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Six Sigma Black Belt

Cert. Prep. Course:MeasureMeasureMeasureMeasureModule V

©2009 ASQ 2

Agenda

This module consists of six lessons:

1. Process characteristics

2. Data collection3. Measurement systems4. Basic statistics5. Probability

6. Process capability

©2009 ASQ 3

Lesson 1 – Process Characteristics

Identify these process variables and evaluate theirrelationships using SIPOC and other tools. (Evaluate)

V.A.1 Input and Output Variables

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©2009 ASQ 4

Input and Output Variables

• What is a Process? It is … – A group of activities which,

together, achieve a specificoutput

– Inputs which aretransformed into outputs.

– A common definition of the

Suppliers, Inputs, Process,Outputs and Customers forthe problem you are trying

to solve

©2009 ASQ 5

Process MappingProcess - Level #1

Step #1

Process -Level #2

Step #1 Step #2

Process - Level #3

Step #3

Step #1 Step #2 Step #3

Step #3

Step #2

•Processes are madeup of sub-processes

•A sub-process in turncan be made up ofanother sub-process

and so on.


©2009 ASQ 6

• A SIPOC diagram defines the boundary of the process


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©2009 ASQ 7

• Typical inputs for a manufacturing process are:

– Man (People) – Machine (Equipment)

– Material (Information / Forms, etc.)

– Method (Procedures)

– Measurement (Data Collection)

– Mother Nature (Environment)

• These are also known as the 6M’s

• Step output or process output

– Product

– Information

– Service

• Customer for output can be either internal or external


©2009 ASQ 9

SIPOC – Example


©2009 ASQ 10

Progress Check

Assume you are a landscape contractor.For the process “mow a customer’s lawn,”

construct a SIPOC diagram. Show all:

• Suppliers

• Inputs

• High-level process steps (process map)

• Outputs

• Customers

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©2009 ASQ 11

Lesson 1-Process Characteristics

Evaluate process flow and utilization to identify wasteand constraints by analyzing work in progress (WIP),work in queue (WIQ), touch time, takt time, cycle time,

throughput, etc. (Evaluate)

V.A.2 Process Flow Metrics

©2009 ASQ 12

Process Flow Metrics

Evaluation of Process Flow

• In Lean, after the “current state” is documented with a ValueStream Map, evaluate the process flow to identify waste and

constraints

• Evaluation should ask these questions:

– Which activities did not add value?

– Which activities can be eliminated?

– Can activities be made simpler?

– Can the next event start before the current one is finished?

– Are there activities that can start sooner, or can they startin parallel?

– Are there similar activities in the process duplicated by morethan one person?

– What are the metrics (such as WIP, WIQ, takt time, etc.)telling us about the process?

©2009 ASQ 13


• Some common metrics used to evaluate process:

– WIP – Work in Progress

– WIQ – Work in Queue

– Touch time

– TAKT time

– Cycle time

– Throughput

• Work in Progress – items that have entered process but notexited

• Work in Queue

– Items that are waiting to be processed by a step in the process

– WIQ is one component of WIP

• Touch Time – actual processing time for the item in a given stepof the process

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©2009 ASQ 14

• Takt time – available work time / number of items to be

processed• Cycle Time – average time for one item to get through a

particular step

• Throughput – number of items output from a process for agiven period of time

• Value-added Time – amount of time used for value-addedactivities

• Value-added Activity

– Physically transforms the item

– Is something for which the customer is willing to pay

– Does the right thing right the first time (DTRTRTFT)

• Set-up Time or Change-over Time – time required to convertfrom producing one product to producing a new and differentproduct


©2009 ASQ 15

Progress Check

• The admissions group at a university requires that25 applications be processed against an admission

criteria in 240 minutes. What is the Takt Time?

• If the cycle time for the above admissions processis 35 minutes, then how many people should be

working in the process?

©2009 ASQ 17

Analyze processes by developing and using value

stream maps, process maps, flowcharts, procedures,work instructions, spaghetti diagrams, circlediagrams, etc. (Analyze)

V.A.3 Process Analysis Tools

Lesson 1 – Process Characteristics

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©2009 ASQ 21

Wait for AvailableSales Person

Initial PhoneContact

C/T = 0W/T = 0VA/T = 0

C/T = 5 minutesW/T = 0VA/T = 0

Sales Pitch

C/T = 10 minutesW/T = 10 minutesVA/T = 10 minutes

Configure System


Fill OutOrder Form


Promise to Ship

C/T = 5 minutesW/T = 5 minutesVA/T = 0

Pending Order“FIFO” Queue

C/T = 7 DaysW/T = 0VA/T = 0

Batch TogetherSimilar Systems

C/T = 6 DaysW/T = 1 DayVA/T = 0

Check Availabilityof Materials

C/T = 3 DaysW/T = 1 hourVA/T = 0

Issue Work Orderto Factory Floor

C/T = 1 DayW/T = 1 hourVA/T = 0

Mtl.Available

?

Yes

No

Change Ship Date

Time Customer is On Telephone

TriggeringEvent

MeasurableDeliverable

While customeris on telephone:

C/T = 60 min.W/T = 55 min.

VA/T = 20 min.

From Contactto Order Launch:

C/T = 17 daysW/T = ~ 1 day

VA/T = 0

Sales Order Processing ValueStream Map – Example C/T = Calendar Time

W/T = Work TimeVA/T = Value-Add Time

©2009 ASQ 22

Sales Order Process FutureState – Example

Time Customer is On Telephone

Wait for AvailableSales Person

Initial PhoneContact

C/T = 0W/T = 0VA/T = 0

C/T = 5 minutesW/T = 0VA/T = 0

Sales Pitch


Configure System


Fill Out Order Form(& Config. System)


Promise to Ship

C/T = 5 minutesW/T = 5 minutesVA/T = 0

Pending Order“FIFO”Queue

C/T = 7 DaysW/T = 0VA/T = 0

Optimize ProductMix

C/T = 6 DaysW/T = 1 DayVA/T = 0

Check Availabilityof Materials

C/T = 3 DaysW/T = 1 hourVA/T = 0

Issue Work Orderto Factory Floor

C/T = 1 DayW/T = 1 hourVA/T = 0

Mtl.Available

?

Yes

NoReconfigure?

While customeris on telephone: 45 min.

45 min.20 min.

From Contactto Order Launch: 7 days

~ 1 day0

Improvement Targets -

C/T = 60 min.W/T = 55 min.VA/T = 20 min.

C/T = 17 daysW/T = ~ 1 dayVA/T = 0

As-Is Target As-Is Target

©2009 ASQ 23

Process Analysis Tools

Process Maps

• Graphically illustrate current process at a high level

• Document all input and output variables

• Identify

– Gaps in current process

– Non-value-added steps

• Use team approach for construction (processowners and stakeholders may help)

Process Maps and Flow Charts

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©2009 ASQ 25

Process Map Construction

Outputto the

Customer, Y

Step 1 Step 2SupplierInputs,

X’s

y1 y2 y3

x1 x2 x3

x1 x2 x3

y1 y2 y3

Product/serviceparameters

InputsOutputs y = f(x)

Process Step

Product/ServiceParameters

y1 y2 y3- - -

Process Parameters x1 x2 X3- - -

C = ControllableCr = CriticalN = NoiseW = Work Inst

Remember the 6 M’s

Man (People)Machine (Equipment)

Method (Procedures)

MaterialMeasurement

Mother Nature (Environment)Five Why’s! Why do we havethis defect. Why . . .


©2009 ASQ 26

Process Map Example

Deposit

Slip, $, & envelope

Document

depositslip

Put slip

& cash intoenvelope

Deposit

readyfor teller

Making a cash deposit in a bank

• Wrong

accountnumber

• Wrong cashamount

• Wrong cash

amount• No cash

• No slip

• Copy error

• Incorrect count

• Incorrect count

• Cash omission• Form omission

X

Y: envelope

with error-free slipand correct cash


©2009 ASQ 27

Flow Chart

• Used to understand the details of the process,

including decision points, rework loops, etc.

• Can be used to create procedures or workinstructions

• Can be hierarchical (detailed maps can haveseveral levels; either a procedural level or work

instruction level)

• Usually developed by walking the process andactually observing what happens (not what people

think happens)


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©2009 ASQ 28

Bracket

Material

Handling

Gusset

Assembly

Bracket

Generator

Op #1

Bracket Cut

Op #2

BracketPunch

Bracket Cut

Move

Op #3

Bracket Bendt

BracketPunch

Move

Bracekt Bend

Move

Gusset

Generator

Op #4

Gusset Cut

Op#5

Gusset Bend

Gusset Cut

Move

GussetBend

Move

Op #6

Assemble

HuskyBracket

Flow Chart Example: Bracket Assembly


©2009 ASQ 31


• Procedures describe work carried out in a process and arecross-departmental

• Work Instructions describe details of activities and are task-

specific

• Both are developed with personnel involved in doing the work

• Advantages

–Capture best practices

–Basis for continuous improvement

–Help reduce variation in processes

Example: The IRS has instructions on how to fill out varioustax forms.

Procedures and Work Instructions

©2009 ASQ 32

• Used to analyze the work flow in a process

• Goal: layout must be conducive to process flow,

so that walking time and travel distances are

minimized.

Spaghetti Diagram


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©2009 ASQ 33

Spaghetti Diagram – CurrentLayout

©2009 ASQ 35


Circle Diagram (Hand-off Map)

• Used to show linkages between various items

• Used to identify predecessor and successorrelationships

• Used to identify bottlenecks: too many inputs

or outputs for a given descriptor around thecircumference can indicate a bottleneck

See CSSBB HB, pages 88 and 89 for an exampleof a Circle Diagram.

©2009 ASQ 36

Progress Check

• Which of the following is not a reason for using

process maps?

a. Supports the identification of disconnectsb. Helps the team better understand the processc. Enables the discovery of problems or

miscommunications

d. To eliminate the planning processe. Helps define the boundaries of the process

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©2009 ASQ 38

Progress Check

• A Spaghetti Diagram is a tool used to evaluate

what type of waste or non-value-added activities?

– Overproduction – Excess Inventory

– Transportation – Repair / Reject

©2009 ASQ 39

Progress Check

A value stream map does not depict

• Material and information flows• The “door to door” flow• Bottleneck activities• Standard times

©2009 ASQ 40

Lesson 2 – Data Collection

Define, classify, and evaluate qualitative and

quantitative data, continuous (variables) and discrete(attributes) data, and convert attributes data tovariables measures when appropriate. (Evaluate)

V.B.1 Types of Data

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©2009 ASQ 41

Types of Data• Qualitative and Quantitative Data

– Continuous or variable data is measurable

(Examples - length, volume, time) – Discrete data or attribute data is countable

or classifiable (Examples – number of defectsor scrap items; go / no-go)

©2009 ASQ 42

Types of Data – Examples

•Continuous Data or Variables Data (most commonlyNormally or Gaussian distributed)

• Height

• Weight• Length• Diameter

•Discrete Data or Attribute Data

• Count data (Poisson distributed)• Number of defects on a sheet of paper• Number of flaws on a bolt of cloth

• Classification data (Binomially distributed)• Go/no-go• Non-defective/defective

©2009 ASQ 43

Progress Check

Classify examples in the table below as Continuous

or Discrete data.

