2007-06-17 tpc dd basicstatistics

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Tech-Pro Consultants

Six Sigma Basic Statistics

March 2005Dr. K.S.Ravichandran


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Bas ic Stat is t ics


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Objectives

Review & Enhance The Basic Statistical & Quality Terms Needed

For Six Sigma Process Improvement

Begin To Enhance Minitab Operating Skills

Politicians Promise: if elected, I'd make certain that everybody gets an above average income


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What is Statistics?

Is the science that develops methods to effectively derive

information from numerical data

Statistics is a collection of scientific methods for collecting,organizing and interpreting data, usually with the goal of inferring

certain properties of the population from a representative sample of

the population

science of collecting and classifying a group of facts according totheir relative number and determining certain values that represent

characteristics of the group

There are three kinds of Lies: Lie, Damned Lie and Statistics Mark Twain


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Types of data

Measures of the Center of the data

Mean

Median

Mode

Measures of the Spread of Data

Range

Variance

Standard Deviation

Normal Distribution and Normal Probabilities

Process Stability and Process Capability

Basic Statistics

Used With Permission

AlliedSignal 1995 -D r. Steve Zinkgraf

Ask a statistician for her phone number... and get an estimate with 95% confidence


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What sorts of data do you see beingcollected around your area?

(List them below)

___________________________________________________

______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

In God we trust. All others must bring data.


7/91Tech-Pro Consultants

wo enera n sof Data(but 3 families)

ATTRIBUTE DATA - The data is discrete (counted).Results from using go/no-go gages, or from the inspection ofvisual defects, visual problems, missing parts, or frompass/fail or yes/no decisions.

VARIABLE DATA - The data is continuous(measured). Results from the actual measuring of acharacteristic such as impedance of a motor winding, tensilestrength of steel, diameter of a pipe, flow rate of a pump, etc.

Statisticians do it discretely and continuously.



ATTRIBUTE DATA (Count Data)

(#1) Number of Items in a Category (Count-Based Proportions) Heads / Tails (i.e., counting # of Heads and # of Tails)

Yes / No (Order Form Filled Out Accurately or Not)

Pass / Fail; Good / Bad (Accurate Billing/Overcharged)

(#2) Counts of Discrete Event Occurrences

# of Scratches on a Car Hood # of Errors on a Form

# of Insulation Breaks in a Spool of Wire

# of times customer hangs up before receiving response

2 General Kinds of Data (but 3 families)

Different Types Of Data Require Different Analysis Tools

VARIABLE DATA (Continuous Measurement Scale) (#3) Continuous Data

Decimal subdivisions are meaningful

Ex: Time to answer the telephone ( Exact # of secs. per call)

Just ask

yourself,Am I

counting

things,

here?

If yes, you

haveattributes

data.

Type-IAttributes

Data

(Binomial)

Type-IIAttributes

Data

(Poisson)



VARIABLES

DATA

TYPE-I

Any Bubbles?(accept / reject

the entire item)

TYPE-II

Number ofBubbles?

Reject Reject Accept Reject

3 2 0 4ATTR

IBUTESDATA

Sample#1 Sample#2 Sample#3 Sample#4

3 Families of Data:

AmIC

ountingThings

?

(D

iscreteData)

(ContinuousD

ata)

(Measurement

Data)

Poisson

Distrib

ution

Binomial

Distribut

ion

NormalDistribu

tion

orOther

Manufacturing Process: Making Sheets of Glass

Weight = 12.2 Weight = 12.4 Weight = 12.1

Glass

Weight

Weight = 11.9



VARIABLES

DATA

TYPE-I

Any Errors?(accept / reject

the entire item)

TYPE-II

Number ofErrors on Form?

Reject Reject Accept Reject

3 2 0 4ATTR

IBUTESDATA

Form#1 Form#2 Form#3 Form#4

3 Families of Data:

AmI

CountingThings

?

