能調整改進至A級。

• C級

• 大於25%，小於等於50%應立即檢討並加以改善

• D級

• 大於50%，小於100%採取緊急措施，並全面檢討，必要時應考停止生產。

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T

=

Cp=

6個估計標準差

=

SU-X

=

Cp=

3個估計標準差

=

X-SL

=

Cp=

3個估計標準差

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製程精密度評價方式

• A+級

• 大於等於1.67可考慮放寛產品變異，尋求管理的簡單化或成本降低方法。

• A級

• 小於1.67，大於等於1.33理想的狀態，故維持現狀。

• B級

• 小於1.33，大於等於1.00確實進行製程管制，儘可能改善為A級。

• C級

• 小於1.00，大於等0.67產生不良品，產品須全數選別，並管理改善製程。

• D級

• 小於0.67採取緊急對策，進行品質改善，探求原因並重新檢討規格。

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{

}

X-SL

SU-X

Cpk=min

=

=

Cpk=(1-|Ca|)Cp

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製程能力指數評價方式

• A級

• 大於等於1.33製程能力足夠

• B級

• 小於1.33，大於等於1.00能力尚可應再努力

• C級

• 小於1.00應加以改善

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製程能力指標的限制

• 如果製程並不穩定，則製程能力指標將沒有意義。

• 若製程產出並非常態分配，則對製程產出的推論將不正確。

• 若製程不準確，則使用Cp指標將造成誤導。

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