Statistics

Statistics and ANOVAME 470Fall 2013Here are some interesting on-the-spot designs from the past and this class.

Fall 2009, 0.27 Cost/Height

Winner, Spring 201015 Tall8$ Cost0.533 Cost/Height

Fall 2011Height = 12Cost = 6Cost/Height = 0.5Fall 2011Height = 24Cost = 12Cost/height = 0.5 I really enjoy the on-the-spot design.What did you learn about the design process?There are many challenges in product developmentTrade-offsDynamicsDetailsTime pressureEconomicsWhy do I love product development?Getting something to workSatisfying societal needsTeam diversityTeam spirit

Design is a process that requiresmaking decisions.4PlanningProduct Development PhasesConceptDevelopmentSystem-LevelDesignDetailDesignTesting andRefinementProductionRamp-UpConcept Development ProcessPerform Economic AnalysisBenchmark Competitive ProductsBuild and Test Models and PrototypesIdentifyCustomerNeedsEstablishTargetSpecificationsGenerateProductConceptsSelectProductConcept(s)Set FinalSpecificationsPlanDownstreamDevelopmentMissionStatementTestProductConcept(s)DevelopmentPlanYou will practice theentire concept development process with your group projectSo we are at the beginning and this slide differentiates between Product Development and Concept DevelopmentWith their projects, students are going to accomplish the entire concept development process.5We will use statistics to make good design decisions!We will categorize populations by the mean, standard deviation, and use control charts to determine if a process is in control.We may be forced to run experiments to characterize our system. We will use valid statistical tools such as Linear Regression, DOE, and Robust Design methods to help us make those characterizations.Cummins asked a capstone group to investigate improvements for turbo charger lubrication sealing.

5.9L High Output Cummins Engine

Cummins Inc. was dissatisfied with the integrity of their turbocharger oil sealing capabilities.Here are pictures of oil leakage.

Oil Leakage into Compressor Housing

Oil Leakage on Impellor PlateThe students developed four prototypes for testing. After testing, they wanted to know which solution to present to Cummins. You will analyze their data to make a suggestion. How can we use statistics to make sense of data that we are getting?Quiz for the dayWhat can we say about our M&Ms?We will look at the results first and then you can do the analysis on your own.Statistics can help us examine the data and draw justified conclusions.What does the data look like?What is the mean, the standard deviation?What are the extreme points?Is the data normal?Is there a difference between years? Did one class get more M&Ms than another?If you were packaging the M&Ms, are you doing a good job?If you are the designer, what factors might cause the variation?11Why would we care about this data in design?

If we designed the manufacturing line, we might be interested in the variation in the number of M&Ms per bag. If we were designing the bags, we would want to know the maximum number of M&Ms that we might hold in a bag. There are lots of reasons that we might be interested as designers in the number of M&Ms in a bag.12

If I am a plant manager, do I like one distribution better than another?Boxplots give us an idea of the variability of the data. Notice the outliers that we have each year. If I am the plant manager, do I like one distribution better than another?13How do we interpret the boxplot?BSNOx2.452.402.352.302.252.20Boxplot of BSNOx(Q2), medianQ1Q3largest value excluding outlierssmallest value excluding outliersoutliers are marked as *Values between 1.5 and 3 times away from the middle 50% of the data are outliers. 14

This is a density description of the data.This is a density description of the data. Some people prefer this arrangement. I am weird in that I look at everything. I dont omit anything.15The Anderson-Darling normality test is used to determine if data follow a normal distribution.

If the p-value is lower than the pre-determined level of significance, the data do not follow a normal distribution. Anderson-Darling Normality TestMeasures the area between the fitted line (based on chosen distribution) and the nonparametric step function (based on the plot points). The statistic is a squared distance that is weighted more heavily in the tails of the distribution. Anderson-Smaller Anderson-Darling values indicates that the distribution fits the data better.

The Anderson-Darling Normality test is defined as: H0: The data follow a normal distribution. Ha: The data do not follow a normal distribution.