(D

iscreteData)

(ContinuousD

ata)

(Measurement

Data)

Poisson

Distrib

ution

Binomial

Distribut

ion

NormalDistribution

orOther

Transactional Process: Converting an expense account forminto a reimbursement check

Time to

Reimburse

Employee36.1 hrs 24.6 hrs 21.0 hrs 29.2 hrs



Sample at 8:00am Sample at 9:00am Sample at 10:00am

Sample

(n)

Number

(np, c)

Proportion

(p,u)

Date

(Shift, Time, etc.)

30%

20%

10%

40%

8:00am

Pass/FailData

9:00am

She tells you are just Average: never mind, she is just being Mean



Sample

(n)

Number

(np, c)

Proportion

(p,u)

Date

(Shift, Time, etc.)

8:00am8:10am

3

2

1

4

Number of

Blemishes

Data8:00am

8:10am8:20am

8:50am

9:00am

9:10am

etc.

etc.8:30am

8:40am



Exercise: Which Type of Data Is It?

(1) Percent defective parts in hourly production

(2) Percent cream content in milk bottles (comes in four-bottle container sets)

(3) Amount of time it takes to respond to a request

(4) Number of blemishes per square yard of cloth, where pieces of cloth may be of variablesize

(5) Daily test of water acidity (pH)

(6) Number of raisins per box of Raisin Bran

(7) Number of defective parts in lots of size 100

(8) Length of screws in samples of size ten from production lots

(9) Number of errors on a purchase order

DIRECTIONS: For each of the following applications, identify the type of data you

would be investigating (Attributes Type-I, Attributes Type-II, or Variables Data)

... AND EXPLAIN YOUR CHOICE



What is the largest probability possible? _______What does this mean?

What is the smallest probability possible? _______What does this mean?

What does a probability of 0.50 mean? _______________

What is the probability you will be struck by lightning during yourlifetime? _____________________

What are your chances of appearing on The Tonight Show?___________________

What is the probability of being killed by terrorists overseas?____________________

What are your chances of being killed by an American in Baltimore?_______________

The Probability Test



What is the largest probability possible?___1.0 = 100%__What does this mean?

What is the smallest probability possible?___0.0 = 0%__What does this mean?

What does a probability of 0.50 mean? 50% Just flip a coin What is the probability you will be struck by lightning during your

lifetime? 0.000001667 = 1/600,000

What are your chances of appearing on The Tonight Show?0.00000204 = 1/490,000

What is the probability of being killed by terrorists overseas?0.000001538 = 1/650,000

What are your chances of being killed by an American in Baltimore?0.00025 = 1/4,000

The Probability Test

Instructor Page

Answers



Roll a fair die once, what is Prob(a six)? ______

Roll a fair die twice, what is Prob(a six on the second roll)?__

Roll two fair dice, what is Prob(get two sixes)?____________

What do you think of the recent headline, Education

research shows 49.5% of all American high school studentsfall below the national average!

The Probability Test (cont.)



The Customer Requirements

Suppose a certain customer permits only those

combinations which yield 3, 4, 5, . . . , or 11.

What is the process capability?

What is the probability of meeting the requirements?

Are capability and probability related?

Probability


6 Sigma Academy Inc. 1 995

The Practical Problem Statement ...



1 2 3 4 5 6

1

2

3

4

5

6

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

5 6 7 8 9 10

6 7 8 9 10 11

7 8 9 10 11 12

Computing the Risks- The Statistical Problem Statement

Ways to form a 2 in =

Ways to form a 12 in =

Probability of Defect



i i i



Deeper Insight Into Probability

Die 1 Die 2 Probability

1 4 .0278

2 3 .0278

3 2 .0278

4 1 .0278

Total .1111

What is the probability of

rolling a 5 using a fair pair

of dice?