Data example Continuous Discrete

1.7 inches

10 scratches

6 rejected parts

10.542 seconds

25 paint runs

32.06 psi

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©2009 ASQ 45

Define and apply nominal, ordinal, interval, and ratiomeasurement scales. (Apply)


V.B.2 Measurement Scales

©2009 ASQ 46

Measurement Scales

©2009 ASQ 47

Progress Check

What type of measurement scale is appropriatefor each example?

Example Measurement scale

A car weighs 3500 lbs

800 people failed an exam

Defects are either critical, major a,major b, and minor

The shipping codes used for lastweek’s orders

The weights of a sample of parts

The temperature of parts after1 hour of cooling

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©2009 ASQ 49

Define and apply the concepts related to sampling(e.g., representative selection, homogeneity, bias,etc.). Select and use appropriate sampling methods(e.g., random sampling, stratified sampling,systematic sampling, etc.) that ensure the integrity

of data. (Evaluate)


V.B.3 Sampling Methods

©2009 ASQ 50

Sampling Methods

PopulationPopulation

Sampling must be: • Random

• Free from bias

• Adequate size

Sampling must be: Sampling must be:

•• RandomRandom

•• Free from biasFree from bias

•• Adequate sizeAdequate size

SampleSample

SampleSample

SampleSample

SampleSample

SampleSample

SampleSample

SampleSampleSampling Concepts

©2009 ASQ 51

• Simple random sample – Ensure all possible samples are equally likely

to be chosen• Example: A voter survey that excludes an

ethnic group is not representative of thepopulation and is biased

• Stratified – Divide population into homogenous groups (strata)

and draw random sample within each group• Example: The population of all hospitals

enrolled in a pharmaceutical research studycan be stratified by therapeutic area and,within each stratum, you can select a randomsample of hospitals to audit

Sampling Methods

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©2009 ASQ 52

Sampling Methods

• Systematic

– Sampling starts with a randomly chosen unit and selectsevery “K‘th” unit thereafter

• Example: A supermarket study on the habits of theircustomers might survey every 20th customer at thecheckout line

• Sampling Errors

– Uncertainty about the timing, methods, and identity ofperson(s) responsible for collecting and reviewing data

– Inadequate description of data collection instruments tobe used

– Failure to identify specific content and strategies fortraining or retraining staff members responsible fordata collection

– Uncalibrated data collection equipment

©2009 ASQ 53

Progress CheckIdentify the type of sampling method used in each example.Is it Random, Stratified, or Systematic?

Example

A customer service call center receives a randomly generatedlist of customers who have interacted with customer servicerepresentatives. The manager selects every fifth name on thelist to conduct a follow-up call for quality assurance

Customer survey results are divided into multiple strata basedon gender and income level.

Random number generator is used by a fast-food chain to print

a survey number on a receipt for customers to use when theycall and answer questions about their experience.

A manufacturer divides data on defects into strata basedon manufacturing location and equipment type used forproduction.

©2009 ASQ 55

Develop data collection plans, including considerationof how the data will be collected (e.g., check sheets,

data coding techniques, automated data collection,etc.) and how it will be used. (Apply)


V.B.4 Collecting Data

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©2009 ASQ 56

Collecting DataA check sheet is a structured, prepared form forcollecting and analyzing data.

Example: check sheet for telephone interruptions

©2009 ASQ 57

Collecting Data

Coded Data: Data coding can be used to improveefficiency and quality of data entry.

Example:

• Measurements such as 1.0003, 1.0002, 1.0009in which the digits 1.000 repeat in all observationscan be recorded as the last digit expressed as an

integer (e.g., 3, 2, and 9).

• Code: Recorded Code = Observation x (10,000)

• Decode: Observation = Recorded Code / (10,000)

©2009 ASQ 58

What tomeasure

TypeofData

OperationalDefinition

Collectingandrecording

Sampling Plan

What How Procedureand Dataform

What Where When Howmany

Name the feature / characteristic of product or service being measured

Variable or Discrete

Specific definition of what Is to be measured

Definition of how it will be measured

Method for data collection,data collection form

Data to be collected

Physical location for data

Timing and frequency of data collection

Number of data points to be collected

Collecting Data

Developing a Data Collection Plan – Example

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©2009 ASQ 59

Progress Check

Identify the data collection technique (Checklist or

Coding) used in each example.

Example

1. A school district wants to ensure that accurate and

relevant information concerning education is madeavailable to the state. High-school completion rates

of the students aged 19 to 20, by gender, from 2000to 2008 is being collected.

2. Runout of a shaft being ground is measured. Thetolerance is + / - 0.0005. The operator documentsonly the last two digits in the inspection report.

©2009 ASQ 61

Lesson 3 – Measurement Systems

Define and describe measurement methods for bothcontinuous and discrete data. (Understand)

V.C.1 Measurement Methods

©2009 ASQ 62

Measurement Methods

• The different types of data was covered underLesson 2 – Data Collection

– Continuous or variable data is measurable

(Examples: length, volume, time)

– Discrete data or attribute data is countable or

classifiable (Examples: number of defects or

scrap items or go/no-go)

– Different tools and measurement methods are

used to obtain data

• gages, calipers, micrometers, coordinatemeasuring machines (CMMs), etc.

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©2009 ASQ 63

Measurement Methods

• Instruments can be used to measure:

– Continuous or variable data

– Discrete or attribute data

• Instruments for measuring continuous or variable data:

– Caliper, micrometer, height gage, dial indicator,drop indicator, CMM, etc.

• Instruments for measuring discrete or attribute data:

– Go / No-Go gages (Plug gages, Thread gages,Bore gages, Ring gages, etc.)

– In transactional processes, people can make thedetermination to accept / reject based on criteria

©2009 ASQ 64

• Measurement tools can be grouped into fivecategories:

– Mechanical – amplifies small movements

– Pneumatic – air pressure used to detect

dimensional variation

– Electronic – changes in resistance, capacitance,or inductance are converted to dimensional

changes – Light Technologies – uses wave interference

– Electron Systems – electron beam microscopeused to make measurements

Measurement Methods

©2009 ASQ 65

Progress Check

1. Which of the following statements are true for anAttribute Gage used for measuring a part?

a. They give you the exact measurement

b. They are usually designed to check a single

dimension or tolerance limit

c. There are usually two members: Go / No-Go

d. They do not tell you how good or how bad theproduct is.

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©2009 ASQ 66

Progress Check

2. Which of the following is not a gage used to collect

attribute data?

a. Ring gage

b. Height gage

c. Plug gage

d. Thread gage

©2009 ASQ 68

Use various analytical methods (e.g., repeatability

and reproducibility (R&R), correlation, bias, linearity,precision to tolerance, percent agreement, etc.)

to analyze and interpret measurement systemcapability for variables and attributes measurementsystems. (Evaluate)


V.C.2 Measurement System Analysis

©2009 ASQ 69

Measurement Systems AnalysisGage R&R Studies: Repeatability and reproducibility (R&R)studies are a method for determining the variation of ameasurement system.

Three methods typically used are:

• The range method quantifies both repeatability andreproducibility together.• The average and range method determines the totalvariability, and allows repeatability and reproducibility to beseparated.

• The analysis of variance method (ANOVA) is the mostaccurate of the three methods. In addition to determiningrepeatability and reproducibility, ANOVA also looks atthe interaction between those involved in looking at the

measurement method and the attributes/parts themselves.

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©2009 ASQ 70

Measurement correlation is typically the

relationship between a measuring instrumentand its standard, since the measurement of bothinvolves variation.

Measurement Systems Analysis

©2009 ASQ 71

Accuracy – Bias: Bias is the difference betweenthe output of the measurement method and the

true value.

The equation

for bias is:

Where:• n = the number of times the standard is measured

• Xi = the ith measurement

• T = the value of the standard


©2009 ASQ 72

Accuracy – Linearity: Linearity measures the bias

across the operating range of a tool or instrument.

Example


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©2009 ASQ 73

• Accuracy – Stability: Stability indicates the total

variation in accuracy readings over time on agiven part.

• Gauge Stability


©2009 ASQ 74

Terminology

Total variation in the measurement system

Measure of variation of repeated measurements

Repeatablenot reproducible

Not repeatable andnot reproducible

Repeatable andreproducible

Precision = σσσσ2 Repeatability + σσσσ2 Reproducibility


©2009 ASQ 75


• Repeatability and Reproducibility represent two aspects ofprecision and help describe the variability of a measurementmethod:

– Repeatability – (a.k.a., “equipment variation”) is thevariation in measurements obtained when one operatoruses the same gauge for measuring the identicalcharacteristics of the same parts.

Precision – Repeatability

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©2009 ASQ 76


– Reproducibility – (a.k.a., “appraiser variation”) is the

variation in the measurements made by differentoperators using the same gage while measuringthe identical characteristic on the same parts.

Precision – Reproducibility

©2009 ASQ 77

The number of significant digits that can be measured bythe system. Increments should be about 0.1 of the productspecification or the process variation.

Poor discrimination

Good discrimination


Terminology Precision – Discrimination

Example: If a recipe calls for adding a certain ingredient with atolerance of + / - one gram, we will want to use a weighing scale

capable of measuring in milligrams.

©2009 ASQ 78


Precision/Tolerance (P/T) is the ratio between the

estimated measurement error (precision) and thetolerance of the characteristic being measured,where 6σE is the standard deviation due tomeasurement variability.

P/T Ratio is the measure of the capability of themeasurement system

Precision – Precision to Tolerance Ratio

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©2009 ASQ 79

% of Measurement Errorto Total Tolerance

Acceptability

Total measurement error

of less than 10% of totaltolerance

Acceptable measuring equipment.

Total measurement errorof 10% to 30% of total

tolerance

Possibly acceptable based on theimportance of the application, cost of the

measuring equipment, cost of repairs, etc.

Total measurement errorof more than 30% of totaltolerance

Generally unacceptable; every effortshould be made to identify and correct theproblem. Customers should be involved

in determining how the problem will beresolved.

