# Lean six sigma (green belt)new

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LEAN

SIX SIGMA

NATIONAL PRODUCTIVITY ORGANIZATION (NPO), LAHORE

MINISTRY OF INDUSTRIES, PAKISTAN

1

SL S

MODULE # 1

2 LEAN SIX SIGMA: AN

OVERVIEW

Understanding about Lean & Six Sigma

Five Lean Principles

Types of Waste

Eight Sources of Waste

A Simple Lean Tool: “5S”

EVOLUTION OF QUALITY

FIELD

Product Insp. to Process to System to Culture to Performance

Control (Opr Mgt ) Change

TQM+ – Wave II

Lean Six

Sigma

Six Sigma

Knowledge

Mgt.

IT

TQM – Wave I

HRM

GROUP

DYNAMICS

Teams

Efficiency

BPR

TPM

JIT/MRP

QA

ISO9000

OPR MGT.

QC

SPC

Quality

Circles

Inspection/

Testing

Metrology

3

WHAT IS LEAN?

4

“A systematic approach to identify and eliminate waste (and non

value-added activities) through continuous improvement by flowing

the product at the pull of the customer in pursuit of perfection.”

Lean Thinking is all about continuous waste elimination !

LEAN is Delivering value to Customers with shortest turn around time

SIX SIGMA DEFINITIONS

A Management driven, scientific methodology for

product and process improvement which creates

breakthroughs in financial performance and

Customer satisfaction.

Source: Motorola

A methodology that provides businesses with the

tools to improve the capability of their business

processes. This increase in performance and

decrease in process variation lead to defect reduction

and improvement in profits, employee morale and

quality of product.

Source: ASQ

5

COMPARISON OF

LEAN & SIX SIGMA6

Six Sigma was developed by Motorola in the 1980s to systematically

improve quality by elimination of defects.

SIX SIGMA LEAN

Objective Deliver value to customer Deliver value to customer

Theory Reduce variation Remove waste

Focus Problem focused Flow focused

Assumptions A problem exists

Figures and numbers are

valued

System output improves if

variation in all processes

inputs is reduced

Waste removal will

improve business

performance

Many small improvements

are better than system

analysis

Six Sigma is a data driven philosophy and process resulting in dramatic

improvement in products/service quality and customer satisfaction.

5 PRINCIPLES OF LEAN

7 In “Lean Thinking” – as summarized by James Womack & D. Jones

1 SPECIFYING VALUE

“Value is only meaningful when expressed in terms of a

specific product or service which meets the customer

needs at a specific price at a specific time.”

2 Identify and Create

Value Streams

“A Value stream is all the actions currently required to

bring a product from raw materials into the arms of the

customer.”

3 Making Value Flow

“Products should flow through a lean organization at the

rate that the customer needs them, without being

caught up in inventory or delayed.”

4

“Only make as required. Pull the value according to the

customer’s demand.

Pull production not

Push

5 Striving for

perfection

“Perfection does not just mean quality. It means

producing exactly what the customer wants, exactly

when the customer requires it, at a fair price and with

minimum waste.

LEAN MANUFACTURING:

ELIMINATING THE WASTE8

Waste caused by overstressing people,

equipment, or systems

Waste due to unevenness or variation

Type–I Muda: Non-Value added, but necessary

for the system to function

Type–II Muda: Non – Value added and

unnecessary for the system to function; the first

targets for elimination

(Unreasonableness)

Muri

Mura

(Inconsistency)

Muda (Waste)

LEAN MANUFACTURING:

ELIMINATING THE WASTE9

Types of Waste (Muda)

Transportation

Inventory

Motion

Waiting

Overproduction

Over processing

Defect

Non-utilized people

Transportation

TYPES OF WASTE (MUDA)

10

Waste (Muda)

Total lead time through the value chain

1. Are they equal or not?

2. If not; Which is the most significant source of

waste?

Which is the most significant source of waste?

Producing

TOO much

ADDITIONAL

transportation cost

Producing

TOO much

to sort, handle and store.

Overproduction is the disease, Defects are the cause?