1 2 3 4 5 6

1 .0278 .0278 .0278 .0278 .0278 .0278

2 .0278 .0278 .0278 .0278 .0278

3 .0278 .0278 .0278 .0278 .0278

4 .0278 .0278 .0278 .0278 .0278

5 .0278 .0278 .0278 .0278 .0278 .0278

6 .0278 .0278 .0278 .0278 .0278 .0278

.0278

.0278

.0278





Establishing the Odds

Value Combinations Probability

2 1 .0278

3 2 .0556

4 3 .0833

5 4 .1111

6 5 .1389

7 6 .1667

8 5 .1389

9 4 .1111

10 3 .0833

11 2 .0556

12 1 .0278Total 36 1.0000

Probability of any given value on Die 1 = 1/6 = .1667

Probability of any given value on Die 2 = 1/6 = .1667

Probability of any given combination = 1/6 x 1/6 = 1/36 = .0278 Used With Permission 6 Sigma Academy Inc. 1 995



Graphing the Results

. . .Hence, the probability of Customer Satisfaction is 94.4 %

Zone of Customer Satisfaction 94.4%

18

16

14

12

10

8

6

4

2

2 1210864 140Total of Dice Values

2.8%2.8% LSL USL

Suppose a certain customer permits only those

combinations which yield 3, 4, 5, . . . , or 11.

Value Combinations Probability

2 1 .0278

3 2 .0556

4 3 .0833

5 4 .1111

6 5 .13897 6 .1667

8 5 .1389

9 4 .1111

10 3 .0833

11 2 .0556

12 1 .0278

Total 36 1.0000





Statistical Distributions

We can describe the behavior of any process or

system by plotting multiple data points for the

same variable

Over time

Across products or business

By different people, machines, etc...

The accumulation of these data can be viewed as

a distribution of values

Represented by: Dot plots

Histograms

Normal curve or other smoothed distributionUsed With Permission

AlliedSignal 1995 - Dr. Steve Zinkgraf



Y = Weight (lbs) 220160 100

Process = Hose

1 Drop = 1 Unit of Output

Histogram is ...a pile of individual values

Dotplot: :

: :

. : . : :

: : : : : : : : : : : . :

. . ::.::::: :.:::.:.:.:.: : : : : . : . .

-----+---------+---------+---------+---------+---------+-C1

100 125 150 175 200 225

D t Pl t



Dot Plots

1st Observation2nd Observation

1.11.01.15

1.21.25

1.31.35

1.41.05

Suppose we have a manufacturing line that is producing shafts.

Diameters range from 1.0 to 1.4 inches. As we make a measurement of a

shaft, we record the value with a dot on the above scale

Ex:

1st Observation = 1.4 inches

2nd Observation = 1.1 inches

Diameter

D t Pl t



And Suppose we continue sampling until 150 shafts have been measured

What Statements Can You Make About Our Process ?

:: :::. . .

:.. :::::: : :

. :.. ..::::::::::::::: :: ::.:..: .::::::::::::::::::::::::.:::..:.: .

Dot Plots

1.11.01.15

1.21.25

1.31.35

1.41.05


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:: :::. . .

:.. :::::: : :

. :.. ..::::::::::::::: :: ::.:..: .::::::::::::::::::::::::.:::..:.: .

Dot Plots

1.11.01.15

1.21.25

1.31.35

1.41.05

Now imagine the same data, grouped into intervals

with bars used to represent how the data looks.


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Histogram Distribution

1.11.051.0

35

30

25

20

15

10

5

0

Frequency

1.15 1.2 1.25 1.3 1.35 1.4

Data represented just with the dots is called a Dot Plot

Using data represented in the above bar format is called a Histogram

Hi


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Histogram

Now weve combined the Histogram with our Lower and Upper Specifications.

Question #1 : What are Specifications ? Where do they come from ?

Question #2: What can you say about our process now ?

1.11.01.15

1.21.25

1.31.35

1.41.05

Upper SpecificationLower Specification

.001 2.0


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Histogram

Suppose the customer has given us new specifications !

Question: What can you say about our process now ?