Interpreting measurement system capability for variables


©2009 ASQ 80

Percent agreement refers to the percent of time in an

attribute Gage R&R study the appraisers agree with: – Themselves (Repeatability) – Other appraisers (Reproducibility)

– A known standard (Bias against expert)

Rating can be done on Nominal or Ordinal scales

Precision – Percent Agreement


©2009 ASQ 81

• There are Statistical programs that will run the dataand provide a graph

– If Kappa value = 1, then everyone agrees

– Kappa = .9 is excellent

– Kappa value < or = .7; means that operatorsneed to be retrained

• If, after retraining, the Kappa value does not showimprovement, then develop a new training method

Interpreting measurement system capability forattributes


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©2009 ASQ 82

Progress Check1. Precision is best described as:

a. a comparison to a known standardb. the achievement of expected ongoing quality

c. the repeated consistency of resultsd. the difference between an average measurement and actual value

2. The overall ability of two or more operators to obtain consistent resultsrepeatedly when measuring the same set of parts and using the same

measuring equipment is the definition of:a. Repeatabilitya. Precision

b. Reproducibilityc. Accuracy

3. In measurement system analysis, which of the following pairs of datameasures is used to determine total variance?

a. Process variance and reproducibility

b. Noise system and repeatabilityc. Measurement variance and process variance

d. System variance and bias

©2009 ASQ 83

Progress Check

4. A calibrated micrometer was used to take 10 replicated measuresof a reference standard. If the mean of the 10 measurements is

0.073, and the true value of the reference standard is 0.075, whatis the bias of the micrometer?

a. 0.001b. 0.002c. 0.073

d. 0.075

5. Repeatability and reproducibility are terms that operationally

definea. biasb. Accuracy

c. Discriminationd. Precision

©2009 ASQ 84

Progress Check

6. The extent to which an instrument replicates its results when

measurements are taken repeatedly on the same unit is called:a. Real biasb. Precisionc. Accuracyd. True value

7. A measurement system analysis is designed to assess thestatistical properties of:a. Gage variationb. Process performancec. Process stability

d. Engineering tolerance

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©2009 ASQ 85

Identify how measurement systems can be appliedin marketing, sales, engineering, research anddevelopment (R&D), supply chain management,

customer satisfaction, and other functional areas.(Understand)


V.C.3 Measurement Systems in the Enterprise

©2009 ASQ 86

Measurement Systems inthe Enterprise• Human Resources

– Performance Approvals / Appraisals / Employee Surveysetc., are conducted by HR

– Employees and Managers need to be familiar with theterms so as a bias may not be formed

• Marketing and Sales

– Customer satisfaction / dissatisfaction data is collectedthrough surveys and other means

– Operational definitions for the data to be collected must

be clear to all parties• Quality Engineering

– Responsible for calibration, Gage R&R, etc.

• Supply Chain

– Evaluates supplier performance and issues supplier reportcards

©2009 ASQ 87

Progress Check

What are some measurement systems that can be

applied in:

• Engineering

• Research and development (R&D)

• Doctor’s office

• Ethics and Compliance

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©2009 ASQ 88

Define and describe elements of metrology, includingcalibration systems, traceability to reference

standards, the control and integrity of standardsand measurement devices, etc. (Understand)


V.C.4 Metrology

©2009 ASQ 89

Metrology

• Simply put, metrology is the science of measurement.Metrology encompasses certain key elements:

– The establishment of measurement standards

that are precise and defined.

– The use of measuring equipment to assessvariability.

– Regular calibration of equipment.

©2009 ASQ 90

• Causes of measurement error can be categorizedunder the 6Ms.

6 Ms example of errors:

• Man (People): Lack of training

• Machine (Equipment): Lack of precision, lack of accuracy

• Method (Procedures): Incorrect tool used / specified

• Material (Information): Instability over time

• Measurement (Data being collected): Incorrect characteristicbeing measured

• Mother Nature (Environment): Vibration, temperature,humidity, etc.

Metrology

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©2009 ASQ 91

• Calibration

– “The set of operations that establish, underspecific conditions, the relationship between

values indicated by the measuring instrumentor system, or values represented by a materialmeasure or reference material and thecorresponding values of a quantity realized

by a reference standard.”

Metrology

©2009 ASQ 92

• In many countries, a National Metrology Institute(NMI) will maintain primary standards ofmeasurement, which will be used to providetraceability to customer’s instruments by calibration.

• Traceability is established in an unbroken chain,from the top level of standards to an instrumentused for measurement.

• In the United States, traceability to standardsis tracked through the NIST # provided by TheNational Institute of Standards and Technology

(www.nist.gov).

Metrology

©2009 ASQ 93

• Calibration system schedules

– Calibration can be called for:

• with a new instrument

• when a specified time period is elapsed

• when a specified usage (operating hours)has elapsed

• when an instrument has had a shock or

vibration which potentially may have put itout of calibration

• whenever observations appear questionable

Metrology

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©2009 ASQ 95

Select one:

Calibration intervals can be adjusted when:a. Defective product is found

b. Some instruments are scrapped whencalibrated

c. A particular characteristic on the instrument isconsistently found to not be within tolerance

d. A new employee is issued a measuringequipment

Progress Check

©2009 ASQ 96

Lesson 4 – Basic Statistics

Define and distinguish between populationparameters and sample statistics (e.g., proportion,

mean, standard deviation, etc.) (Apply)

V.D.1 Basic Terms

©2009 ASQ 97

Basic TermsBasic TermsBasic TermsBasic Terms

• Statistics has two areas of interest1. Descriptive2. Inferential

• Descriptive Statistics (and Parameters)

o Use well-defined mathematical quantifies to describe

sample statistics and population parameterso Provide a common language to communicate concepts

like location and spread (central tendency and dispersion)

• Inferential Statisticso Use sample data to make inferences or predictions about

population parameterso Statistical inferences are made with defined confidence

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©2009 ASQ 98

Population • A population is an entire group of objects that

have been made, or will be made, containing a

characteristic of interest• A population parameter is a quantity that describes

some characteristic of a population.

o Examples of a population parameters are the

population mean and population standard

deviationo Population parameters are represented by

Greek letters

Basic Terms (continued)Basic Terms (continued)Basic Terms (continued)Basic Terms (continued)

©2009 ASQ 99

Sample • A sample is a part or fraction of a population

(generally drawn at random)

• Samples are frequently used because data onevery member of a population is often impossible

or too costly to collect

• A sample statistic is a quantity that describes somecharacteristic of a sampleo Examples of a sample statistics are the sample

mean and sample standard deviation

o Sample statistics are represented by Romanletters

Basic Terms (continued)Basic Terms (continued)Basic Terms (continued)Basic Terms (continued)

©2009 ASQ 100

Basic TermsBasic TermsBasic TermsBasic Terms

Definitions

Statistics infer information about the parameters of the population.

Populat ion SamplesSize N nLocation Average (Mean) µ x

Dispersion: Variation

Variance σ2 s2

Std dev σ sRange R = max - min

Note: Greek letters are used for population parameters; and English letters or Greek letterswith 'hat' are used for sample statistics. Sample statistics estimate population parameters.

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©2009 ASQ 101

BasicBasicBasicBasic Terms

Basic properties of Data Distributions:

• Number of observations – N

• Central tendency – measures of where the center,or most typical value, of the data set lies (Mean,Median and Mode)

• Dispersion – measures of the amount of variation

or spread in the data set (Range, Variance andStandard Deviation)

©2009 ASQ 102

Basic Terms

• Mean is a measure of where the center of the

distribution lies. It is the sum of all observationsdivided by the number of observations.

• Median (50th percentile) is the middle observation in

the data set. It is determined by ranking the data and

finding that value that is half above and half below.

• Mode is the most frequently occurring number in the

data set

• Note: the term “average” includes the mean, median,and mode. Over time, mean and average havebecome synonymous.

©2009 ASQ 103

• Range of a set of data is the difference between the highestand lowest values in the set.

• Variance denoted by Sigma Squared (σ2) is a measure ofdispersion of the population about t he mean.

• Standard deviation (σ) is a measure of how far all theobservations in a data set deviate from the mean.

– The standard deviation is the most commonly reportedmeasure of dispersion.

– It is commonly called the “RMS value” – that is, theSQUARE ROOT of the MEAN of the SUM of deviationscores.

Basic Terms

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©2009 ASQ 104

• Proportion (p): In a sample taken from a population,units of interest are counted, and a proportion iscalculated relative to the total sample.

• Proportion (p) = Number of units of interest in thesample / Total number of units in the sample

• For example, a two-proportion test is used to check ifthere is a statistical difference between:

– Proportion defective before and after a processimprovement

– Percent accuracy of form completion betweentwo different office locations

Basic Terms

©2009 ASQ 105

Describe and use this theorem and apply the

sampling distribution of the mean to inferentialstatistics for confidence intervals, control charts,

etc. (Apply)


V.D.2 Central Limit Theorem

©2009 ASQ 106

Central Limit Theorem

Central limit theorem states that the distributionof sample averages will tend toward a normal

distribution as the sample size, n, increases.

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©2009 ASQ 107

Significant points:• Sample means curve is narrower (extreme values are “averaged out”).• Sample means tends to be normal, regardless of the distribution of

individuals. This tendency increases as the sample size increases.• We can usually approximate “distributions of means” with a normal

distribution.

Distribution ofSample Means Distribution of

Individual Observations

Individual observations

represent the distribution of the

population; e.g., actual values ofall observations in all subgroups

Sample means

represent the distribution of the

averages (means); e.g., the valuesof the averages of the subgroups


©2009 ASQ 108

Example:• Sample five parts each hour for 20 hours and measure a d imension.• Calculate the average dimension for the five parts sampled each

hour.• Plot the 100 measurements (5 x 20) and observe the distribution of

all 100 (these are “individual observations”).• Plot the 20 averages (that were calculated based on each hour of

production) and observe the distribution of the 20 averages (theseare “sample means”).

• Result should look like the following diagram:

Distribution ofSample Means

Distribution ofIndividual Observations


©2009 ASQ 109

σ

n

x xσ =

is also called the Standard Error of the Mean

Standard deviation of the distribution ofsample means, X

Standard deviation of individual Xs

Sample size used to calculaten xσ

xσ

X

xσ NOTE:

Standard Deviation of Averages


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©2009 ASQ 110

Central Limit Theorem – Example

Given: The factory collects nine observations per hour,

for 100 hours, from production line #1.

Results were:= 500 pounds (the average of the averages)

= 25 (based on a sample = 9)

Question: What is the standard deviation of the producton line #1?

xσ x

n

xσ

σ =From:

x

We calculate:

75)3(25925 ===

= n xσ σ

x

©2009 ASQ 111


Key Points of the Central Limit Theorem and Six Sigma

• Using ± 3 sigma control limits, the central limit theorem is the

basis of the prediction that, if the process has not changed, asample mean falls outside the control limits an average of only0.27% of the time.

©2009 ASQ 112


Key Points of the Central Limit Theorem and Six Sigma(Continued)

• Most points on the chart tend to be near the average.

• The curve's shape tends to be bell-shaped and the sides tend tobe symmetrical.

• The theorem allows the use of smaller sample averages to evaluateany process because distributions of sample means tend to form anormal distribution.

• The theorem appears when the process is in control (predictable).

• The theorem leaves variations from common causes to chance(thus distributing according to the central limit theorem).

• The theorem identifies and removes variations from special causes.

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©2009 ASQ 113


• The Central Limit Theorem (CLT) is the basis for

calculating confidence intervals and hypothesis tests

• The X bar – R chart depends on the CLT, sinceeach point X bar plotted on the chart is the averageof a subgroup of samples and represents thesample mean.

• The X bar in the X bar – R chart evaluates theprocess’s central tendency over a period of time

©2009 ASQ 114

Progress Check

1. The average cycle time for approval of a sample of 49purchase orders is 40 hours with a standard deviation of14 hours. What is the standard error of the mean?

2. If instead of 49 Purchase Orders, the sample size was 196,with the same mean and standard deviation, what is thestandard error of the mean?

3. What can you conclude looking at both the results?a. As sample size increases, standard error increases

b. As sample size decreases, standard error decreasesc. As sample size increases, standard error decreases

©2009 ASQ 116

Calculate and interpret measures of dispersionand central tendency, and construct and interpret

frequency distributions and cumulative frequencydistributions. (Evaluate)

Lesson 4 – Basic Statistic

V.D.3 Descriptive Statistics

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©2009 ASQ 117

Descriptive Statistics

• Mean, median, mode

• Calculation of std. deviation

• Kurtosis, skewness

• Cumulative frequency distribution

©2009 ASQ 118


Use descriptive statistics to describe data, usuallysample data, with math or graphics to defineelements such as:

• Central tendency or location: median, mean, andmode

• Dispersion or spread: range, variance, and standard

deviation

©2009 ASQ 119


Central tendency is a

measure of most of the data'slocation. Central tendencyrefers to a variety of keymeasurements like mean,

median, and mode – mean

being the most common ofthe three.