Another quantitative measure for reporting the result of the normality test is the p-value. A small p-value is an indication that the null hypothesis is false. (Remember: If p is low, H0 must go.)

P-values are often used in hypothesis tests, where you either reject or fail to reject a null hypothesis. The p-value represents the probability of making a Type I error, which is rejecting the null hypothesis when it is true. The smaller the p-value, the smaller is the probability that you would be making a mistake by rejecting the null hypothesis.

It is customary to call the test statistic (and the data) significant when the null hypothesis H0 is rejected, so we may think of the p-value as the smallest level at which the data are significant.Note that our p value is quite low, which makes us consider rejecting the fact that the data are normal. However, in assessing the closeness of the points to the straight line, imagine a fat pencil lying along the line. If all the points are covered by this imaginary pencil, a normal distribution adequately describes the data. Montgomery, Design and Analysis of Experiments, 6th Edition, p. 39If you are confused about whether or not to consider the data normal, it is always best if you can consult a statistician. The author has observed statisticians feeling quite happy with assuming very fat lines are normal.http://www.statit.com/support/quality_practice_tips/normal_probability_plot_interpre.shtml For more on Normality and the Fat Pencil

You can use the fat pencil test in addition to the p-value.Walter Shewhart

www.york.ac.uk/.../ histstat/people/welcome.htm Developer of Control Charts in the late 1920sYou did Control Charts in DFM. There the emphasis was on tolerances. Here the emphasis is on determining if a process is in control. If the process is in control, we want to know the capability.19(READ SLIDE TEXT)

A caveat about this definition: We do not use errors and mistakes as synonyms. However, in this presentation we draw on the work of many people, and some authors will use the word mistake as a synonym with error. Where one of these is quoted, we have not changed their words. We do indicate their less precise use of the word mistake by italicizing it on the slide. Adapted from M. Hinckley, Quality by Design, 1996What does the data tell us about our process?SPC is a continuous improvement tool which minimizes tampering or unnecessary adjustments (which increase variability) by distinguishing between special cause and common cause sources of variationControl Charts have two basic uses:Give evidence whether a process is operating in a state of statistical control and to highlight the presence of special causes of variation so that corrective action can take place.Maintain the state of statistical control by extending the statistical limits as a basis for real time decisions.If a process is in a state of statistical control, then capability studies my be undertaken. (But not before!! If a process is not in a state of statistical control, you must bring it under control.)SPC applies to design activities in that we use data from manufacturing to predict the capability of a manufacturing system. Knowing the capability of the manufacturing system plays a crucial role in selecting the concepts.Voice of the ProcessControl limits are not spec limits.Control limits define the amount of fluctuation that a process with only common cause variation will have.Control limits are calculated from the process data.Any fluctuations within the limits are simply due to the common cause variation of the process.Anything outside of the limits would indicate a special cause (or change) in the process has occurred.Control limits are the voice of the process.The capability index depends on the spec limit and the process standard deviation.Cp = (allowable range)/6s = (USL - LSL)/6s

USL (Upper Specification Limit)LSLLCLUCL (Upper Control Limit)http://lorien.ncl.ac.uk/ming/spc/spc9.htm

Lower Control Limit for 2008Upper Control Limit for 2008If there is no difference in this year, students should be getting between 6 and 10 M&M with the average being 8.23Minitab prints results in the Session window that lists any failures.Test Results for I Chart of StackedTotals by C4

TEST 1. One point more than 3.00 standard deviations from center line.Test Failed at points: 129

TEST 2. 9 points in a row on same side of center line.Test Failed at points: 15, 110, 111, 112, 113

TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on one side of CL).Test Failed at points: 52, 66, 119, 160, 161

TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on one side of CL).Test Failed at points: 91, 97