1.11.01.15

1.21.25

1.31.35

1.41.05

Lower Specification

1.1

Upper Specification

1.3


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Dotplot Distribution

Imagine a customer service help line in which the business knows that to

stay competitive, it must return the customers telephone calls in less

than 30 minutes. The actual response time was measured 150 times and

plotted above.

: : :::. . .

:.. :::::: : :

. :.. ..::::::::::::::: :: ::.:..: .::::::::::::::::::::::::.:::..:.: .

-+---------+---------+---------+---------+-------

28.0 29.0 30.0 31.0 32.0


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3432302826

Upper SpecLower Spec

Time

Smoothed (Normal) Distribution

Finally, we can view the data as a smoothed distribution (red line), in this

example using the normal distribution assumption. It provides an

approximation of how the data might look if we were to collect an infinite

number of data pointsUsed With Permission



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Forming the Normal Curve

Uni ts of M easurem

Center of the bar

Smooth curve interconnecting

the center of each bar

Area of Yield

Performance

Limit

Probability

of a Defect

p(x > a) = 1 2

e-(1/2)[(x - m)/]2

a

dx

+ infinity- infinity

Given that 100% of the area

under the normal curve liesbetween , we may

calculate that area which lies

beyond the performance limit.

Doing so would reveal the

random chance probability of

creating a defect.

Note: The tails of the normal curve will touch the baseline at infinity. Used With Permission


a

Basic Statistics


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Types of data

Measures of the Center of the Data

Mean

Median

Mode


Range

Variance

Standard Deviation

Shape: Normal Distribution and Normal Probabilities


Basic Statistics



D t E l


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Data Example(Actual # of Days from Order to Ship)

140 170 215 130 136 130 150

145 175 150 155 123 120 110

160 175 145 150 155 130 116

190 170 155 148 140 131 108 155 180 155 155 120 120 95

165 135 150 150 130 118 125

150 170 155 140 138 125 133

190 157 150 180 121 135 110

195 130 180 190 125 125 150 138 185 160 145 116 118 108

160 190 135 150 145 122

155 155 160 164 150 115

153 170 140 112 102

145 155 142 125 115

Where is the Center of the Data?


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Mean = The average value(the Center of Gravity)

Where is the Center of the Data?

Decribed in 2 ways:

- Uses all data points- Heavily influenced byextreme values

X =Sum of the data points

Number of data points

Median = the 50% point,(or the middle number)

To find the median of a data set,

(1) arrange data in order fromsmallest to largest

(2) the middle number is the median!

1, 2, 3, 14, 85

The median is 3

- Not heavily influenced byextreme values

As head of the universitys Communications Dept. you are asked


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What is the average income

(or center of gravity)?

$10, 20, 30, 40, 50 ($ in thousands)

What is the median

income?

to summarize the average starting salaries of Communications

graduates.

$10, 20, 30, 40, 5000 ($ in thousands)

What is the average income

(or center of gravity)?

What is the median

income?

However, under the advice of the Public Relations Dept. you consider

to including one of your former Communications majors:

Shaquille ONeal (a rather wealthy rookie basketball star)

Where is the Center?


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Mode (not used as much): The value that occurs most often.

The Mode may not exist; and if does exist, it may

not be unique.

-Can be used with categorical/attribute data

Where is the Center?

What is the mode for the following set of defect data?

# of change notices issued:

-Price change: 13

-Spec change: 112

-Ship to address change: 40

-Delivery date changed: 79

What doesBimodal

mean?

rea ou


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rea ouExample

Suppose your son or daughter isconsidering going to work for a small, familyowned business after graduation. The

owner of the business proudly states that,of the last 7 college graduates hired, themean salary was $25,000; the salaries werebimodal, with modes of $18,000 and

$20,000; and the median salary was$19,000. He refuses to identify theindividual salaries

From Introductory Statistics William D. Ergle

Exercise


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Exercise

Minitab can easily calculate the Mean and Median

1. Open up Minitab

2. Open file: Distskew.mtw

3. Perform The Following

Stat>Basic Statistics>

Descriptive Statistics>

4. Enter The Variables Names

5. Evaluate Results

D i ti St ti ti F 3 Di t ib ti


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TABULAR FORM

Variable N Mean Median TrMean StDev

Normal 500 70.000 69.977 70.014 10.000

Pos Skew 500 70.000 65.695 68.554 10.000

Neg Skew 500 70.000 73.783 71.368 10.000

Descriptive Statistics For 3 Distributions

Look For This In Your Session Window !