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©2009 ASQ 120

Mode – the most frequently occurring or most likely value

Median – the 50th percentile

(half the values are above and half below the median)

Definitions for Central Tendency

Mean – the sum of all members divided by the population size(average)


N

X ++ X + X + X =

N

X =µMean,Population N321

N

1=ii K∑

n

X = X Mean,Sample

n

1=ii∑

©2009 ASQ 121

• Range of a set of data is the difference between thehighest and lowest values in the set.

• Variance denoted by Sigma Squared is a measure

of dispersion of the population about the mean.

• Variances are additive; standard deviations are not

additive...

…so σ12 + σ2

2 + σ32 is OK,

but, σ1 + σ2 + σ3 is NOT OK

Definitions – Spread


©2009 ASQ 122


X

x1

x2

xn

We could add the differences between each value x and theaverage of the values x; however, that would always yield

zero. Therefore, we square the difference between each xand x, to eliminate the negatives and emphasize the outliers,then take the average of the results. This is defined as the

variance or σ2. Obviously, σ = 2σ


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©2009 ASQ 123


σσσσ = the units of interest and its population standard deviation

µ = population mean

N = total population

s = estimate of standard deviation

n = sample size

( )( ) ( ) ( ) ( )

N

X X X X X

N

22

3

2

2

2

1 ... −−+−+−=σ

µ µ µ µ

( ) ( ) ( ) ( )n - 1

X X X X X X X X n

22

3

2

2

2

1 ... −−+−+−= s =σ̂

Standard Deviation is a measure of dispersion of the population

about the mean


©2009 ASQ 124

Definitions – Shape

Skewness: Indicates a lack of symmetry. A distribution is

skewed if one tail extends farther than the other.

• A value close to 0 indicates symmetric data.

• Negative values indicate negative/left skew.

• Positive values indicate positive/right skew.

Kurtosis: Indicates how sharply peaked a distribution is.

• Values close to 0 indicate normally peaked data.

• Negative values indicate a distribution that is flatterthan normal.

• Positive values indicate a distribution with a sharperthan normal peak.


©2009 ASQ 125

• A cumulative frequency distribution is createdfrom a frequency distribution by adding anadditional column to the table called “CumulativeFrequency.” For each value, the cumulative

frequency for that value is the frequency up toand including the frequency for that value.

• The cumulative frequency column shows the

number of data at or below a particular variable


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©2009 ASQ 126

Cumulative Frequency

Distribution Table – Example

– For data point 45, add thecumulative frequency for theprevious data point 44 (6),plus the frequency for datapoint 45 (4).

– This gives you a cumulativefrequency of 10 for datapoint 45.

– Finally, notice that thecumulative frequency forthe highest data point 51 is30 – the same as the totalof the frequency column.

Cumulative FrequencyDistribution Table

Temperature Frequency Cumulat iveFrequency

43 3 3

44 3 6

45 4 10

46 3 13

47 3 16

48 0 16

49 6 22

50 4 26

51 4 30

n = 30


©2009 ASQ 127

A frequency distribution (histogram) is the pattern or

shape formed by a group of measurements in a distributionsummarizing data.

Frequency distribution - Example


©2009 ASQ 128

Progress Check

For the following data sets, calculate the mean, median andmode.

Data Sets

Point Data Set 1 Data Set 2 Data Set 3

A 3 4 10

B 3 3 1

C 6 5 7

D 7 4 1

E 4 16 10

F 7 4 6

G 5 3 1

H 5 4 8

I 4 3 1

J 6 6 1

K 5 3 9

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©2009 ASQ 130

Construct and interpret diagrams and charts,including box-and-whisker plots, run charts, scatter

diagrams, histograms, normal probability plots, etc.(Evaluate)


V.D.4 Graphical Methods

©2009 ASQ 131

Graphical Methods

• A dot plot is a statistical

chart consisting of a groupof data points plotted ona simple scale.

• When dealing with largerdata sets (around 20-30or more data points), a

histogram may be moreefficient, as dot plots maybecome too cluttered afterthis point.

***

****

**

**

*

**

***

**

*

****

** *

*

**

.10 .12 .14 .16 .18

.2

0 .22

Dot plot – Example

©2009 ASQ 132

Graphical Methods

Box-and-whisker plots are a five-number data

summary.

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©2009 ASQ 133

Graphical Methods

• A box-and-whisker plot can be useful for handling many data

values.• They allow people to explore data and to draw informal

conclusions when two or more variables are present.

• It shows only certain statistics rather than all the data.Five-number summary is another name for the visualrepresentations of the box-and-whisker plot.

• The five-number summary consists of the median, thequartiles, and the smallest and greatest values in thedistribution.

• Immediate visuals of a box-and-whisker plot are the center,the spread, and the overall range of distribution.

©2009 ASQ 134

Graphical Methods

Constructing a Box-and-Whisker plot

The first step is to find the median, lower quartile,

and upper quartile of a given set of data.

Box-and-Whisker plot – Example:

– The following set of numbers shows the numberof surgeries performed in a hospital every month,

for the past 15 months (data are arranged fromlow to high).

– 18 27 34 52 54 59 61 68 78 82 85 87 91 93 100

©2009 ASQ 135

Graphical Methods

Box-and-Whisker plot – Example (Continued)• First find the median. The median is the value exactly in the

middle of an ordered set of numbers. This is 68, because thedata set has an odd number of values. If the data set has aneven number of values, use the mean of the two values oneither side of the split.

• Next, consider only the values to the left of the median:18 27 34 52 54 59 61.

– Find the median of this set of numbers, which is 52.This is the lower quartile.

• Now consider only the values to the right of the median:78 82 85 87 91 93 100.

– Find the median of this set of numbers, which is 87.This is the upper quartile.

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©2009 ASQ 136

Graphical Methods

Box-and-Whisker plot – Example (Continued)

• Now find the interquartile range (IQR). The IQR is thedifference between the upper quartile and the lower quartile.

– In our example the IQR = 87 - 52 = 35. – The IQR is a very useful measurement because it is less

influenced by extreme values, and it limits the range to themiddle 50% of the values.

• Finally, the whiskers extend out to the data’s smallest number18 and largest number 100

©2009 ASQ 137

Graphical Methods

Box-and-Whisker plot – Example (Continued)•Put it all together in a graph.

©2009 ASQ 140

Graphical Methods

Run charts display how a process performs over time.

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©2009 ASQ 141

Graphical methods

Scatter diagrams graph pairs of continuous data, with one variable

on each axis, to examine the relationship between them.

Scatter diagrams can show correlation that may exist between thevariables as is shown in the scatter diagrams showing strong (High

Positive) and weak (Low Positive)

©2009 ASQ 142

Graphical Methods Scatter Diagram – Continued

Note: Correlation does not necessarily mean causation, i.e.,a Cause and Effect relationship may not exist between thevariables.

WARNING: DO NOT MAKE “IMPORTANT” INFERENCES ABOUTSCATTER DIAGRAMS UNTIL YOU HAVE COMPLETED ACORRELATION ANALYSIS. LOOKS CAN BE DECEIVING!

©2009 ASQ 143

Graphical MethodsPareto charts identify the top or key (“the vital few”) areas to beaddressed.

• A Pareto chart is a vertical bar chart: frequencies on theordinate and descriptions (categories) on the abscissa.

• Dr. Juran postulated that 80% of the cumulative frequenciesare attributable to 20% of the categories. If your data don’t obeythis rule, look for additional categories or combine categories.

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©2009 ASQ 146

Graphical Methods

• Histogram is a graphical display of tabulated frequenciesshown as bars. It shows what proportion of cases fall intoeach of several categories: it is a form of data binning.

• The graph reveals information about the sample data such as:

– The spread

– The shape

– The approximate center

• Benefits

– Easy to read

– Visual display of the data set’s location and variation

– Quick communication method

©2009 ASQ 147

Graphical Methods

Normality Probability Plots, also called Normal Test Plots,are used to investigate whether process data exhibit thestandard normal bell curve or Gaussian distribution.

©2009 ASQ 148

Graphical Methods

• Karl Friedrich Gauss first presented the theorybehind the normal curve.

– The normal probability plot determines if a set

of data came from a population that is normallydistributed.

– Before computers, statisticians designed normal

probability paper to graph the data.

– Normal distributions follow a linear pattern when

plotted on normal probability paper; therefore, if

the data plots along a straight line, it is normallydistributed.

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©2009 ASQ 149

Progress Check

©2009 ASQ 150

A. Off target, but withinspecs

B. On target within theleast variation

C. On target within the

widest dispersion

D. Bimodal

1.

2.

3.

4.

Progress Check

Match the description with the correct histogram.

©2009 ASQ 151

Define and distinguish between enumerative(descriptive) and analytic (inferential) statisticalstudies, and evaluate their results to draw valid

conclusions. (Evaluate)


V.D.5 Valid Statistical Conclusions

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©2009 ASQ 152

Valid Statistical Conclusions

Enumerative vs. Analytical Studies

In 1975, W. E. Deming defined enumerative studies asstudies in which action will be taken on the universe,

and analytical studies as studies in which action will

be taken on a process to improve performance in thefuture.

©2009 ASQ 153

• Statistical studies provide tools for obtaininginformation based on data

• There are two principal types of studies:

– Descriptive or enumerative

– Inferential or analytical

• Descriptive or enumerative statistics usually havesample data, with math or graphics to define

elements such as:

– Central tendency: median, mean, and mode

– Variation: range and variance

– Graphs: histograms, box plots, and dot plots


©2009 ASQ 154

• Inferential or analytical statistics involve theevaluation of ratio or measured data.

• Analytical statistics are usually performed toestimate the population parameters:

– To determine the difference between twopopulations (hypothesis testing)

– To determine the differences among a number

of populations (analysis of variance)

– To evaluate the degree of relationship betweentwo or more variables (correlation and

regression).


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©2009 ASQ 155

• Analytical statistics are usually performed using the

scientific process of:

– Making a hypothesis of what we expect to find.

– Collecting data.

– Analyzing the data.

– Drawing a conclusion about the validity of thehypothesis.

• So, analytical statistics describes what thepopulation should be in order to have given

rise to the sample that was obtained.


©2009 ASQ 156

Progress Check

Match the example to the type of study:• Descriptive or enumerative• Inferential or analytical

1. If a sample of four that are taken from a box of

bonbons are found to have three orange and

one vanilla, it might be concluded that the boxcontains 75% orange bonbons.

2. Four bonbons are taken from a box of bonbons:two are orange; two are vanilla flavored.

©2009 ASQ 157

Progress Check (Continued)

• Match the example to the type of study:

– Descriptive or enumerative

– Inferential or analytical

3. We want to investigate whether women get better scores onASQ’s Certified Six Sigma Black Belt certification examination.We count the number of women and men passing the first timethey take the exam, and their level of education. Results fromthe study is placed into a contingency table.