TYPES OF WASTE (MUDA)

11

5S – A Simple “LEAN TOOL”

12

Implementing the 5S is often the first step in

Lean Transformation

5S – A Framework to create and maintain your workplace

1. S: SORT (Organization)

Distinguish between what is and is not needed

2. S: SET IN ORDER (Orderliness)

A place for everything and everything in its place

3. S: SHINE (Cleanliness)

Cleaning and looking for ways to keep it clean

4. S: STANDARDIZE (Adherence)

Clearly define Tasks and Procedures

5. S: SUSTAIN/SYSTEMIZE (Self-Discipline)

Stick to the rules, conscientiously

5S GAME

13

The “Numbers Game” is an

exercise that illustrates the

power of 5s.

The game consist of 4 quick

rounds. You must not look at

the sheets until instructed and

must finish when time is up

5S GAME: ROUND#1

14

We will apply 5S to a workplace and measure

the improvement in executing our job.

During each 30 second shift, your job is to

strike out the numbers 1 to 49 in order

The first page of numbers represents our

current workplace

Ready… Set…

5S – A Simple “LEAN TOOL”

(Cont…)15

Decide what is needed and what is not,

and dispose of all items that are not necessary

1 SORT / SIFTING

CAR

PARKING

AREA

Should these barrels

be in car parking area?

5S GAME: ROUND#2

16

Japanese concept for

house keeping

Sort (Seiri)

Straighten (Seiton)

Shine (Seiso)

Standardize (Seiketsu)

Sustain (Shitsuke)

The first “S” is Sort

We have removed

numbers between 50

and 90 which are not

needed

Ready… Set…

What sort of

improvement does this

yield?

5S – A Simple “LEAN TOOL”

(Cont…)17

2 STRAIGHTEN / SET IN

ORDER

Provide orderly storage in the right place for

all necessary items so that they can be easily

found and used when needed.

EQUIPMENT

STORAGE AREA

Easy to determine

equipment location

5S GAME: ROUND#3

18

Japanese concept for

house keeping

Sort (Seiri)

Straighten

(Seiton)

Shine (Seiso)

Standardize (Seiketsu)

Sustain (Shitsuke)

The second “S” is

Straighten or Set in Order

We have installed a rack

system to help locate

the numbers.

Numbers go from

bottom to top, left to

right

Ready… Set…

What sort of improvement

does this yield?

5S – A Simple “LEAN TOOL”

(Cont…)19

3 SHINE / SWEEPING

Maintain a clean worksite at all times in order

to make work easier, safer, healthier and more

satisfying

I am motivated to

work in this standards

Service

Workshop

5S – A Simple “LEAN TOOL”

(Cont…)20

4 STANDARDIZE

Continuously keep work

area orderly and clean

5S GAME: ROUND#4

21

Japanese concept for

house keeping

Sort (Seiri)

Straighten (Seiton)

Shine (Seiso)

Standardize

(Seiketsu)

Sustain (Shitsuke)

The fourth “S” is

Standardize

We’ve created a system of

ordering the numbers from

lowest to highest from left

to right and top to bottom

We’ve put one number in

each box to standardize

Ready… Set…

What sort of improvement

does this yield?

5S – A Simple “LEAN TOOL”

(Cont…)22

5 SUSTAIN / SELF-DISCIPLINE

Make it habit to engaging 5S activities

daily basis by establishing standards.

The fifth “S” is Sustain

This is your challenge: Sustain your lean activities

Often the hardest to achieve

MODULE # 2

23

SIX SIGMA: AN OVERVIEW

What is Six Sigma?

Different opinions on the definition of six sigma

Six Sigma is a Philosophy

Six Sigma is a Set of Tools

Six Sigma is a Methodology

Six Sigma as a Measure

Six Sigma as a Metric

Six Sigma Structure

WHAT IS SIX SIGMA?

In a narrow sense…

A metric based on Statistical Measure called

Standard Deviation

In a broader, business sense…

WORLD CLASS QUALITY providing a BETTER

product or service, FASTER, and at a LOWER

COST than our competitors.

VARIATION… “the enemy of the customer

satisfaction”

24

WHAT IS SIX SIGMA?

DIFFERENT OPINIONS ON THE DEFINITION OF SIX SIGMA:

Six Sigma is a PHILOSOPHY:

This is generally expressed as y = f(x).

