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Statistics and ANOVA

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Statistics and ANOVA

ME 470

Fall 2013

Here are some interesting on-the-spot designs from the past and this class.

Winner, Spring 2010

15” Tall

8$ Cost

0.533 Cost/Height

Fall 2009, 0.27 Cost/Height

Fall 2011

Height = 12

Cost = 6

Cost/Height = 0.5

Fall 2011

Height = 24

Cost = 12

Cost/height = 0.5

- What did you learn about the design process?
- There are many challenges in product development
- Trade-offs
- Dynamics
- Details
- Time pressure
- Economics

- Why do I love product development?
- Getting something to work
- Satisfying societal needs
- Team diversity
- Team spirit

Design is a process

that requires

making decisions.

Concept

Development

System-Level

Design

Detail

Design

Testing and

Refinement

Production

Ramp-Up

Planning

Concept Development Process

You will practice the

entire concept

development

process with

your group project

Mission

Statement

Development

Plan

Identify

Customer

Needs

Establish

Target

Specifications

Generate

Product

Concepts

Select

Product

Concept(s)

Test

Product

Concept(s)

Set Final

Specifications

Plan

Downstream

Development

Perform Economic Analysis

Benchmark Competitive Products

Build and Test Models and Prototypes

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.

5.9L High Output Cummins Engine

Cummins Inc. was dissatisfied with the integrity of their turbocharger oil sealing capabilities.

Oil Leakage into Compressor Housing

Oil Leakage on Impellor Plate

The 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.

- Quiz for the day
- What can we say about our M&Ms?
- We will look at the results first and then you can do the analysis on your own.

- 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?

Why would we care about this data in design?

If I am a plant manager, do I like one distribution better than another?

How do we interpret the boxplot?

largest value excluding outliers

B

o

x

p

l

o

t

o

f

B

S

N

O

x

Q3

2

.

4

5

2

.

4

0

(Q2), median

2

.

3

5

x

O

N

S

B

2

.

3

0

Q1

2

.

2

5

2

.

2

0

outliers are marked as ‘*’

smallest value excluding outliers

Values between 1.5 and 3 times away from the middle 50% of the data are outliers.

This is a density description of the data.

The 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 Test

Measures 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 α at which the data are significant.

You can use the “fat pencil” test in addition to the p-value.

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. 39

If 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.

For more on Normality and the Fat Pencil

http://www.statit.com/support/quality_practice_tips/normal_probability_plot_interpre.shtml

Walter Shewhart

Developer of Control Charts in the late 1920’s

You 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.

www.york.ac.uk/.../ histstat/people/welcome.htm

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 variation

Control 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.

Control 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.

Cp = (allowable range)/6s = (USL - LSL)/6s

LSL

USL (Upper Specification Limit)

LCL

UCL (Upper Control Limit)

http://lorien.ncl.ac.uk/ming/spc/spc9.htm

Upper Control Limit for 2008

Lower Control Limit for 2008

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

X Data

Multiple Xs

Single X

X Data

X Data

Discrete

Continuous

Discrete

Continuous

Discrete

Logistic Regression

Multiple Logistic Regression

Multiple Logistic Regression

Single Y

Chi-Square

Discrete

Y Data

Y Data

Y Data

Continuous

One-sample t-test

Two-sample t-test

ANOVA

Simple Linear Regression

Multiple Linear Regression

Continuous

ANOVA

Multiple Ys

When to use ANOVA

- The use of ANOVA is appropriate when
- Dependent variable is continuous
- Independent variable is discrete, i.e. categorical
- Independent variable has 2 or more levels under study
- Interested in the mean value
- There is one independent variable or more

- We will first consider just one independent variable

ANOVA Analysis of Variance

- Used to determine the effects of categorical independent variables on the average response of a continuous variable
- Choices in MINITAB
- One-way ANOVA
- Use with one factor, varied over multiple levels

- Two-way ANOVA
- Use with two factors, varied over multiple levels

- Balanced ANOVA
- Use with two or more factors and equal sample sizes in each cell

- General Linear Model
- Use anytime!

- One-way ANOVA

Practical Applications

- Determine if our break pedal sticks more than other companies
- Compare 3 different suppliers of the same component
- Compare 6 combustion recipes through simulation
- Determine the variation in the crush force
- Compare 3 distributions of M&M’s
- And MANY more …

The null hypothesis for ANOVA is that there is no difference between years.

General Linear Model: StackedTotals versus C4

Factor Type Levels Values

C4 fixed 3 2008, 2010, 2011

Analysis of Variance for StackedTotals, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

C4 2 6.6747 6.6747 3.3374 4.71 0.010

Error 203 143.8559 143.8559 0.7086

Total 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!

What are some conclusions that you can reach?

Is there a statistical difference between years?

The p value indicates that there is a difference between the years. The Tukey printout tells us which years are different.

Grouping Information Using Tukey Method and 95.0% Confidence

C4 N Mean Grouping

2010 57 7.9 A

2008 86 7.7 A B

2011 63 7.4 B

Means that do not share a letter are significantly different.

The averages for 2010 and 2008 are not statistically different. The averages for 2008 and 2011 are not statistically different.

Why would we care about this data in design?

If I am a plant manager, do I like one distribution better than another?

This is a density description of the data.

Select 2008

The 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.

When doing control charts for ME470, select all tests.

It may be hard to see, but highlight the “tests” tab.

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

Upper Control Limit for 2008

Lower Control Limit for 2008

Command:

>Stat>ANOVA>General Linear Model

What are some conclusions that you can reach?

Is there a statistical difference between years?

The null hypothesis for ANOVA is that there is no difference between years.

General Linear Model: StackedTotals versus C4

Factor Type Levels Values

C4 fixed 3 2008, 2010, 2011

Analysis of Variance for StackedTotals, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

C4 2 6.6747 6.6747 3.3374 4.71 0.010

Error 203 143.8559 143.8559 0.7086

Total 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!

Command:

>Stat>ANOVA>General Linear Model

The p value indicates that there is a difference between the years. The Tukey printout tells us which years are different.

Grouping Information Using Tukey Method and 95.0% Confidence

C4 N Mean Grouping

2010 57 7.9 A

2008 86 7.7 A B

2011 63 7.4 B

Means that do not share a letter are significantly 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.com

This recommendation is thanks to Dr. DeVasher.

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

General Linear Model: stackedTotal versus StackedYear

Factor Type Levels Values

StackedYear fixed 4 2004, 2005, 2006, 2009

Analysis of Variance for stackedTotal, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

StackedYear 3 1165.33 1165.33 388.44 149.39 0.000 Look at low P-value!

Error 266 691.63 691.63 2.60

Total 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 R

209 40.0000 21.7917 0.1700 18.2083 11.36 R

215 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% Confidence

StackedYear N Mean Grouping

2004 60 23.5 A

2006 90 21.8 B

2005 60 20.7 C

2009 60 17.5 D

Means that do not share a letter are significantly different.

Tukey 95.0% Simultaneous Confidence Intervals

Response Variable stackedTotal

All Pairwise Comparisons among Levels of StackedYear

StackedYear = 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 0.0

Zero 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 0.0

- Is there a difference in production performance between the plain and peanut M&Ms?

Individual Quiz

Name:____________Section No:__________CM:_______

You will be given a bag of M&M’s. Do NOT eat the M&M’s.

Count the number of M&M’s 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&M’s. Note: You are not required to eat the M&M’s.