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MER301: Engineering Reliability LECTURE 2: Chapter 1: Role of Statistics in Engineering Chapter 2: Data Summary and Presentation L Berkley Davis Copyright 2009 MER301: Engineering Reliability 1 Summary of Lecture 2 Topics Summary of Chapter 1 Topics Engineering Method Statistics in Engineering Collection of Engineering Data Observing Processes over Time Populations and Samples Data Displays Summary of Chapter 2 Topics Central Point and Spread L Berkley Davis Copyright 2009 Dot Diagrams Histograms Box and Whisker Plots Scatter plots Median,Quartiles,Interquartile range Means, Variances and Standard Deviations MER301: Engineering Reliability Lecture 2 2 Engineering Method L Berkley Davis Copyright 2009 Successful design and introduction of a new product is dependent on a rigorous engineering process that is executed with discipline and attention to detail Design for Six Sigma is one such process that allows the designer to explicitly account for the effects of variation MER301: Engineering Reliability Lecture 2 3 Elements of Design for Six Sigma Flowdown of Customer Requirements(CTQ’s) to Engineering Measurement System Analysis(Gage R&R) Statistical Design Methods(Probabilistic Analyses) rather than Deterministic(Mathematical) Analysis Quantitative Transfer Functions linking CTQ’s(Y’s) to x’s Disciplined Risk Assessment Process Design Optimization and Robust Design allow products to be minimally sensitive to design, operating and manufacturing variation Design for Manufacturability/Process Capabilityto ensure product CTQs are met in light of manufacturing capability Validation of product performance L Berkley Davis Union College Copyright 2009Engineering Mechanical MER301: Engineering Reliability Lecture 2 4 Critical to Quality Variables(CTQ’s) Products/Processes have measures of performance, operational flexibility, reliability, and cost that are directly seen by the end customer These are called CTQ variables(Big Y’s) and are the ultimate measurement of an engineered product or process Big Y’s are functions of other variables that the engineer must control in the design(control variables) or allow to be uncontrolled(noise) Y fn( x1 , x2 ,...xn ) L Berkley Davis Copyright 2009 The Designer must understand MER301: Engineering Reliability Product and Process CTQ’s Lecture 2 5 Measurement System Errors… Total Error in a measurement is defined as the difference between the True Value and the Measured Value of Y Accuracy of Measurement System is defined as the difference between a Standard Reference and the Average Observed Measurement Two general categories of error – Bias or Accuracy Error and Precision Error (excluding gross blunders) Total Error = Bias Error + Precision Error for independent random variables Measurement System Error is described by Average Bias Error (Mean Shift)and a statistical estimate of the Precision Error (Variance) Measurement System Analysis is a Fundamental Part of Every Experiment L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 6 Experimental Gage R&R Precision and Accuracy Not Accurate, Not Precise Not Accurate, Precise L Berkley Davis Copyright 2009 Accurate, Not Precise Accurate, Precise MER301: Engineering Reliability Lecture 2 7 Engineering Models Mathematical Model:Quantitative description of a system/event with descriptive equations Physics Based(Mechanistic) Models built from first principles Empirical Models built from Data and Engineering Knowledge Both Physics Based and Empirical Models can be either Deterministic or Statistical/Probabilistic Deterministic For Y=fn(x’s) , model does not explicitly account for variation Accounts for variation in x’s, by letting each x be described by a mean value and a variation Probabilistic/Statistical