Six Sigma is a SET OF TOOLS:

The Six Sigma expert uses qualitative and quantitative

techniques to drive process improvement.

Six Sigma is a METHODOLOGY:

DMAIC Vs DMADV

Six Sigma is a METRIC:

it uses the measure of sigma, DPMO (Defect Per Million

Opportunities), RTY (Rolled Throughput Yield) etc.

Six Sigma is a MEASURE:

Short Term Vs Long Term

25

SIX SIGMA FRAMEWORKS

26

SIX

SIGMA

Lean Six

Sigma DFSS

VARIATION

Defects

Cost of Poor

Quality

WASTE / SPEED

Cycle Time,

Delivery

Cost of Operation

RELIABILITY &

ROBUSTNESS

Design Features

DMAIC DMAIC DMADV

SIPOC,

CTQ, SPC,

FMEA, DOE,

QFD, CoQ,

ANOVA,

Hypothesis,

Regression,

MSA (R & R)

5S, Value

Mapping,

Time Study,

TPM,

Cellular

Prod.,

Takt Time,

Poke Yoke

VOC, QFD,

FMEA, CTQ,

Gage R & R,

DOE,

Reliability

Analysis, SPC,

Systems

Engineering

PROGRAM

FOCUS /

THEME

METHODOLOGY

TOOLS

DMAIC METHODOLOGY

27

DEFINE: “What is important to the business?”

The problem is defined, including who the customers are and what they want, to

determine what needs to improve.

Expected benefits for the project sponsor & Time line

MEASURE: “How are we doing with the current process?”

The process is measured, data are collected, and compared to the desired state.

ANALYZE: “What is wrong with the current process?”

The data are analyzed in order to determine the cause of the problem.

IMPROVE: “What needs to be done to improve the process?”

The team brainstorms to develop solutions to problems; changes are made to the

process, and the results are measured to see if the problems have been eliminated.

If not, more changes may be necessary.

CONTROL: “How do we guarantee performance so that the improvements are

sustained over time?”

If the process is operating at the desired level of performance, it is monitored to

make sure the improvement is sustained and no unexpected and undesirable

changes occur.

SIX SIGMA IS A METRIC

WHAT IS SIGMA LEVEL?

A metric that indicate how well a process is performing. A

higher sigma level means higher performance . A Statistical

measure of the capability of a process.

28

1. Defects

2. Defects Per Unit (DPU)

3. Parts Per Million (PPM)

4. Defects Per Million Opportunities (DPMO)

5. Yield

6. First Time Yield

7. Rolled Throughput Yield (RTY)

8. Sigma Level

Each of these metrics serves a different purpose and may be used at different

levels in the organization to express the performance of a process in meeting the

organization’s (or customer’s) requirements.

SIX SIGMA: METRICS …

29

DPU

(Defects / Unit)

(# of Defects / # of Units)

Say:

10 Defects, 100 Pairs

DPU = 10/100 = 0.1 (10%)

DPO

(Defects / Opportunity)

(# of Defects) / (# of Units

X # of Defect Opportunities

/ Unit)

Say:

10 Defects, 100 Pairs,

2 Opportunities / Carton

DPO = 10/(100 X 2) = 0.05

or 5% for each type

SIX SIGMA: METRICS …

30

DPMO

(Defects / M.

Opportunities)

DPO X 106

Say:

10 Defects, 100 Pairs

2 types of defects

DPMO = 0.05 X 106 = 50,000

SIGMA

Consult Z–Table or Excel

Sigma Level

Yield =1–DPO =1–0.05 = 95 %

From M.S. Excel:

=Normsinv(%Yield)+1.5

50,000 DPMO = 3.145σ

SIX SIGMA IS A MEASURE

1.5 Sigma

Shift

Theory

31

SIX SIGMA STRUCTURE

32

Quality Council / Steering Committee

Champions

Master

Black Belt

Black Belt Black Belt

Green Belt Green Belt Green Belt Green Belt

HOD’S /

Owners

Sponsors

Process Owner

Coach

Trainers

Team

Leaders

Team

Members

Project

Managers

MODULE # 3

33 DMAIC METHODOLOGY:

DEFINE PHASE

Project Charter

IPO & SIPOC Diagram

Process Flow Diagram

Lean Process Metrics

Cost of Quality

PROJECT CHARTER

34 Project Title Project Title

Business Case Why should you do this project?