Graphical Form


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Graphical Form

Different Distributions


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1101009080706050403020

100

50

0

C1

Frequency

Comparison of Distributions.

Sketch in the Means and Medians on each Distribution.

Negative Skew Positive Skew

Symmetric

Distribution

80706050403020100

300

200

100

0

C3

Frequency


Tail

13012011010090807060

300

200

100

0

C2

Frequency


Tail

Different Distributions



Graphical Reminder


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Graphical Reminder

* The 3 Charts On The Previous Page

Were Created Under The Minitab Histogram OptionGraph>Histogram

Relationship Of The Mean & Median


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1101009080706050403020

100

50

0

Normal

Frequency

Mean, Median

80706050403020100

300

200

100

0

Neg Skew

Frequency

MedianMean

13012011010090807060

300

200

100

0

Pos Skew

Frequency

Median Mean

Relationship Of The Mean & Median



Basic Statistics


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Types of data


Mean

Median

Mode


Range

Variance

Standard Deviation



Basic Statistics



Population Parameters vs Sample Statistics


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Population Parameters vs Sample Statistics

m = Population Mean = Population Standard Deviation

Examples of

POPULATION:

Entire United States

Yrs. Worth of Acct. Payable

Every Grain of Sand On The Beach

Examples of SAMPLE:

1000 US Citizens

Hrs. Worth of Acct.

Pay

Handful of Sand

^ = Sample Standard DeviationX= Sample Mean

s =

3 a s to describe ho far the


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Range = R the difference between largest

and smallest observations

Standard Deviation = s

Variance = s2 (just the square of the std dev!)

3 ways to describe how far the

data is spread:


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Avg = ___

Sum of thelast column

= _______

Divide theSum by (n-1):= Variance = S2

= __________



X5

4

3

1

2

X2

-1

X X4

1

X X2

Square Root ofthe Variance= Std.Dev. = S= _________

S S 2

Calculate manually the Variance and Standard

Deviation of These 5 Data Points

S2

CLASS EXERCISE


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Avg = 3

Sum of thelast column

= 10

Divide theSum by (n-1):= Variance = S2

= 2.5



X5

4

3

1

2

X2

1

0

-2

-1

X X4

1

0

4

1

X X2

Square Root ofthe Variance= Std.Dev. = S= 1.58

S S 2

Calculate manually the Variance and Standard

Deviation of These 5 Data Points

S2

CLASS EXERCISE

Instructor Page

Computational Equations


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Computational Equations

Population Mean

m =

X

N

i

i

N

1

Sample Mean

Population Standard

Deviation

m

=

(X )

N

i

2

i=1

N

Sample Standard

Deviation

x =

x

n

i

i=1

n

s =

(X )

n -1

i

2

i=1

N

X



The Standard Deviation


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The Standard Deviation

mPoint of Inflection

1

T USL

p(d)

Upper Specification Limit (USL)

Target Specification (T)

Lower Specification Limit (LSL)Mean of the distribution (m)Standard Deviation of the distribution () 3

The distance between the point of inflection and

the mean constitutes the size of a standard

deviation. If three such deviations can be fit

between the target value and the specification limit,

we would say the process has three sigma

capability.



Basic Statistics


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Types of data


Mean

Median

Mode


Range

Variance

Standard Deviation



Basic Statistics



The Normal Distribution


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The Normal Distribution is a distribution ofdata which has certain consistent properties

These properties are very useful in our

understanding of the characteristics of the

underlying process from which the data wereobtained

Most natural phenomena and man-made

processes are distributed normally, or can be

represented as normally distributed





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Property 1: A normal distribution can bedescribed completely by knowing only the:

mean, and

standard deviation


Distribution One

Distribution

Two

Distribution Three

What is the difference among these three normal distributions?