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©2009 ASQ 158

Lesson 5 – Probability

Describe and apply probability concepts suchas independence, mutually exclusive events,multiplication rules, complementary probability,

joint occurrence of events, etc. (Apply)

V.E.1 Basic Concepts

©2009 ASQ 159

Basic Concepts

• Definition and formula

• Venn diagram

• Rules of probability

• Contingency tables

• Conditional Probability

• Mutually exclusive events

• Multiplication Rule of Probabilities

©2009 ASQ 160

Basic Concepts

• Definition and formula

– Probability – The chance of somethinghappening. Expressed as a decimal or fraction.

– Outcome – The result

– Sample space – Set of all possible outcomes

(heads, tails)

– Event – A collection of all outcomes

– Frequency – The number of observations foreach sample

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©2009 ASQ 162

Basic Concepts

• Venn diagram

– Venn diagrams are a way of picturing relationshipsbetween different groups of things or “a set.”

– Venn diagrams can be used to illustraterelationships.

• To draw a Venn diagram, first draw a rectangle

which is called “universe.”

• In the context of Venn diagrams, the universe

is not “everything,” but “whatever we areinterested in.”

• A circle represents the probability of an event

we are interested in.

©2009 ASQ 163

Basic Concepts

• Venn diagram

– The probability that an event “A” willoccur is shown in a Venn diagram:

– The area inside the rectangle is

called the universe and has aprobability of 1.

– The area inside the circle

represents the probabilitythat “A” will occur.

– The probability of event “A”not occurring is the shaded

area as shown: “Not A”

A

Not “A”

©2009 ASQ 167

Basic Concepts

Rules of probability

• There are seven fundamental rules of probability:

1. The probability of an event is between 0 and 1

2. The sum of all possible probabilities of definedevents is equal to 1.00

3. The complementary law

4. The additive law

5. The multiplicative law

6. The combination law

7. The conditional law

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©2009 ASQ 168

Rule 1:

Probability Rule

The probability of an event is between 0 and 1

• Probability of an event that CANNOT occur is 0.0

• Probability of an event that IS CERTAIN to occur

is 1.0

• The formula is:

• P(A) = 0.00 to 1.00

©2009 ASQ 169

P1 + P2 + P3 + . . . Pn = 1.00

Example: In roll of a single die, P(1) = 1/6, P(2) = 1/6,

P(3) = 1/6, P(4) = 1/6, P(5) = 1/6, P(6) = 1/6;

P(1, or 2, or 3, or 4, or 5, or 6) = 1/6 + 1/6 +1/6 + 1/6

+1/6 + 1/6 = 1.0

The sum of all possible probabilities of defined

events is equal to 1.00

Rule 2:

Probability Rule

©2009 ASQ 170

Rule 3:

The complementary law:Since the sum of all possible probabilities ofdefined events is equal to 1.00, the probabilityof an event NOT occurring is equal probabilityof the event occurring subtracted from 1.0

If, P(A) = probability an event will occurThen, 1 - P(A) = probability an event will NOT occur

Example:P(draw a heart from a deck of cards) = 13/52P(draw a non-heart from a deck of cards) = 1-(13/52)

= 39/52

Probability Rule

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©2009 ASQ 171

Rule 4:

Probability Rule

The additive law:

If events are mutually exclusive (events cannot occur atthe same time), the sum of the probability of occurrenceof these events is equal to 1.00

If A and B are the only possible outcomes and are mutuallyexclusive events, then: P(A) + P(B) = 1.0

Example:In a single flip of a coin. the only two reasonable outcomes

are a head or a tail (neglecting the possibility of landing on edge);

these are mutually exclusive events since observing a headmeans a tail cannot be observed, and vice versa

Therefore: P(head or tail) = P(head) + P(tail) = 1.0

©2009 ASQ 172

Rule 5:

If, A , B, and C are independent events (not influenced by eachother), then:

P(A and B and C) = P(A) x P(B) x P(C)

Example: The probability of rolling a “6” in consecutivethree rolls of a single die (note: outcome of one roll does

not influence the other)P(roll “6” three times in a row) = 1/6 x 1/6 x 1/6= 1/216

The multiplicative law:The probability of the joint occurrence of independentevents is the product of the probability of each event(“independent” events do not influence likelihood of the

occurrence of any of the other events)

Probability Rule

©2009 ASQ 173

Rule 6:

The combination law:The probability of occurrence of either or both non-independent events is the sum of the probability of eachindependent events minus the probability of joint events.

P(A or B or both A and B) = P(A) + P(B) - {P(A) x P(B)}

Example: The probability of drawing an ace or club from adeck of cards:P(ace or club) = P(ace) + P(club) - P(ace of clubs)

= 4/52 + 13/52 - {4/52 x 13/52}

= 17/52 - {1/52}= 16/52 = 0.31

Probability Rule

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©2009 ASQ 174

Rule 7:Probability Rule

The conditional law:The probability of observing two dependent events is theproduct of the probability of the first event and theconditional probability of the second event, given the firsthas occurred.

Given that A and B are dependent:P(A and B) = P(A) x P(B|A)

Example: The probability of drawing a second ace from a deckof cards, given that the first draw was an ace (assume twocards are drawn, without replacement)

P(two aces) = P(1st ace) x P(2nd ace|1st ace) = 4/52 x 3/51=0.0769 x 0.0589 = 0.00452

©2009 ASQ 177

Progress Check

1. What is the probability of flipping three heads ina row?

2. What is the probability of drawing three acesin a row from a deck of cards if the cards are

replaced and reshuffled after each draw?

3. What is the probability of drawing three aces in

a row from a deck of cards if the cards are NOTreplaced after each draw?

©2009 ASQ 185

Describe, apply, and interpret the followingdistributions: normal, Poisson, binomial, chi square,

Student’s t and F distributions. (Evaluate)


V.E.2 Commonly used Distributions

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©2009 ASQ 186

Commonly Used DistributionsNormal distribution is the spread of information (such asdemographics) where the most frequently occurring value

is in the middle of the range and other probabilities tail offsymmetrically in both directions.

The normal probability

density function is:

Examples

π σ 2

1)( = x f

( )22 / 1σ

µ −−

x

e

©2009 ASQ 187

Standard Normal Distribution•Mean = 0 and standard deviation= 1•One table for all pairs of mean andstandard deviation

•Requires transformation: Z-score

•Z is an ordinate that cuts off acumulative normal probability (areaunder the standard normal curve)from minus infinity to the value of Z

•For example, an ordinate at t hemean = 0 cuts off 50% of the areaunder the curve or a cumulativeprobability of 0.50

σ

X - X =

σ

µ- X =Z

Commonly Used Distributions

©2009 ASQ 188

Progress Check

Consider the following example:The average time it takes for an emergency room administrativeclerk to process a new patient form is 25 minutes, and the dataindicate the standard deviation is three minutes. From this data,we can compute the percentage of time that the clerk performsthe task in under 30 minutes as follows:

See CSSBB HB, page 499 (Appendix 13 CumulativeNormal Distribution Table), for the area under theStandard Normal curve for a specific Z-value. ForZ=1.67, the area under the curve = 0.9525

67.13

2530=

−=

−=

σ

µ X Z

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©2009 ASQ 190


Binomial distributions are used to model discrete

(attribute) data having only two possible outcomes(i.e., pass or fail, yes or no).

The binomial probability distribution equation willshow the probability of getting X defectives in asample of n units:

For a binomial distribution,

the mean is µ = np and

BINOMIAL PROBABILITY DISTRIBUTION

( ) xn xn

xp pC xP

−−= 1)(

( ) pnp −= 1σ

©2009 ASQ 191


Poisson distributions are used to estimate the

probability of a discrete event where values arex = 0, 1, 2, 3, etc.

The Poisson distribution equation shows P(x) which is

the probability of x occurrences in the sample wheremu is the average number of defects per unit, x is thenumber of defects in the sample, and e is a constant

approximately equal to 2.7182818:

The mean and standard deviationfor the Poisson distribution are

calculated as follows:

!)(

x

e xP

x µ µ −

=

pn= µ

np== µ σ

©2009 ASQ 192


Chi-square, Student’s t, and F distributions areused in Six Sigma to test hypotheses, constructconfidence intervals, and compute control limits.

CHI SQUARE

( )2

2

E

E O −∑= χ

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©2009 ASQ 193


A Student's t distribution (t statistic) is commonly

used to test hypotheses regarding confidenceintervals for means when a sample size is small(less than 30) and the population standarddeviation is unknown.

The equation is the student’s t distribution (t score)where x-bar is the sample mean, s is an estimateof the population standard deviation, and n is the

sample size.

ns

xt

/

0 µ −=

©2009 ASQ 194


The f distribution (f-test) is a tool used to test forequality of variances (two population variances) from

two normal populations, specifically showing whetherthere is statistical significance between two samples.

The f-statistic is theratio of two sample

variances given bythe formula: ( )( )2

2

2

1

S S F =

F-STATISTIC

©2009 ASQ 195

Progress Check

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©2009 ASQ 197

Describe when and how to use the followingdistributions: hypergeometric, bivariate, exponential,lognormal and Weibull. (Apply)


V.E.3 Other Distributions

©2009 ASQ 198

Other Distributions

• The hypergeometric distribution is used whenitems are drawn from a population withoutreplacement. That is, the items are not returned tothe population before the next item is drawn out.

– The items must fall into one of two categories,such as good/bad or conforming/nonconforming.

– The hypergeometric distribution is similar in

nature to the binomial distribution, except thesample size of the hypogeometric is largecompared to the population.

– The hypergeometric distribution is appropriatewhenever the sample size is greater than 10% ofthe population (n > 0.1N ).

©2009 ASQ 199

Other Distributions

• The hypergeometric distribution determines the probability ofexactly x number of defects when n items are samples from apopulation of N items containing D defects. The equation is:

Where:

x = number of nonconforming units in the sample (r issometimes used here if dealing with occurrences)D = number of nonconforming units in the populationN = finite population sizen = sample size

x

N

x-n

D-N

x

D

=f(x)

Combination of D things, x at a time

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©2009 ASQ 200

Other Distributions

• Hypergeometric distribution – Example

• A group of 12 cellular telephones is being shipped to alocal retailer. While the phones are much in demand, themanufacturer has been having some problems with phonesbeing shipped with the wrong type of battery. Because thephones are in demand, the retailer agrees to accept theshipment of 12 phones, but only if the shipment has fewerthan three defective phones. Because time is of the essence,

the retail manager decides to only inspect four phones(meaning the manager should find one or fewer defectivephones). Checking the sample of four, the manager findsone phone with the wrong battery. Should the remainderof the shipment be rejected?

©2009 ASQ 201

Other Distributions

• Hypergeometric distribution – Example – Answer:

• Given the information provided:

– N = population of 12

– D = number of defectives allowed at three

– n = sample size of four

– x = number of defectives in the sample of n

– f(x) = probability of getting x defectives in thesample

– For this example, it is necessary to solve the

equation for both probability of 0 and 1 sincethe shipment would be accepted if it also hadno defectives.

©2009 ASQ 202

Other Distributions

Answer: There is a 0.764 probability of one or fewer bad phones.

For most retailers, this risk level would be unacceptable.