What are the benefits of doing this project?

Problem Statement What is the problem, issue and/or concern?

Goal What are your improvement objectives and targets?

Metrics (CTQ’s) PRIMARY Metric(s): Key measures to be used for the

objectives

SECONDARY Metric(s): Those measures which indicates

impacts on secondary concerns and which indicates that

problem is not shifted to other key areas.

Project Scope What authority do you have?

Which processes/products you are addressing?

What is not within this project?

Project Team Who are the team leader, sponsor, and members?

What are their roles and responsibilities in this project?

Project Plan How and when are you going to get this project done (DMAIC

stages)

Communication

Plan

What are your interfaces with each other?

What are your meeting & reporting times?

35

PROJECT CHARTER

EXAMPLE

WHAT IS A PROCESS?

(IPO DIAGRAM)

An Input-Process-Output (IPO) diagram, also known as a general process

diagram,

provides a visual representation of a process by defining a process and

demonstrating the relationships between input and output elements.

The input and output variables are known as ‘factors’ (X) and ‘responses’ (Y)

, respectively.

36

INPUTS OUTPUTS

BILLING

PROCESS

Data Entry Method

Amount of Personnel

Training

Method for obtaining bill

from information

Time to complete a bill

Number of errors / bill

TYPES OF PROCESS MAPS:

SIPOC DIAGRAM

Suppliers Inputs Process Outputs Customers

6) Who are the

Suppliers of the

Inputs?

5) What

are the

Inputs of

the

Process?

2)a. What is the

start of the

process?

1) What is the

process?

2) b. What is the

end of the

process?

3) What are

the outputs

of the

process?

4) Who are

the customers

of the

outputs?

37

TYPES OF PROCESS MAPS: SIPOC DIAGRAM…

Example SIPOC Diagram of Husband making wife a

cup of tea.

38

PROCESS FLOW DIAGRAM

(FLOW CHART)

A Flowchart is a diagram that uses graphic symbols to represent the

nature and flow of the steps in a process / system.

Deciding when & where to collect data

FEW SYMBOLS USED IN FLOW DIAGRAM

Process Symbol

“An Operation or Action step”

Terminator Symbol

“Start or Stop Point in a process”

Inventory / Buffer

“Raw Material / Finished Goods Storage”

Inventory / Buffer

“Partial Finished Goods

“Work In Process” Storage” Document Symbol

“A Document or Report”

Database Symbol

“Electronically Stored

Information”

Flow Line

Decision Point

39

PROCESS FLOW DIAGRAM ….

1.What you THINK it is …

2.What it ACTUALLY is…

3. What it SHOULD be…

40

PROCESS MAPPING LEVELS

41 LEVEL–1: The Macro Process Map, sometimes called a Management

Level or viewpoint.

LEVEL–2: The Process Map, sometimes called the worker level or

viewpoint. This example is from the perspective of the pizza chef.

LEVEL–3: The Micro Process Map, sometimes called the Improvement

level or viewpoint. Similar to a level–2, it will show more steps and

tasks and on it will be various performance data; yields, cycle time,

value and non-value added time, defects, etc.

TYPES OF PROCESS MAPS

42

THE LINEAR FLOW PROCESS MAP

THE DEPLOYMENT FLOW or SWIM LANE PROCESS MAP

LEAN PROCESS: Value Added

& Non–Value Added43

Value Added Activity

Transforms or shapes material or information or people

And it’s done right the first time

And the customer wants it

Non-Value Added Activity – Necessary Waste

No value is created, but cannot be eliminated based on

current technology, policy, or thinking

Examples: project coordination, company mandate, law

Non-Value Added Activity – Pure Waste

Consumes resources, but creates no value in the eyes of

the customer

Examples: idle/wait time, rework, excess checkoffs

LEAN PROCESS METRICS

Processing times or activity time: how long does the worker or

process spend on the task?