TEST 7. 15 points within 1 standard deviation of center line (above and below CL).Test Failed at points: 193, 194, 195, 196, 197, 198, 199, 200This chart is extremely helpful for deciding what statistical technique to use.X DataSingle XMultiple Xs Y DataSingle Y Multiple Ys X DataDiscrete Continuous Y DataDiscrete Continuous One-sample t-testTwo-sample t-testANOVAX DataDiscrete Continuous Y DataDiscrete Continuous Chi-SquareSimple Linear RegressionLogistic RegressionANOVAMultiple Linear RegressionMultiple Logistic RegressionMultiple Logistic RegressionNote: our Y data is really not continuous. You should not use this analysis without consulting a statistician. I actually talked to Dr. DeVasher and he said, when you have a lot of data, it is ok to use this as a first pass.25When to use ANOVAThe use of ANOVA is appropriate whenDependent variable is continuousIndependent variable is discrete, i.e. categoricalIndependent variable has 2 or more levels under studyInterested in the mean valueThere is one independent variable or more

We will first consider just one independent variableOk, the dependent variable is not really continuous. We normally dont have portions of M&Ms. However, it is ok for this in class example, AND we let you eat M&Ms. 26ANOVA Analysis of Variance

Used to determine the effects of categorical independent variables on the average response of a continuous variableChoices in MINITABOne-way ANOVAUse with one factor, varied over multiple levelsTwo-way ANOVAUse with two factors, varied over multiple levelsBalanced ANOVAUse with two or more factors and equal sample sizes in each cellGeneral Linear ModelUse anytime!I always use the General Linear Model because the software then figures out if you have any of the special cases. 27Practical ApplicationsDetermine if our break pedal sticks more than other companiesCompare 3 different suppliers of the same componentCompare 6 combustion recipes through simulationDetermine the variation in the crush forceCompare 3 distributions of M&MsAnd MANY more General Linear Model: StackedTotals versus C4

Factor Type Levels ValuesC4 fixed 3 2008, 2010, 2011

Analysis of Variance for StackedTotals, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PC4 2 6.6747 6.6747 3.3374 4.71 0.010Error 203 143.8559 143.8559 0.7086Total 205 150.5306

S = 0.841813 R-Sq = 4.43% R-Sq(adj) = 3.49%

This p value indicates that the assumption that there is no difference between years is not correct!The null hypothesis for ANOVA is that there is no difference between years.

What are some conclusions that you can reach?Ok, we can see that there is one data point that doesnt fit anything! It also looks like we have one year with lower variation.30Is there a statistical difference between years?

Wow there looks like there is a big difference until you read the numbers then there isnt much difference between years. 31Grouping Information Using Tukey Method and 95.0% Confidence

C4 N Mean Grouping2010 57 7.9 A2008 86 7.7 A B2011 63 7.4 B

Means that do not share a letter are significantly different.

The p value indicates that there is a difference between the years. The Tukey printout tells us which years are different.The averages for 2010 and 2008 are not statistically different. The averages for 2008 and 2011 are not statistically different.

This is what Minitab should look like when students open it. Column C1 has the individual totals from students in 2008. Column C2 has the totals from 2010. C3 has the totals from spring 2011.There are previous years in other columns.33

Command:>Stat>Basic Statistics>Display Descriptive Statistics

Have students open the student supplied file for today. It is a Minitab file called M&Mtotals, student file.34Why would we care about this data in design?

If we designed the manufacturing line, we might be interested in the variation in the number of M&Ms per bag. If we were designing the bags, we would want to know the maximum number of M&Ms that we might hold in a bag. There are lots of reasons that we might be interested as designers in the number of M&Ms in a bag.35

If I am a plant manager, do I like one distribution better than another?Boxplots give us an idea of the variability of the data. Notice the outliers that we have each year. If I am the plant manager, do I like one distribution better than another?36

This is a density description of the data.This is a density description of the data. Some people prefer this arrangement. I am weird in that I look at everything. I dont omit anything.37>Stat>Basic Statistics>Normality TestSelect 2008

This test is for normality of all of the data. This is because, at this point, I dont know whether the years are different or not. If I have reason to suspect that the data are different, I would do a normality test on each year. This will be taken care of later in the ANOVA test.38The Anderson-Darling normality test is used to determine if data follow a normal distribution.