Statistical Number Line


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X Axis

m3 m2 m1 m m+1 m+2 m+3

300

Suppose the weights of players on a footballteam had m=300 lbs and =10 lbs

You fill in the X-axis values (weights) above

Exercise(pounds)

add 10 add 10 add 10



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X Axis

m3 m2 m1 m m+1 m+2 m+3

300 310 320 330270 280 290

Suppose the weights of a football teamhad m=300 lbs and =10 lbs

You fill in the X-axis values (weights)

Exercise

Instructor Page

(pounds)


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X Axism3 m2 m1 m m+1 m+2 m+3300 310 320 330270 280 290 (pounds)

68%

m + 1= 68%ofthe individuals

Instructor Page


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X Axism3 m2 m1 m m+1 m+2 m+3300 310 320 330270 280 290 (pounds)

95%

m + 2= 95%of the individuals

Instructor Page


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X Axism3 m2 m1 m m+1 m+2 m+3300 310 320 330270 280 290

m + 3= 99.7%of the individuals

(pounds)

99.7%

Instructor Page

The Normal Curve and Probability Areas


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Associated with the Standard Deviation

43210-1-2-3-4

40%

30%

20%

10%

0%

68%

95%

Probabilityofsampleva

lue

Number of standard deviations from the mean

99.73%

Property 2: The area under sections of the curve

can be used to estimate the cumulative probability

of a certain event occurring

Cumulative probability

of obtaining a valuebetween two values



Empirical Rule of Standard Deviation


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p

Number of

StandardDeviations

TheoreticalNormal

EmpiricalNormal

+/- 168% 60-75%

+/- 2 95% 90-98%

+/- 3 99.7% 99-100%

The previous rules of cumulative probability apply even when a set of data is

not perfectly normally distributed. Lets compare the values for a theoretical

(perfect) normal distributions to empirical (real-world) distributions




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How can I tell if my data is bell-shaped?(i.e., Normally Distributed)

Normal Probability Plots


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We can test whether a given data set can be described as normal

with a test called a Normal Probability Plot

If a distribution is close to normal, the normal probability plot will be astraight line.

Minitab makes the normal probability plot easy. Using Distskew.Mtw.Choose: Stat>Basic Stats>Normality Tests

Produce a normal plot of each of the first 3 columns. Which appear tobe normal?

3 Ways To See If Your Data Is NormallyDistributed


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Distributed

80706050403020100

300

200

100

0

C3

Frequency


13012011010090807060

300

200

100

0

C2

Frequency


1101009080706050403020

100

50

0

C1

Frequen

cy


1069686766656463626

.999

.99

.95

.80

.50

.20

.05

.01

.001

Probability

Normal

p-value: 0.328

A-Squared: 0.418

Anderson-Darling Normality Test

N of data: 500

Std Dev: 10

Average: 70

Normal Distribution

13012011010090807060

.999.99

.95

.80

.50

.20

.05

.01

.001

Probability

Pos Skew

p-value: 0.000

A-Squared: 46.447


N of data: 500

Std Dev: 10

Average: 70

Positive Skewed Distribution

80706050403020100

.999

.99

.95

.80

.50

.20

.05

.01

.001

Probability

Neg Skew

p-value: 0.000

A-Squared: 43.953


N of data: 500

Std Dev: 10

Average: 70

Negative Skewed Distribution


AlliedSignal 1995 -Dr. Steve Zinkgraf

If the NormalityTest shows a

P-value that is

lessthan 0.05,then the data is

NOT

represented

well by anormal

distribution

P Value for Normality Test


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y

If your P value is lessthat than .05, thenthe data is NOT approximately normal.