Capacity: how many units / customers can the worker or process

make / deal per unit of time

Bottleneck is the process step with the lowest capacity

Process Capacity is the capacity of the bottleneck

Cycle time is the time interval between the completion of two

consecutive units (or batches)

Flow rate (Throughput rate) is the output rate that the process is

expected to produce

Flow Time (Throughput time) = The amount of time for a unit /

customer to move through the system

Inventory: The number of flow units / customers in the system

Utilization is the ratio of the time that a resource is actually being

used relative to the time that is available for use

Work Load/Implied Utilization = Capacity requested by demand /

Available Capacity

Takt Time: (Available Time) / Demand

44

QUALITY COST

Feigenbaum defined quality costs as:

“Those costs associated with the definition, creation, and

control of quality as well as the evaluation and feedback of

conformance with quality, reliability, and safety

requirements, and those costs associated with the

consequences of failure to meet the requirements both

within the factory and in the hands of customers.”

“QUALITY IS FREE” (Crosby)

45

COST OF QUALITY

COST OF ACHIEVING GOOD QUALITY

PREVENTION COSTS

The cost of any action taken to investigate, prevent or reduce the risk of a non-

conformity

Include quality planning costs, designing products with quality characteristics,

Training Costs, etc.

APPRAISAL COSTS

The costs associated with measuring, checking, or evaluating products or services

to assure conformance to quality requirements

Include inspection & Testing Costs, Test Equipment Costs, Operator Costs, etc.

COST OF POOR QUALITY

INTERNAL FAILURE COSTS

The costs arising within the organization due to non-conformities or defects

include scrap, rework, process failure, downtime, and price reductions

EXTERNAL FAILURE COSTS

The costs arising after delivery of product or service to the customer due to non-

conformities or defects

include complaints, returns, warranty claims, liability, and lost sales

46

MEASURING AND REPORTING

QUALITY COSTS

INDEX NUMBERS

ratios that measure quality costs against a base

value

LABOR INDEX

ratio of quality cost to labor hours

COST INDEX

ratio of quality cost to manufacturing cost

SALES INDEX

ratio of quality cost to sales

PRODUCTION INDEX

ratio of quality cost to units of final product

47

MODULE # 4

48 DMAIC METHODOLOGY:

MEASURE PHASE

Types of Data

Data Collection (Check Sheet)

Pareto Analysis

Cause & Effect Diagram & Matrix

Descriptive Analysis

Qualitative Data Analysis

Quantitative Data Analysis

Process Capability Studies (Cp,Cpk)

TYPES OF DATA

49 Attribute Data (Qualitative)

Is always binary, there are only two possible values (0, 1)

1. Yes, No

2. Success, Failure

3. Go, No Go

4. Pass, Fall

Variable Data (Quantitative)

Discrete (Count) Data:

Can be categorized in a classification and is based on

counts.

1. Number of defects

2. Number of defective units

3. Number of Customer Returns

Continuous Data:

Can be measured on a scale, it has decimal subdivisions

that are meaningful

1. Time, Pressure

2. Money

3. Material feed rate

CHECK SHEET

A check sheet is a Form, in Diagram or Table format, prepared in

advance for Recording/Collecting Data. You can thus gather necessary

Data by just making a Check mark on the Sheet.

50

TYPES OF CHECK SHEET

DEFECTIVE ITEM (ATTRIBUTE)

CHECK SHEET:

VARIABLE CHECK SHEET:

DEFECTIVE LOCATION CHECK

SHEET:

PARETO ANALYSIS

A bar graph used to arrange information in such a way that

priorities for process improvement can be established.

Count

Percent

Fault Desc.

Count 43 107

Percent 31.9 30.4 9.3 7.4 4.4 4.3 3.5

707

2.0 1.9 4.8

Cum % 31.9 62.3 71.6 79.0 83.3 87.7

674

91.2 93.2 95.2 100.0

205 164 97 96 78 45

Other

M

isc. Defects

Trim

Not As

Specified

Raw

Edge

Seam

Const. Not As

Specified

O

pen

Seam

Uneven

St.

Puckering

Skip

St.

Broken

St.

2500

2000

1500

1000

500

0

100

80

60

40

20

0

Pareto Chart of Fault Desc.