If the p-value is lower than the pre-determined level of significance, the data do not follow a normal distribution.

Command:>Stat>Control Charts>Variable Charts for Individuals>IndividualsNote: This is assuming that the packages are coming in sequence off of the manufacturing line. If this is actually true it is a coincidence.40

When doing control charts for ME470, select all tests.It may be hard to see, but highlight the tests tab.Why do we care about 15 points in a row within 1 standard deviation of center line? It isnt normal!At Honda, there was a suspension problem car always pulled left. They were always up near the upper spec limit41

Minitab prints results in the Session window that lists any failures.Test Results for I Chart of StackedTotals by C4

TEST 1. One point more than 3.00 standard deviations from center line.Test Failed at points: 129

TEST 2. 9 points in a row on same side of center line.Test Failed at points: 15, 110, 111, 112, 113

TEST 5. 2 out of 3 points more than 2 standard deviations from center line (on one side of CL).Test Failed at points: 52, 66, 119, 160, 161

TEST 6. 4 out of 5 points more than 1 standard deviation from center line (on one side of CL).Test Failed at points: 91, 97

TEST 7. 15 points within 1 standard deviation of center line (above and below CL).Test Failed at points: 193, 194, 195, 196, 197, 198, 199, 200

Lower Control Limit for 2008Upper Control Limit for 2008If there is no difference in this year, students should be getting between 6 and 10 M&M with the average being 8.44Command:>Stat>ANOVA>General Linear Model

45

What are some conclusions that you can reach?Ok, we can see that there is one data point that doesnt fit anything! It also looks like we have one year with lower variation.46Is there a statistical difference between years?

Wow there looks like there is a big difference until you read the numbers then there isnt much difference between years. 47General Linear Model: StackedTotals versus C4

Factor Type Levels ValuesC4 fixed 3 2008, 2010, 2011

Analysis of Variance for StackedTotals, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PC4 2 6.6747 6.6747 3.3374 4.71 0.010Error 203 143.8559 143.8559 0.7086Total 205 150.5306

S = 0.841813 R-Sq = 4.43% R-Sq(adj) = 3.49%

This p value indicates that the assumption that there is no difference between years is not correct!The null hypothesis for ANOVA is that there is no difference between years. Command:>Stat>ANOVA>General Linear Model

Grouping Information Using Tukey Method and 95.0% Confidence

C4 N Mean Grouping2010 57 7.9 A2008 86 7.7 A B2011 63 7.4 B

Means that do not share a letter are significantly different.

The p value indicates that there is a difference between the years. The Tukey printout tells us which years are different.The averages for 2010 and 2008 are not statistically different. The averages for 2008 and 2011 are not statistically different. Here is a useful reference if you feel that you need to do more reading.http://www.StatisticalPractice.comThis recommendation is thanks to Dr. DeVasher.

You can also use the help in Minitab for more information.

Lets look at what happened with plain M&MsWe are going to look at these for fun. See if students can get these same results on their own.52What do you see with the boxplot?

The boxplot makes me think that 2009 is lower and that maybe 2004 is higher53Do we see anything that looks unusual?

Except for that strange outlier, I really dont see anything weird at all.54

Both 2004 and 2005 look remarkably in control. 2006 looks awful, 2009 isnt too bad.55

General Linear Model: stackedTotal versus StackedYear

Factor Type Levels ValuesStackedYear fixed 4 2004, 2005, 2006, 2009

Analysis of Variance for stackedTotal, using Adjusted SS for TestsSource DF Seq SS Adj SS Adj MS F PStackedYear 3 1165.33 1165.33 388.44 149.39 0.000 Look at low P-value!Error 266 691.63 691.63 2.60Total 269 1856.96