Mystery Distribution


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y y

Generate a Normal Probability Plot for the Mystery variable

in Mystery.mtw

What is your conclusion? Is this a normal distribution?

15010050

.999

.99

.95

.80

.50

.20

.05

.01

.001

Probability

Mystery

p-value: 0.000

A-Squared: 27.108

Anderson-Darling Normality Tes t

N of data: 500

Std Dev: 32.3849

Average: 100

Mystery Distribution



Central Limit Theorem


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The central limit theorem states that the distribution of the sample means, our estimate of m, can beapproximated with a normal distribution even though the original population may be non-normal.

Given this, we may say that the grand average (resulting from averaging sets of samples) approachesthe universe mean as the number of sample sets approaches infinity. This property is at the core ofmany statistical tests and is very important for resolving a wide array of industrial problems.

Random sample of g sets with n measurements assigned to each set

Various sampling distributions of individual measurements

XX


6 Sigma Academy Inc. 1995

For more detail, see

the next few pages.


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The Distribution of Averages

The Distribution of Individuals

VS

Important Distinctions:

What would the Distribution ofIndividuals look like?


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Individuals look like?

= Individual Measurement

= Average of the SubgroupFlashlight

Y = Lifetime(Hrs)96 85 74


? ?

The Distribution

of Individuals

What would the Distribution of

I di id l l k lik ?


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Individuals look like?


= Average of the SubgroupFlashlight



The Distribution

of Individuals

What would the Distribution ofAverages look like?


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Averages look like?


= Average of the Subgroup

Y = Weight (lbs)

10.5 10 9.5

The Distribution of Averages?

What would the Distribution of Averages look like?


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What would the Distribution of Averages look like?


= Average of the Subgroup

Y = Weight (lbs)

10.5 10 9.5

The Distribution of Averages

Distribution of Individuals Distribution of Averages


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X

Distribution of Individuals Distribution of Averages

A Pile of Individuals A Pile of X-Bars

Spread is...

X

X

n

Histogram is...

1 Individual 1 Avg (i.e., 1 X-Bar)1 point is ...

What is the probability that theaverage lifetime of an n=20 samplewill exceed 87 hours?

What is the probability thatan individual battery will lastbeyond 87 hours?

The questionmight be...

8574 96 8574 96

Compressed by n

Graphically...

SE(Mean)

Dist of Avgs spread compresses by factor of n


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97

95

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__n=20 n=50n=12n=4n=2n=1

Dist. of Avgs spread compresses by factor of n

X

X

n

Individ

uals

97

95

93

91

89

87

85

83

81

79

77

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Basic Statistics


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Types of data

Measures of the Center of the Data Mean

Median

Mode


Range

Variance

Standard Deviation





Basic Statistics


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Variability Is the process on target with minimum variability?

We use the mean to determine if process is on target.

We use the Standard Deviation determine variability Stability

How does the process perform over time?Represented by a constant mean and predictable variability over time.

Which process is the best process? Used With PermissionAlliedSignal 1995 - Dr. Steve Zinkgraf

2520151050

80

70

60

50

Sample Number

Sample

Mean

X-Bar Chart for Process B

X=70.98

UCL=77.27

LCL=64.70

2520151050

75

70

65

Sample Number

Sample

Mean

X-Bar Chart for Process A

X=70.91

UCL=77.20

LCL=64.62

Variation


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While every process displays Variation, some processes display

controlled variation, while other processes display uncontrolledvariation (Walter Shewhart).

. Controlled Variation is characterized by a stable and consistentpattern of variation over time. Associated with Common Causes.

Uncontrolled Variation is characterized by variation that changesover time. Associated with Special Causes.

Process A shows controlled variation.

Process B shows uncontrolled variation

Special Causes

2520151050

75

70

65

SampleNumber

SampleMean

X-Bar Chart for Process A

X=70.91

UCL=77.20

LCL=64.62

2520151050

80

70

60

50

SampleNumber

Sample

Mean

X-Bar Chart for Process B

X=70.98

UCL=77.27

LCL=64.70



Can We Tolerate Variability ?