( )

−

−

==

n

N

xn

D N

x

D

x X P

( ) 255.0495

1261

4

12

04

312

0

3

0 =×

=

−

−

= f

( ) 509.0495

843

4

12

14

312

1

3

1 =×

=

−

−

= f

1)=P(x+0)=P(X =1)≤P(X

0.7640.509+0.255=1)P(X =≤

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©2009 ASQ 204

Other Distributions• When two variables are distributed jointly, the resulting

distribution is a bivariate distribution.

• Bivariate distributions may be used with either discrete orcontinuous data.

• The variables may be completely independent or a covariancemay exist between them.

• Example: The lengths of a manufactured part’s two dimensionsare important characteristics to be measured.

– Let X represent the length of one dimension of the part,and let Y represent the length of the second dimension ofthe same part.

– In general, these two lengths are not necessarilyindependent of one another.

– An important property of the dimensions is that they bothmeet specifications.

©2009 ASQ 205

Other Distributions

• The exponential distribution is a continuousprobability distribution often used to model problemsin reliability such as “time between events.”

• Exponential distributions are frequently used toanalyze reliability, and often model items with aconstant failure rate.

• The exponential distribution is closely related to

the Poisson distribution, and is used to determinethe average time between failures or average timebetween a number of occurrences.

©2009 ASQ 206

Other Distributions

Where:

µ = the mean

λ = failure rate which is the same as 1/ µ x = x-axis values

The exponential distribution equation is:

The following equation gives the cumulative probabilities without the need for a table:

( ) ∞≤≤=−

x x f

x

01 µ

µ l ( ) x

x f λ λ −= lOR

µ

x

x X P−

−=< l1)(x

x

x X Pλ µ ll ==>

−

)(OR

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©2009 ASQ 207

Other Distributions

• Exponential Distribution – Example

– A Florida electric company experiences an average of 500electrical outages each year due to storms and hurricanes.What is the probability that the weekend crews, who workfrom 6:00 PM on Friday evening to 6:00 AM on Mondaymorning, will not receive a call?

– Data summary:

• µ = 500 electrical outages each year.

• Since there are 365 days in each year and 24 hoursper day, there are 8760 hours each year.

• The time between each outage is 8760/500 = 17.52hours.

• The weekend shift works 60 hours (1800 Fridaythrough 0600 Monday); therefore x = 60.

©2009 ASQ 208

Other Distributions

Exponential Distribution – Example

Using the equation:

Answer: The chance that the weekend crew will not get a callis 3.3%, since 96.7% of the time a call will be received duringthe 60 hours.

µ

x

x X P−

−=< l1)(x

x

x X Pλ µ −

−

==> ll)(OR

0326.0)60( 52.17

60

==>−

l X P

©2009 ASQ 209

Other Distributions

• Lognormal distribution is a skewed-rightdistribution with most data in the left tail, andconsisting of the distribution of the randomvariable whose natural logarithm follows thenormal distribution.

• The lognormal distribution assumes only positivevalues.

• When the data follows a lognormal distribution,a transformation of data is done to make the datafollow a normal distribution.

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©2009 ASQ 210

Other Distributions

Example: The first column of the following table contains datathat is lognormally distributed. The second column contains the

natural logarithm of the first column. The second column isnormally distributed.

©2009 ASQ 211

Other Distributions

• Weibull distribution: The Weibull distribution isone of the most widely used lifetime distributionsin reliability engineering.

• It is a versatile distribution that can take on thecharacteristics of other types of distributions, basedon the value of the shape parameter, β.

• Used when modeling failure rates, particularly whenthe response of interest is percent of failures as a

function of usage (time).

©2009 ASQ 212

Progress Check

Match the distribution type to the typical application

Distribution

Type

Typical Application

Exponential Created with the joint frequency distributions ofmodeled variables.

Weibull Used when raw data is skewed and the log of thedata follows a normal distribution. This distributionis often used for understanding failure rates orrepair times.

Lognormal Used for instances of examining the time betweenfailures.

Bivariate Used when modeling failure rates, particularly whenthe response of interest is percent offailures as a function of usage (time).

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©2009 ASQ 213

Lesson 6 – Process Capability

Define, select, and calculate Cp and Cpk to assessprocess capability. (Evaluate)

V.F.1 Process Capability Indices

©2009 ASQ 214

Process Capability Indices

Natural Process Limits vs. Specification Limits:Process limits are the voice of the process resulting from theproduct variations produced.

Process capability indices (Cp and Cpk) and process performanceindices (P

p, P

pk, and C

pm) identify the current state of the process,

and provide statistical evidence for comparing after-adjustment

results to the starting point. Although these indices have acommon purpose, they differ in their approach.

©2009 ASQ 215


Cp assumes the process is centered in the specification width.It is also known a “Process Entitlement.”Useful when:• Identifying the process’ current state• Measuring the actual capability of a process to operate within

customer-defined specification limits• The data set is from a controlled, continuous process

Using Cp to Assess Process Capability

ST p σ ˆ 6

LSL - USL C =

Short-term estimate for the processstandard deviation: this iswithin-subgroup variation

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©2009 ASQ 216


When it is compared to Cp, Cpk measures the de-centering ofthe process within the specification width.

Useful when:• You have a data set from a controlled, continuous process• Cpk does tell about the process’ ability to align with the target

(centered on the customer requirement)

Note: Cp is often called “Process Capability,” and Cpk “ProcessCapability Index.”

Using Cpk to Assess Short-term Process Capability

=

ST ST

pk LSL x xUSLC σ σ ˆ3-;

ˆ3-min

©2009 ASQ 217

Progress CheckCalculating Capability

For the data shown below, use the formulas given in “definitions”to calculate estimates of:

• Cp

• Cpk

Part 1 Short-term data:X = 5

σst= 2LSL = 0

USL = 10

Part 2: Presume the LSL is changed to = 1. Recalculate processcapabilities:

• Cp,

• Cpk.

©2009 ASQ 219

Define, select, and calculate Pp, Ppk and Cpm toassess process performance. (Evaluate)


V.F.2 Process Performance Indices

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©2009 ASQ 220

Process Performance Indices

Pp measures the ratio between the specificationtolerance and process spread.

Useful when:• The type of data collected is continuous

Using Pp to Assess Long-term Process Capability

LT p σ 6

LSL - USL P = Long-term estimate for the

process standard deviation:this is overall variation

©2009 ASQ 221


Ppk measures the absolute distance of the mean to

the nearest specification limit.

Useful when:• The type of data collected is continuous

Using Ppk to Assess Long-term Process Capability

−−=

σ σ 3;

3min

LSL X X USLP

pk

©2009 ASQ 222


Cpm is also referred to as the Taguchi index.

Useful when:

• The target is not the center or mean of the USL - LSL• Establishing an initial process capability during the

Measure phase

Using Cpm to Assess Process Performance

( )2

2

1σ

µ T

C C

p

pm

−+

=

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©2009 ASQ 223

Progress Check

If a process has a long-term variance of four units

and a specification of 96 +/- 4, what is the long-term process capability, Pp?

a. 0.33

b. 0.66c. 1.00d. 1.5

©2009 ASQ 225

Describe and use appropriate assumptions and

conventions when only short-term data or attributesdata are available and when long-term data are

available. Interpret the relationship between long-term and short-term capability. (Evaluate)


V.F.3 Short-term and Long-term Capability

©2009 ASQ 226

Short-term vs. Long-term Capability

Process capability may be examined as both

short-term and long-term capability. Short-termcapability is measured over a very short time period,since it focuses on the machine’s ability based ondesign and quality of construction.

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©2009 ASQ 227

This is a processcapability outputfrom Minitab

• Short-term uses withinstandard deviation(“Potential” or

“Entitlement” meaning itdoesn’t get any better)

• Long-term uses overallstandard deviation

• Typically, withinstandard deviation

is less than overall

• If within subgroup

variation is very large,within may exceedoverall—not typical

604.5603.0601.5600.0598.5597.0595.5

LSL USL

LSL 595

Target *

USL 605

Sample Mean 600.23

S ample N 100

StDev(Wi th i n) 1 .70499

StDev(Overall) 1.87388

Process Data

C p 0.98

C P L 1 .0 2

C P U 0 .9 3

C pk 0 .9 3

Pp 0.89

P PL 0 .9 3

P PU 0 .8 5

P pk 0 .8 5

C pm *

Overall Capability

Potential (Within)C apability

P P M < L S L 0 .0 0

P P M > U SL 0 .0 0

P P M T o ta l 0 .0 0

Observed Performance

P P M < L S L 1 07 9. 43

PPM > USL 2573.67

P P M T o ta l 3 65 3 .1 0

Exp. Within Performance

PPM < LSL 2627.23

PPM > USL 5455.68

P P M T o ta l 8 08 2. 9 1

Exp. Overall Performance

Within

Overall

Process Capability of Supplier 2

Short-term vs. Long-term Capability(Minitab)

©2009 ASQ 228

Process capabilityis typically cited forshort-term (withinsubgroup standard

deviation)• Using the withinsubgroup standarddeviation, there are

1079.43 ppm below

the LSL and 2573.67above the USL

• Putting the entire fractiondefective (3653.10) in the right tail of the standard normaldistribution shows Process Sigma (Z.bench or sigma level)is 2.68

604.5603.0601.5600.0598.5597.0595.5

LSL USL

LSL 595

Target *

USL 605

Sampl e Mean 600. 23

S ample N 100

StDev(Within) 1.70499

StDev(Overal l) 1.87388

ProcessData

Z .Bench 2. 68

Z .L SL 3 .0 7

Z .U SL 2 .8 0

Cpk 0.93

Z .Bench 2. 41

Z .L SL 2 .7 9

Z .U SL 2 .5 5

Ppk 0.85

Cpm *

Overall Capability

Potential(Within)Capability

P P M < LS L 0. 00

P P M > USL 0. 00

P P M T ot al 0 .0 0

Observed Performance

P P M < LS L 1079. 43

P P M > USL 2573. 67

P P M Total 3653. 10

Exp. Within Performance

P P M < LS L 2627. 23

P P M > USL 5455. 68

P P M Total 8082. 91

Exp. Overall Performance

Within

Overall

Process Capability of Supplier 2

Short-term vs. Long-term Capability(Minitab)

©2009 ASQ 229

Short term vs Long term capability

Process width

Specificationwidth

USL-LSL

Process Z =2 σ

LSL USL

µ

Process Z = half of thenumber of standarddeviations between USLand LSL

Process Z or Process Sigma

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©2009 ASQ 230

3

1.5

3

ShortTerm

p

LongTerm Short Term

Long Term

p

Z C

Z Z

Z P

=

= −

=

2ShortTerm

st

USL LSL Z σ −=

Using Z to Calculate Capability

Remember a Six Sigmaprocess has Zst = 6 andZlt = 4.5 (assuming a

shift of 1.5 σ)

Short-term vs. Long-term Capability

©2009 ASQ 231

Progress Check


If the Cp of a manufacturing process capturingvariables data is 1.0; calculate the following:

• Process Sigma Short term

• Process Sigma Long term• Pp

©2009 ASQ 233

Identify non-normal data and determine when it isappropriate to use Box-Cox or other transformationtechniques. (Apply)


V.F.4 Process Capability for non-normal data

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©2009 ASQ 234

Process Capability for Non-normal Data

USL

average

What percentagefalls here?