The 80–20 theory was first developed in 1906, by Italian economist,

Vilferdo Pareto, who observed an unequal distribution of wealth and

power in a relatively small proportion of the total population.

Joseph M. Juran is credited with adapting Pareto’s economic

observations to business applications.

Separates the “vital few” from the “trivial many” (Pareto Principle)

51

80 – 20 Rule:

80% of your phone calls go to

20% of the names of your list

20% of the roads handle 80% of

the traffic

80% of the meals in a restaurant

come from 20% of the menu

20% of the people causes 80% of

the problems

PARETO ANALYSIS …

Pareto Chart Using Minitab– EXAMPLE:

Suppose you work for a company that

manufactures motorcycles. You hope to reduce

quality costs arising from defective

speedometers. During inspection, a certain

number of speedometers are rejected, and the

types of defects recorded.

You enter the name of the defect into a

worksheet column called Defects, and the

corresponding counts into a column called

Counts.

You know that you can save the most money by

focusing on the defects responsible for most of

the rejections.

A Pareto chart will help you identify which

defects are causing most of your problems.

Open the worksheet EXH_QC.MTW

52

CAUSE & EFFECT DIAGRAM

Step 1: Identify the problem and enter in effect box

Step 2: Draw in the spine of fishbone

Step 3: Identify Main Causes

Step 4: Identify sub-causes influencing the effect

Step 5: Identify detailed causes and analyze diagram

53

OPEN THE FILE: SURFACEFLAWS.MTW

The “Y”

The

“Problem”

The “X’s”

“Causes”

DEFINITION OF X–Y MATRIX

The (X–Y) CAUSE & EFFECT Matrix is:

A tool used to identify potential X’s and assess their relative

impact on multiple Y’s (including all Y’s that are customer

focused)

Based on the team’s collective “opinions”

Created for every project

Updated whenever a parameter is changed

To summarize, the X–Y is a team based prioritization tool for the

potential X’s

WARNING! This is not real data, this is organized brainstorming!!

At the conclusion of the project you may realize that the things

you thought were critical are in fact not as important as was

believed.

The X–Y Matrix is this Prioritization Tool!

54

THE X–Y MATRIX …EXAMPLE: Let’s take the example of the newspaper printing process. After

transforming the VOC we find the CTQ (Y’s) as below:

Clearly readable print

Good quality photo

Harmless to health

Upon brainstorming, the input process parameters (X’s) have been found as below:

Good quality ink

Less vibration during operation of printing press

Paper quality

For the above sets of X’s and Y’s the X-Y matrix table will look like the example

below:

Wherever there is a strong relation between X’s and Y’s, put 9. For weak relations put 3 or 1. Keep

the intersection field blank if there is significant relation.

Weighted sum for “Good quality ink” is calculated as 15*9 + 10*9 + 10*1 = 235.

55

Output Parameters (Y’s)

Clearly readable

print

Good Quality

Photo

Harmless to

health

Weighted

Sum of X’s

Weightage of Y’s 15 10 10

Input

Parameters

(X’s)

Good quality ink 9 9 1 235

Vibration less

operation of printing

press

9 3 165

Paper quality 1 1 9 115

SIX SIGMA

METHODOLOGY

DMAIC

MEASURE PHASE

SIX–SIGMA STATISTICS

56

WHAT IS STATISTICS?

1. Collecting Data

e.g. Survey

2. Presenting Data

e.g., Charts & Tables

3. Characterizing Data

e.g., Average

57

Why?Data

Analysis

Decision-

Making

© 1984-1994 T/Maker Co.

TYPES OF DATA58

Attribute Data (Qualitative)

Is always binary, there are only two possible values (0, 1)

1. Yes, No

2. Success, Failure

3. Go, No Go

4. Pass, Fall

Variable Data (Quantitative)

Discrete (Count) Data:

Can be categorized in a classification and is based on counts.