L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 8 Engineering Models Physics Based Models Physics Based Fluid Mechanics Models Conservation of Mass, Momentum, Continuity and Energy Fluid Mechanics/Heat Transfer V 0 t Continuity,Navier-Stokes, Momentum Energy, Acoustics, Lubrication, Turbulence DV V V V P F 2 V Elasticity Dt t Stress/ Strain,isotropic media, Energy Beam/Column Theory Electromagnetic Theory Maxwell’s Laws, Ohm’s Law, Q De k 2 T q r Wave equations, Plasma t Dt dynamics Dynamics Kinematics,Inertia, Rigid Bodies Unio n Coll eg e Mec ha nic al Engi ne eri ng L Berkley Davis Copyright 2009 Empirical Modeling- Regression Analysis The Big Y is the Pull Strength.. Wire Length and Die Height are the independent variables The goal here is to use the data to create an empirical model that relates the value of Y to the values of the x’s The methodology is to conduct a regression analysis… L Berkley Davis Copyright 2009 L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 13 Statistics in Engineering… Engineers work with data sets and need methods and tools to summarize data and draw conclusions Descriptive statistics to present data in an understandable manner Measures of central points and variation to characterize and data Engineers deal with variation in all of their work. Variation arises from: Real variation caused by parts tolerance, materials property variations or operational differences Apparent or Gage R&R variation from measurement system error A consequence of variation is that engineers must deal with probability in product assembly, product performance, and product reliability Statistical Design Methods are needed to deal with probabilistic design L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 10 Statistical Methods/Tools… Probability –The Laws of Chance Descriptive Statistics- Analytical and graphical methods that allow us to describe or picture a data set Inferential Statistics- Methods by which conclusions can be drawn about a large group of objects based on observing only a portion of the objects Model Building- Development of prediction equations(transfer functions) from experimental data L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 11 Uses of Statistical Tools Establishing design targets from CTQ’s Data collection(sampling,gage R&R,DOE) Sampling strategy Analysis of data(means,variances, generation of transfer functions, descriptive statistics) Statistical Inference/hypothesis testing Model Building/Optimization/Validation Statistical Design/Process Control L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 12 Collection of Engineering Data Retrospective Study Uses existing data to model existing processes/designs in order to make predictions about future performance Observational Study Quality of data often an issue with this kind of study Process or phenomenon is recorded Insufficient data set(too few x’s or too narrow a range of variation of x’s) watched and data is Not enough samples for statistical validity variables All relevant Validity of measurements in question are measured Designed Measurements are made with the Retrospective Studies often used in required rigor failure RCA’s Union College Mechanical Engineering Experiment System Output (big Y’s)observed There is no intervention in the under controlled conditions process/phenomenon on the part Y=fn(control x’s, noise x’s) of those making the study Control variables are manipulated MER301: Engineering Reliability Lecture 1 Union College Mechanical Engineering 26 MER301: Engineering Reliability Lecture 1 Noise variables must be identified Study environment is regulated 27 Used to establish “cause and effect” between x’s and Y’s L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 Union College Mechanical Engineering MER301: Engineering Reliability Lecture 1 13 28 Designed