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There will always be variability present in any process We can tolerate variability if

The total variability of the Output is relatively small compared to theprocess specifications and the process is on target

The process is stable over time

LSL USLNom USL

LSL USLNom

Acceptable

Cost

Cost

OLD

New

Traditional

Goal Post

Mentality



Expanding On The Goal Post Mentality


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LSL USLNom

UNDER THE OLD RULES,

The field goal kicker gets 3 points for his team as long as

the ball falls between the LSL and USL.

3 Points

Expanding On The Goal Post Mentality


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LSL USLNom

UNDER THE NEW RULES,

The Field Goal Kicker Might Get...3 points Target & +/-12 points Between +/-1 & +/-21 point > +/-2 Out To The LSL & USL

321 2 1Points

Data Analysis Tasks For Improvement


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Determine If Process Is Stable

If process is no tstable, identify and remove causes of

instability

Determine The Location Of The Process Mean.

Is It On Target?

If not, identify the variables which affect the mean and

determine optimal settings to achieve target value

Estimate The Magnitude Of The Total Variability. Is

i t acceptable with respect to the c ustom er requirements (spec l imi ts)? If not, identify the sources of the variability and eliminate or

reduce their influence on the process



Visualizing the Process Dynamics - Is TheProcess Stable ?


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Inherent Capability of the

Process

General Assumptions::

Over time, a typical process

will shift and drift by approx. 1.5

. . . also called short-term capability

Time 1

Time 2

Time 3

Time 4

TLSL USL

Sustained Capability of theProcess . . . also called long-term capability



The Goal Is ...


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Variables Data

0% Rejected

Target

Attributes Data

How We Progress Toward The Goal


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PHASE ONE - Unpredictable Performance

- VARIATION (SPECIAL / NATURAL CAUSES)

- UNPREDICTABLE (HOURLY, DAILY)

- DETECT AND ELIMINATE SPECIAL CAUSES

PHASE TWO - Stability

- IN CONTROL

- NATURAL VARIATION ONLY

Not capable of getting

all the water output into

the clowns mouth?

How We Progress Toward The Goal UpperSpecificationLower Specification


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IN CONTROL, BUT NOT CAPABLE

(Variation from common causes excessive)

IN CONTROL AND CAPABLE

(Variation from common causes reduced)

SIZE

LOWERSPECIFICATION

LIMIT UPPER

SPECIFICATION

LIMIT

Now it is capable of

getting all the water output

into the clowns mouth

1.11.01.15

1.21.25

1.31.35

1.41.05.001 2.0

Is The Process on Target ? - Accurate ?


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USL

Part

T

LSL

Recognize that the process center (m) is

independent of the design center (T). In

other words, the ability of a process to

repeat any given centering condition is

independent of the design specifications.

1.233 1.235 1.239 1.241 1.243 1.245 1.2471.237

m ManufacturingDistribution of the Widget

Part

54321

Increase in nonconformance due

to shift in process centering



Is The Process on Target ? - Precise?


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1.235 1.237 1.239 1.241 1.243 1.245 1.247

USL

Part

T

LSL

Recognize that the process width is

independent of the design width. In

other words, the inherent precision of

a process is not determined by the

design specifications.

Manufacturing Distribution

of the Widget Part



Is The Variability AcceptableTo Customer Requirements ?


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USL Y = f (X1 . . . XN)

The variation inherent to any dependent variable (Y) is determined by

the variations inherent to each of the independent variables.

LSL

Poor Process

Capability

LSL USL

Very High

Probabilityof Defects

Very High


LSL USL

ExcellentProcess

Capability

Very Low


Very Low




Summary


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Reviewed & Enhanced The Basic Statistical & Quality TermsNeeded For Six Sigma Process Improvement

Began to Build Up Minitab Operating Skills


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Six Sigma

Q&A


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Six Sigma

Thank You

2007-06-17 tpc dd basicstatistics

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