Consequences of Using Non-

normal Data• Example: Calculating Process

Sigma with a Normal curve

To determine Process Sigma,find the defect area beyond thespecification limits:[Find Fraction Defective in Z table ]

If the data is not Normal, thedefect area will be incorrectlyestimated using this method andwill misrepresent Process Sigma

The percentage isdifferent for theNormal curve

USL

©2009 ASQ 235

• Transform the data

– Mathematically transform the raw data into an approximatelynormal distribution

– Calculate process capability of the transformed data andspecification limits

– Use known techniques

• Box Cox Transform

• Johnson Transform

• Fit the data to another distribution

– Use the parameters of the distribution and the specificationlimits

– For example: Exponential, Weibull, Chi-Square, etc.

• Use non-linear regression to fit a curve to the data and find area

of the tails beyond spec limits


©2009 ASQ 236

Johnson Transformation

• Johnson Transformation optimally selects a function fromthree families of distributions of a variable, which are easily

transformed into a standard normal distribution.

• If the Johnson transformation does not adequately transformyour data, the Box-Cox Transformation may work better.

• The Box-Cox Transformation simply finds a powertransformation.

• In other words, Box-Cox determines if your non-normal data,raised to a power between -5 and 5, become normal, and ifthe natural log of your non-normal data is normal.

Process Capability for

Non-normal Data

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©2009 ASQ 237

(Power)Common Namesλλλλ Yλ

-2

-1

-0.5

0

0.5

1

2

Reciprocal (inverse) squared

Reciprocal (inverse)

Reciprocal square root (inverse)

Natural Log1

Square root

---

Squared

1

Y2

1

Y

1

Y

Log e (Y)

Y

No

transformation

Y2

• A Box-Cox powertransformationraises Y to the

power of λ1

• Powertransformationsinclude those

we’ve already

seen:

1. When Lambda = 0 the natural log is used as the transform


©2009 ASQ 239

Progress Check

When calculating Process Capability for non-normal

data, what are some of the acceptable methods?

1. Use the properties of the distribution that fitsthe data

2. Transform the data using Box-Cox technique3. Calculate capability as you would for a normaldistribution

a. 2 and 3

b. 1, 2, and 3c. 1 and 2d. 3 and 1

©2009 ASQ 240

Calculate the process capability and process sigmalevel for attributes data. (Apply)


V.F.5 Process Capability for Attributes Data

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©2009 ASQ 241

Process Capability for Attributes Data

In the case of attribute data (discrete data), theprocess capability is simply the average of proportiondefective. This is called binomial process capability.

For “p” chart and “np” chart, it is p barFor “c” and “u” chart, it is c bar and u bar.

©2009 ASQ 242

Calculating Process Capability for Attribute Data:

• Calculate the average of proportion defective; this is theYield Y.

• (1-Yield) = Proportion of Defectives

• Put entire fraction defective in right tail of normal distribution(right tail is by convention).

• Locate Z, from the normal table, the ordinate that “cuts off” theright tail.

• Attribute data is long-term data (unless otherwise stated),

• So Process Sigma Z(short term) =(Z long term +1.5)


©2009 ASQ 243

• Attribute Data is typically considered Long-Term

Data

• Therefore:

Z (short term) = [ Z (long term) + 1.5 ]

Remember a Six Sigma process has Zst = 6 and Zlt = 4.5(assuming a shift of 1.5 s)


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©2009 ASQ 244

Progress Check

Process Capability for Attributes Data: Example

Consider a process with a 95% yield (5% defective)

What is the process sigma?

• Long Term

• Short Term

FractionDefective

(5%)

Yield 95%

Z

©2009 ASQ 245

Describe and apply elements of designing andconducting process capability studies, includingidentifying characteristics and specifications,developing sampling plans, and verifying stabilityand normality. (Evaluate)


V.F.6 Process Capability Studies

©2009 ASQ 246

Process Capability Studies

• Purpose: To determine whether a process can meet customerrequirements

– Take appropriate action if requirements are not met

• Steps to conducting a study:

– Select a quality characteristic for the study

• Determine customer specs for the characteristic under study

– Conduct a gage R&R study to confirm the capability of themeasurement system

– Collect data.

• Approximately 25 to 30 subgroups

• Subgroup size typically five

– Plot data on control charts.

• Eliminate special cause variation, if any

– Test normality of data

– Calculate process capability

– Update control plan

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©2009 ASQ 247

Distinguish between natural process limits andspecification limits, and calculate process performancemetrics such as percent defective, parts per million

(PPM), defects per million opportunities (DPMO),defects per unit (DPU), process sigma, rolled

throughput yield (RTY), etc. (Evaluate)


V.F.7 Process Performance vs. Specification

©2009 ASQ 248

Process Performance vs.Specification• Process limits are the voice of the process

resulting from the product variations produced.The supplier collects data over time to form aprocess curve for determining the variation inthe units against the customers’ specification.

• Process limits are + /- 3 standard deviations fromthe mean

©2009 ASQ 249

• Specification limits are set by the customer, and resultfrom either customer requirements or industry standards.The amount of variance (process spread) the customer iswilling to accept sets the specification limits.

– Example: A customer wants a supplier to produce 12-inchrulers. Specifications call for an acceptable variation of +/-0.03 inches on each side of the target (12.00 inches). Thecustomer is saying acceptable rulers will be from 11.97 to12.03 inches.

• If the process is not meeting the customer’s specification limits,two choices exist to correct the situation:

– Change the process’ behavior.

– Change the customer’s specification (requires customer

approval).

Process Performance vs.Specification

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©2009 ASQ 250

Percent Defective is the number of values of a variable

(expressed as a percentage) that fall outside someuser-defined specification limits.

PercentDefective

(5%)

Yield 95%

USLLSL


©2009 ASQ 251

Parts per million (PPM or ppm) is a measurement

that is expressed by dividing the data set into

1,000,000

PPM is used:• when defect rates are low

PPM = Fraction Defective x 1,000,000


©2009 ASQ 253

Defects per million opportunities (DPMO) is the

measure of capability for discrete (attribute) dataor continuous data (which is the more commonapplication).

DPMO is used when there are multiple opportunitiesper unit.

DPMO = DPO x 10^6


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©2009 ASQ 254

DPMO Example: Calculate the DPMO. The bulleted list below

indicates the number of opportunities for a defect during each

step in the process:

• Step 1 – Apply paint: two opportunities for defect

• Step 2 – Affix decals: three opportunities for defect• Step 3 – Apply clear coat/varnish: five opportunities for defect

Apply paint Affix decals Apply clear coat

Units = 100 Units = 100 Units = 100

Defects = 2 Defects = 1 Defects = 1

Opportunities/unit = 2 Opportunities/unit = 3 Opportunities/unit = 5

DPMO = (Total defects

/ Total opportunities

)(1,000,000)Total defects = 2 + 1 + 1 = 4 Total opportunities = (Total

opportunities/unit )(Total

units )

Total opportunities/unit = 2 + 3 + 5 = 10 Total units = 100 Total opportunities = (Total

opportunities/unit )(Total

units ) = (10)(100) = 1,000

DPMO = (4/1,000)(1,000,000) = 4,000


©2009 ASQ 255

Progress Check

Calculate the DPMO. There are 15 defects found in100 letters to be sent to customers. There are two

opportunities per unit.

©2009 ASQ 257

Defects per unit (DPU) is the measure of capabilityfor discrete (attribute) data, and is found by dividing

the number of defects by the number of units:

DPU = Defects / Units

DPU is used when:

• Defining the project and throughout the entireproject• Gaining an understanding of the problem• Generating a measurable statistic (number)

for evaluating a process


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©2009 ASQ 258

DPU Example – Defects per Unit:Olivia’s Toy Manufacturer produces toy cars . The companyplans to analyze the finishing process and will start by

measuring DPU. The finishing process involves three steps:• Apply paint• Affix decals

• Apply clear coat/varnish

A sample of 100 cars is used for the observation. DPU ismeasured for each step and then calculated for the entireprocess:

Apply paint Affix decals Apply clear coat Final

Units = 100 Units = 100 Units = 100

Defects = 2 Defects = 1 Defects = 1

DPU = 2 ÷÷÷÷ 100 =

.02

DPU = 1 ÷÷÷÷ 100 =

.01

DPU = 1 ÷÷÷÷ 100 = .01 DPU = .02+.01+.01

= .04


©2009 ASQ 259

Progress Check

Calculate DPU or recalls per car. For 100 cars thereare: one recall for brakes, five recalls for engineproblems, and two recalls for power train problems.

©2009 ASQ 260

Yield is defined as the percentage of products thatsuccessfully complete the production process.

First pass yield (FPY) is the percentage of units

that successfully complete a process with no rework.

Rolled throughput yield (RTY) is the probabilitythat a single unit can pass through a series ofprocess steps free of defects.

RTY = yield of process step 1 * yield of process step 2 * …..* yield of process of step n


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©2009 ASQ 261

Rolled Throughput Yield, RTY:

Inspection

Rework

Scrap

N SOperation

Classic and Rolled Throughput Yield

steps processof numbern

yieldpassfirstFPY

FPY FPY FPY FPY RTY n321

=

=

••••= L

StartedNumber

ShippedNumber

N

S YieldClassic ========

Ignores

rework

Takes rework

into account

FPY includes

no defects

©2009 ASQ 262

Process Performance vs. Specification – RTY Example

Step 2

Step 3

Step1

Output

Step 4

Input

100units

90units

80 units

75 units70 units

Accept 85, Rework 5 unitsScrap 10 unitsFPY = 100 – (10+5) / 100 = 85/100 = .85

Accept 73, Rework 7 units

Scrap 10 units

FPY = 90 – (10+7) / 90 = .81


Scrap 5 units

FPY = 80 – (5+3) / 80 = .90


Scrap 5 units

FPY = 75 – (5+10) / 75 = .8

Considering both Scrap and ReworkTrue RTY = .85*.81*.9*.8 = .49572 = 49.57%Classic Yield = 100 input; 70 output = 70%

©2009 ASQ 263

Step 1 Step 2 Step 3 Step 4 Output

Start 100 90 80 75 70

Accept 85 73 72 60

Rework 5 7 3 10

Scrap 10 10 5 5

First Pass Yield 85.00% 81.11% 90.00% 80.00%

Rolled Throughput Yield 49.64%

Classic Yield 70.00%

Classic Yield and Rolled-throughput Yield

Process Performance vs.

Specification – RTY Example

85/10070/100

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©2009 ASQ 264

Progress Check

Calculate the RTY of a process with 100 steps

where each step of the process has a 99% firstpass yield.

©2009 ASQ 266


-dpuY = e

Thus, if we know the number ofdefects per unit, we can calculate RTY.

“e” is a constant 2.718281828….

The natural log (ln) of this constant is 1.

Formula: for conversion of DPU into yield

To Calculate DPU from Yield:DPU= - Natural log (RTY) = -LN(RTY)

Formula: for conversion of yield into defects per unit

©2009 ASQ 267

Operation 2Operation 1

Progress Check

Operation 3

Y1=99.8% Y2=97.4% Y3=96.4%

The first pass yield at each of the three operationsin a process is given above.