Factorial Experiments Several process variables(factors) and their ranges are identified as being significant in a Factorial Study L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 14 Observing Processes over Time Observing Processes over Time All processes exhibit variation over time…variation may be caused by random factors or by system degradation(wear) Control Charts can be used to Process Variation over Time monitor/correct process - Run or Control Charts performance Union College Mechanical Engineering MER301: Engineering Reliability Lecture 1 25 Union College Mechanical Engineering L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 MER301: Engineering Reliability Lecture 1 27 15 Summary of Chapter 2 Topics Populations and Samples Data Displays Dot Diagrams Histograms Box and Whisker Plots Scatter plots Central Point and Spread Median,Quartiles,Interquartile range Means, Variances and Standard Deviations L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 16 Populations and Samples Population- entire group of objects being studied Sample- collection of objects from which data are actually gathered Sample may be all or part of the entire population Sample Data are used to make predictions about the Population Validity of the predictions depends on how the Sample is taken and how big it is… Both Populations and Samples are characterized by the Central Point and the Spread of the variables being studied Populations are what we want to know aboutSample data are what we get….. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 17 Data Displays Dot Diagrams Histograms Dotplot for Weight Frequency 15 100 110 120 130 140 150 160 170 180 10 5 190 Weight 0 100 110 120 130 140 150 160 170 180 190 200 Weight Box and Whisker L Berkley Davis Copyright 2009 Scatter Plots MER301: Engineering Reliability Lecture 2 18 Pareto Charts Widely used in process analysis to identify the most frequent failures L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 19 Measures of Central Point and Spread Percentile Ordered ranking of Data Median – measure of central tendency Not sensitive to Outliers Quartiles – divides data into 4 equal parts First or lower, second, third or upper Interquartile Range – measure of Spread L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 20 Central point-Population Mean For a population of size N…. Descriptive Statistics Variable: L2MeanEx Anderson-Darling Normality Test A-Squared: P-Value: 21.45 22.20 22.95 23.70 24.45 25.20 25.95 26.70 27.45 28.20 Mean StDev Variance Skewness Kurtosis N 24.9898 1.0143 1.02887 5.96E-02 -5.6E-02 5000 95% Confidence Interval for Mu Minimum 1st Quartile Median 3rd Quartile Maximum 21.1995 24.2996 24.9634 25.6760 28.4057 N x i 1 0.801 0.038 i 95% Confidence Interval for Mu N 24.9616 24.94 24.96 24.98 25.00 25.02 25.0179 95% Confidence Interval for Sigma 0.9948 1.0346 95% Confidence Interval for Median 95% Confidence Interval for Median L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 24.9317 25.0001 21 What is Variance? Variance is a quantitative measure of the square of the difference between each measurement in a sample and the mean of the sample. Comparison of the(square root of)variance to the mean gives information as to how well a process is controlled L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 22 Spread-Population Variance Measure of variation in the population Descriptive Statistics Variable: L2MeanEx Anderson-Darling Normality Test N 2 ( xi ) i 1 A-Squared: P-Value: 2 0.801 0.038 21.45 22.20 22.95 23.70 24.45 25.20 25.95 26.70 27.45 28.20 Mean StDev Variance Skewness Kurtosis N 24.9898 1.0143 1.02887 5.96E-02 -5.6E-02 5000 95% Confidence Interval for Mu Minimum 1st Quartile Median 3rd Quartile Maximum 21.1995 24.2996 24.9634 25.6760 28.4057 N 95% Confidence Interval