S = 1.61249 R-Sq = 62.75% R-Sq(adj) = 62.33%

Unusual Observations for stackedTotal

Obs stackedTotal Fit SE Fit Residual St Resid 25 27.0000 23.4667 0.2082 3.5333 2.21 R 34 20.0000 23.4667 0.2082 -3.4667 -2.17 R209 40.0000 21.7917 0.1700 18.2083 11.36 R215 21.0000 17.4917 0.2082 3.5083 2.19 R

R denotes an observation with a large standardized residual.Grouping Information Using Tukey Method and 95.0% ConfidenceStackedYear N Mean Grouping2004 60 23.5 A2006 90 21.8 B2005 60 20.7 C2009 60 17.5 D

Means that do not share a letter are significantly different.Tukey 95.0% Simultaneous Confidence IntervalsResponse Variable stackedTotalAll Pairwise Comparisons among Levels of StackedYearStackedYear = 2004 subtracted from:

StackedYear Lower Center Upper -------+---------+---------+---------2005 -3.531 -2.775 -2.019 (---*---)2006 -2.365 -1.675 -0.985 (-*--)2009 -6.731 -5.975 -5.219 (--*--) -------+---------+---------+--------- -5.0 -2.5 0.0Zero is not contained in the intervals. Each year is statistically different. (2004 got the most!)

StackedYear = 2005 subtracted from:

StackedYear Lower Center Upper -------+---------+---------+---------2006 0.410 1.100 1.790 (-*--)2009 -3.956 -3.200 -2.444 (--*--) -------+---------+---------+--------- -5.0 -2.5 0.0

StackedYear = 2006 subtracted from:

StackedYear Lower Center Upper -------+---------+---------+---------2009 -4.990 -4.300 -3.610 (--*--) -------+---------+---------+--------- -5.0 -2.5 0.0Implications for designIs there a difference in production performance between the plain and peanut M&Ms?What do you think? The plain M&Ms have different means, but maybe this is simply because we have a narrower resolution. 60Individual QuizName:____________Section No:__________CM:_______You will be given a bag of M&Ms. Do NOT eat the M&Ms.Count the number of M&Ms in your bag. Record the number of each color, and the overall total. You may approximate if you get a piece of an M&M. When finished, you may eat the M&Ms. Note: You are not required to eat the M&Ms.ColorNumber%BrownYellowRedOrangeGreenBlueOtherTotalInstructions for Minitab Installation

Minitab on DFS:

Lets Look at Toyota RecallsNov 02, 2009 US: 3.8 million Toyota and Lexus vehicles again recalled due to floor mat problem, this time for all driver's side mats.[5]Nov 26, 2009 US: floor mat recall amended to include brake override[32] and increased to 4.2 million vehicles.[citation needed]Jan 21, 2010 US: 2.3 million Toyota vehicles recalled due to faulty accelerator pedals[6] (of those, 2.1 million already involved in floor mat recall).[3]Jan 27, 2010 US: 1.1 million Toyotas added to amended floor mat recall.[33]Jan 29, 2010 Europe, China: 1.8 million Toyotas added to faulty accelerator pedal recall.[7]

Lets consider the Toyota problem.What was the first clue that there was a problem?Starting in 2003, NHSTA received information regarding reports of accelerator pedals that were operating improperly.

How many reports causes the manufacturer to suspect a problem?To issue a recall NHTSA would need to prove that a substantial number of failures attributable to the defect have occurred or is likely to occur in consumers use of the vehicle or equipment and that the failures pose an unreasonable risk to motor vehicle safety.ODI conducted a VOQ-based assessment of UA rates on the subject Lexus incomparison to two peer vehicles and concluded the Lexus LS400t vehicles were not overrepresented in the VOQ database.

How might we look at two populations and decide this?

Office of Defects InvestigationVehicle Owner QuestionnaireUnintended AccelerationVOQ Vehicle Owner questionnaire; UA unintended acceleration, ODI - Office of Defects Investigation65