Calculate the following:

• Rolled Throughput Yield• Defects per Unit

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©2009 ASQ 268

Module Status

1. Process characteristics

2. Data collection3. Measurement systems

4. Basic statistics5. Probability6. Process capability

©2009 ASQ 269

Module 5

Exercise Solutions

©2009 ASQ 270

Answers- Process Flow Metrics

1. Takt time = available work time / customer demandAvailable work time = 240 minutesCustomer demand = 25 applications be processed

Takt Time = 240 / 25= 9.6 minutes

Answer: Takt time = 9.6 minutes per application. Each applicationshould take no longer than 9.6 minutes.

Note: If an application takes longer than 9.6 minutes to process,then the required output cannot be met.

2. Number of people = Cycle time / Takt timeCycle time = 35 minutesTakt time calculated from above = 9.6 minutesTherefore Number of people = 35 / 9.6 = 3.64

Answer: Number of people = 4 (Rounded up)

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©2009 ASQ 271

Answers – Process Analysis Tools

• Which of the following is not a reason for using process maps?a. Supports the identification of disconnects

b. Helps the team better understand the processc. Enables the discovery of problems or miscommunicationsd. To eliminate the planning processe. Helps define the boundaries of the process

• A Spaghetti Diagram is a tool used to evaluate what type of

waste or non-value-added activities?Overproduction

Excess Inventory

TransportationRepair / Reject

• A value stream map does not depictMaterial and information flows

The “door to door” flow

Bottleneck activitiesStandard times

©2009 ASQ 272

Answer – Types of Data

Classify examples in the table below as Continuous

or Discrete data.

Data example Continuous Discrete

1.7 inches X

10 scratches X

6 rejected parts X

10.542 seconds X

25 paint runs X

32.06 psi X

©2009 ASQ 273

Answer – Measurement Scales

Example Measurement scale

A car weighs 3500 lbs Ratio

800 people fa iled an exam Nomina l

Defects are either critical,major a, major b, and minor

Ordinal

The shipping codes used forlast week’s orders

Nominal

The weights of a sample of parts Ratio

The temperature of parts after1 hour of cooling

Interval

What type of measurement scale is appropriate for each example?

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©2009 ASQ 274

Answer Identify the type of sampling method used in each example.Is it Random, Stratified, or Systematic?

ExampleA customer service call center receives a randomly generated listof customers who have interacted with customer servicerepresentatives. The manager selects every 5th name on the list

to conduct a follow-up call for quality assurance. SystematicCustomer survey results are divided into multiple strata based ongender and income level. StratifiedRandom number generator is used by a fast-food chain to print asurvey number on a receipt for customers to use when they calland answer questions about their experience. RandomA manufacturer divides data on defects into strata based onmanufacturing location and equipment type used for production.Stratified

©2009 ASQ 275

Answer

Example

A school district wants to ensure that accurate and relevantinformation concerning education is made available to the state. Highschool completion rates

of the students aged 19 to 20, by gender, from 2000 to 2008 is beingcollected.Answer: Checklist

Runout of a shaft being ground is measured. The tolerance is + / -0.0005. The operator documents only the last two digits in the

inspection report.

Answer: Coding

Identify the data collection technique (Checklist or Coding) usedin each example.

©2009 ASQ 276

Answer – Measurement Methods

1. Which of the following statements are true for anAttribute Gage used for measuring a part?

Answer: (b, c and d)

• They are usually designed to check a singledimension or tolerance limit

• There are usually two members: Go / No-Go

• They do not tell you how good or how bad thepart is.

2. Which of the following is not a gage used to

collect attribute data

Answer: b. Height Gage

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©2009 ASQ 277

Answers – Measurement SystemAnalysis

1. Precision is best described as:a. a comparison to a known standardb. the achievement of expected ongoing quality

c. the repeated consistency of resultsd. the difference between an average measurement and actual value

2. The overall ability of two or more operators to obtain consistent resultsrepeatedly when measuring the same set of parts and using the same

measuring equipment is the definition of:a. Repeatabilityb. Precision

c. Reproducibilityd. Accuracy

3. In measurement system analysis, which of the following pairs of datameasures is used to determine total variance?

a. Process variance and reproducibility

b. Noise system and repeatabilityc. Measurement variance and process variance

d. System variance and bias

©2009 ASQ 278

Answers

4. A calibrated micrometer was used to take 10 replicated measuresof a reference standard. If the mean of the 10 measurements is

0.073, and the true value of the reference standard is 0.075, whatis the bias of the micrometer?

a. 0.001b. 0.002c. 0.073

d. 0.075

5. Repeatability and reproducibility are terms that operationally

definea. biasb. Accuracy

c. Discriminationd. Precision

©2009 ASQ 279

Answers

6. The extent to which an instrument replicates its results when

measurements are taken repeatedly on the same unit is called:a. Real biasb. Precisionc. Accuracyd. True value

7. A measurement system analysis is designed to assess thestatistical properties of:a. Gage variationb. Process performancec. Process stability

d. Engineering tolerance

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©2009 ASQ 280

Select one:

Calibration intervals can be adjusted when:a. Defective product is found

b. Some instruments are scrapped whencalibrated

c. A particular characteristic on the instrument isconsistently found to not be within tolerance

d. A new employee is issued a measuringequipment

Answer - Metrology

©2009 ASQ 281

Answer – Central Limit Theorem1. The average cycle time for approval of a sample of 49 purchase orders is

40 hours with a standard deviation of 14 hours. What is the standard error

of the mean?AnswerStandard error of the mean: Sigma X bar = S/ Square root of the sample

size.S = 14 hours

Sample size = 49Therefore: Sigma X bar = S/ Square root of the sample size = 14 / sq

root(49) = 14/7 = 2

2. If instead of 49 Purchase Orders, the sample size was 196, with the samemean and standard deviation, what is the standard error of the mean?

AnswerSigma X bar = S/ Square root of the sample size = 14 / sq root(196) =

14/14 = 1

3. What can you conclude looking at both the results?

a. As sample size increases, standard error increasesb. As sample size decreases, standard error decreasesc. As sample size increases, standard error decreases

©2009 ASQ 282

Answers

For the following data sets, calculate the mean, median andmode.

Statistic Data Set 1 Data Set 2 Data Set 3

Mean 5 5 5

Median 5 4 6

Mode 5 3 and 4(Bimodal)

1

Notice that data set 2 has a bimodal distribution in which twovalues (3 and 4) occur more frequently in the data set than therest of the values.

From the data sets, you can see that the mean is stronglyaffected by extreme values (value = 16 in data set 2), while themedian is not.

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©2009 ASQ 283

Progress Check

Answers:1. B

2. B

©2009 ASQ 284

A. Off target, but withinspecs

B. On target within theleast variation

C. On target within the

widest dispersion

D. Bimodal

1.

2.

3.

4.

Progress Check

Match the description with the correct histogram.

Answers:A – 4

B – 2C – 1D – 3

©2009 ASQ 285

Progress Check

Match the example to the type of study:• Descriptive or enumerative

• Inferential or analytical

1. If a sample of four that are taken from a box of bonbons are found tohave three orange and one vanilla, it might be concluded that the

box contains 75% orange bonbons. Inferential

2. Four bonbons are taken from a box of bonbons: two are orange; twoare vanilla flavored. Descriptive

3. We want to investigate whether women get better scores on ASQ’sCertified Six Sigma Black Belt certification examination. We count

the number of women and men passing the first time they take theexam, and their level of education. Results from the study is placed

into a contingency table. Descriptive or Enumerative

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©2009 ASQ 286

Answers - Probability

1. What is the probability of flipping three heads in a row?

P(3H) = 1/2 x 1/2 x 1/2 = 1/8 (multiplicative law)

2. What is the probability of drawing three aces in a row from a deck ofcards if the cards are replaced and reshuffled after each draw?

P(3 aces| replacement) = 4/52 x 4/52 x 4/52 = (4/52)^3 = (.077)^3= 0.0004552 (multiplicative law)

3. What is the probability of drawing three aces in a row from a deck ofcards if the cards are NOT replaced after each draw?

P(3 aces| NO replacement) = 4/52 x 3/51 x 2/50= 0.0769 x 0.05882 x 0.04= 0.000181 (conditional law)

©2009 ASQ 287

Answer – Commonly Used Distributions

Emergency room example.

The calculated Z value of 1.67 (find 1.6 going downthe left-hand side of the table, then 0.07 goingacross the table) for the area under the curve up to

30 is 0.9525. This means that in 95.25% of the time,an emergency room administrative clerk canprocess a new patient form in less than 30 minutes;

or 4.75% of the time it is greater than 30 minutes.

©2009 ASQ 288

Answers

Answers:

1. D2. C3. B

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©2009 ASQ 289

Answers

Match the distribution type to the typical application

DistributionType

Typical Application

Bivariate Created with the joint frequency distributions of

modeled variables.

Lognormal Used when raw data is skewed and the log of thedata follows a normal distribution. This distributionis often used for understanding failure rates orrepair times.

Exponential Used for instances of examining the time between

failures.

Wiebull Used when modeling failure rates, particularly when

the response of interest is percent offailures as a function of usage (time).

©2009 ASQ 290

Answer Calculating Capability

Part 1:

Cp = (USL-LSL)/6σST = (10-0)/6x2 =0.833

Cpk = lesser of: (USL - X-bar) / 3σST or, (X-bar - LSL) / 3σST

= (10 - 5)/ 3x2 = 5/6 = .8333 or, (5 - 0)/ 3x2 = 5/6 = .8333

Part 2:

Cp = (USL-LSL)/6σST = (10-1)/6x2 =.75

Cpk = lesser of: (USL - X-bar) / 3σST or, (X-bar - LSL) / 3σST

= (10 - 5)/ 3x2 = 5/6 = .8333 or, (5 - 1)/ 3x2 = 4/6 = .666

©2009 ASQ 291

Answer

If a process has a long-term variance of four units

and a specification of 96 +/- 4, what is the long-term process capability, Pp?

a. 0.33b. 0.66

c. 1.00d. 1.5

Variance = 4; Standard Dev = sq.rt of variance = 2

Pp = USL-LSL / 6* std dev = 4 / 6*2 = 0.33

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©2009 ASQ 292

Answers


If we have calculated Z for the short term and for the longterm, it is a simple matter to calculate Pp.

The important thing to remember is that the relationshipbetween process sigma and process capability is 3.Answer: Cp = Zst / 31 = Zst / 3Therefore Zst = 3

Zlt = Zst-1.5Zlt = 3-1.5 = 1.5

Pp = Zlt/3 = 1.5/3 = .5Answer: Zst = 3Zlt = 1.5

Pp = 0.5

©2009 ASQ 293

Progress Check

When calculating Process Capability for non-normal

data, what are some of the acceptable methods?

1. Use the properties of the distribution that fitsthe data

2. Transform the data using Box-Cox technique3. Calculate capability as you would for a normaldistribution

a. 2 and 3

b. 1, 2, and 3c. 1 and 2d. 3 and 1

©2009 ASQ 294

Answer

Calculate the DPMO. There are 15 defects found in100 letters to be sent to customers. There are twoopportunities per unit.

Answer:

DPMO = 15/(2*100)*1000000. = 75,000.

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©2009 ASQ 295

Progress Check

Calculate DPU or recalls per car. For 100 cars there

are: one recall for brakes, five recalls for engineproblems, and two recalls for power train problems.

Answer:8 recalls per car.

©2009 ASQ 296

Answer

Calculate the RTY of a process with 100 steps

where each step of the process has a 99% firstpass yield.

Answer:

The RTY is 99**100 = 36.6%.

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