for Mu 24.9616 24.94 24.96 24.98 25.00 25.02 25.0179 95% Confidence Interval for Sigma 0.9948 1.0346 95% Confidence Interval for Median 95% Confidence Interval for Median L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 24.9317 25.0001 23 Data Central Point-Sample Mean Yi 68.4 66.4 69.5 71.6 71.4 72.5 64.6 68.5 71.2 66.8 67.6 65.6 65.3 67.1 67.5 64.8 67.9 68.2 69.1 67.8 67.4 68.3 71.7 68.8 68.1 n observations in a sample are denoted by x1, x2, …, xn, n x L Berkley Davis Copyright 2009 xi n 25 Descriptive Statistics Mean Standard Error 68.244 0.432707754 Median 68.1 Mode #N/A Standard Deviation 2.163538768 Sample Variance Kurtosis Skewness 4.6809 -0.395792379 0.316647157 Range i 1 7.9 Minimum 64.6 Maximum 72.5 n Sum n 25 1706.1 Count 25 Largest(1) 72.5 Smallest(1) 64.6 Confidence Level(95.0%) MER301: Engineering Reliability Lecture 2 0.893064904 24 Data Central Point-Sample Median Yi 68.4 66.4 69.5 71.6 71.4 72.5 64.6 68.5 71.2 66.8 67.6 65.6 65.3 67.1 67.5 64.8 67.9 68.2 69.1 67.8 67.4 68.3 71.7 68.8 68.1 n observations in a sample are denoted by x1, x2, …, xn, n x xi i 1 n Descriptive Statistics Mean 68.244 Standard Error 0.432707754 Median 68.1 Data Rank Percent 6 72.5 1 100.00% 23 71.7 2 95.80% 4 71.6 3 91.60% 5 71.4 4 87.50% 9 71.2 5 83.30% 3 69.5 6 79.10% 19 69.1 7 75.00% 24 68.8 8 70.80% 8 68.5 9 66.60% 1 68.4 10 62.50% #N/A 22 68.3 11 58.30% 2.163538768 18 68.2 12 54.10% 25 68.1 13 50.00% 17 67.9 14 45.80% 20 67.8 15 41.60% 11 67.6 16 37.50% 7.9 15 67.5 17 33.30% Minimum 64.6 21 67.4 18 29.10% Maximum 72.5 14 67.1 19 25.00% 1706.1 10 66.8 20 20.80% Mode Standard Deviation Sample Variance 4.6809 Kurtosis -0.395792379 Skewness 0.316647157 Range Sum Count Largest(1) n 25 25 72.5 Smallest(1) 64.6 Confidence Level(95.0%) L Berkley Davis Copyright 2009 Point 0.893064904 MER301: Engineering Reliability Lecture 2 2 66.4 21 16.60% 12 65.6 22 12.50% 13 65.3 23 8.30% 16 7 64.8 64.6 24 25 4.10% 0.00% 25 Spread-Sample Variance Measure of variation in the sample Note n-1 rather than N as divisor n 2 ( xi x ) 2 i 1 s n 1 Descriptive Statistics Mean Standard Error 68.244 0.432707754 Median Mode Standard Deviation Sample Variance Kurtosis Skewness 68.1 #N/A 2.163538768 4.6809 -0.395792379 0.316647157 Range 7.9 Minimum 64.6 Maximum 72.5 Sum Count 25 Largest(1) 72.5 Smallest(1) 64.6 Confidence Level(95.0%) L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 2 1706.1 0.893064904 26 Sample Mean and Variance…Rank Order Median..Histogram and Box Plot… n Point Data Rank Percent 6 72.5 1 100.00% 23 71.7 2 95.80% 4 71.6 3 91.60% 5 71.4 4 87.50% 9 71.2 5 83.30% 3 69.5 6 79.10% 19 69.1 7 75.00% 24 68.8 8 70.80% 8 68.5 9 66.60% 1 68.4 10 62.50% 22 68.3 11 58.30% 18 68.2 12 54.10% 25 68.1 13 50.00% 17 67.9 14 45.80% 20 67.8 15 41.60% 11 67.6 16 37.50% 15 67.5 17 33.30% 21 67.4 18 29.10% 14 67.1 19 25.00% 10 66.8 20 20.80% 1706.1 2 66.4 21 16.60% Count 25 12 65.6 22 12.50% Largest(1) 72.5 13 65.3 23 8.30% Smallest(1) 64.6 16 7 64.8 64.6 24 25 4.10% 0.00% x xi i 1 n s 2 n ( xi x ) i 1 n 1 Descriptive Statistics Mean Standard Error 68.244 0.432707754 Median Mode Standard Deviation Sample Variance Kurtosis Skewness 68.1 #N/A 2.163538768 4.6809 -0.395792379 0.316647157 Range 7.9 Minimum 64.6 Maximum 72.5 Sum Confidence Level(95.0%) L Berkley Davis Copyright 2009 0.893064904 MER301: Engineering Reliability Lecture 2 2 27 Summary of Lecture 2 Topics Summary of Chapter 1 Topics Engineering Method Statistics in Engineering Collection of Engineering Data Observing Processes over Time Populations and Samples Data Displays Summary of Chapter 2 Topics Central Point and Spread L Berkley Davis Copyright 2009 Dot Diagrams Histograms Box and Whisker Plots Scatter plots Median,Quartiles,Interquartile range Means, Variances and Standard Deviations MER301: Engineering Reliability Lecture 2 28