1. Number of defects

2. Number of defective units

3. Number of Customer Returns

Continuous Data:

Can be measured on a scale, it has decimal subdivisions that are

meaningful

1. Time, Pressure

2. Money

3. Material feed rate

DEFINITIONS OF SCALED DATA

59

TYPE OF

DATA

OPERATOR DESCRIPTION EXAMPLES

Nominal =, ≠ Categories Types of defects,

Types of colors

Ordinal <, > Rankings Severity of

defects: critical,

major, minor

Interval +, – Differences but

no absolute zero

Temperature of a

ship

Ratio / Absolute zero Pressure, Speed

STATISTICAL METHODS

60

STATISTICAL

METHODS

DESCRIPTIVE

STATISTICS

INFERENTIAL

STATISTICS

1. Involves

Collecting Data

Presenting Data

Characterizing Data

2. Purpose

Describe Data

1. Involves

Estimation

Hypothesis Testing

2. Purpose

Make Decisions About

Population Characteristics

DESCRIPTIVE ANALYSIS OF

QUALITATIVE DATA61

QUALITATIVE DATA

TABLES GRAPHS NUMBERS

One Way Table

Two–Ways Table

.

.

.

N – Ways Table

Bar Chart

Pie Chart

Multiple Bar Chart

Component Bar

Chart

Percentages

DESCRIPTIVE ANALYSIS OF

QUANTITATIVE DATA62

QUANTITATIVE DATA

TABLES GRAPHS NUMBERS

Frequency Distribution

Stem and Leaf Plot

Histogram

Box and Whisker’s Plot

Center

Important

Points

Variation

Mean

Median

Mode

Trimmed Mean

Median

Quartiles

Deciles

Percentiles

Range

Inter-Quartile Range

Variance

Standard Deviation

Skewness

Kurtosis

Distribution

Open “Pulse.mtw” ; Conduct Descriptive Analysis on the pulse1 data.

BOX & WHISKER’S PLOT

USING MINITAB

CONSTRUCTING BOX PLOT (One Y):

You want to examine the overall durability of your carpet products.

Samples of the carpet products are placed in four homes and you

measure durability after 60 days. Create a Box Plot to examine the

distribution of durability scores.

Open worksheet Carpet.mtw

63

Constructing Box Plot: (One Y–with Groups)

You want to assess the durability of four experimental carpet products.

Samples of the carpet products are placed in four homes and you

measure durability after 60 days. Create a box plot with median labels

and color-coded boxes to examine the distribution of durability for each

carpet product.

Open the worksheet CARPET.MTW

NORMAL DISTRIBUTION

64

Characteristics of the normal distribution:

Continuous distribution – Line does not break

Symmetrical distribution – Each half is a mirror of the other half

Asymptotic to the horizontal axis – it does not touch the x axis and goes

on forever

Unimodal – means the values mound up in only one portion of the graph

Area under the curve = 1; total of all probabilities = 1

Normal distribution is characterized by the mean and the Std Dev

Values of μ and σ produce a normal distribution

…2.71828

…3.14159=

Xofdeviationstandard

Xofmean

:

2

1

)(

2

2

1

e

Where

x

xf e

X

NORMALITY TEST

65

Open the worksheet CRANKSH.MTW

NORMALLY TEST:

o Generate a normal probability plot and performs a hypothesis test

to examine whether or not the observations follow a normal

distribution. For the normality test, the hypothesis are,

o Ho: Data follow a normal distribution Vs H1: Data do not follow a

normal distribution

o If ‘P’ value is > alpha; Accept Null Hypothesis (Ho)

PROCESS CAPABILITY66

The inherent ability of a process to meet the expectations of the

customer without any additional efforts. (or)

The ability of a process to meet product design/technical

specifications

– Design specifications for products (Tolerances)

upper and lower specification limits (USL, LSL)

– Process variability in production process

natural variation in process (3 from the mean)

Provides insight as to whether the process has a :

Centering Issue (relative to specification limits)

Variation Issue

A combination of Centering and Variation

Allows for a baseline metric for improvement.

TWO KINDS OF VARIABILITY

67

Common Cause / Inherent variability:-

Inherent in machine/process (design, construction and nature of

operation).

Special Cause / Assignable variability: –

Variability where causes can be identified.

Assignable variability eliminated / minimized by Process Capability

Study.

FOR A CAPABLE PROCESS:

INHERENT + ASSIGNABLE < TOLERANCE

PROCESS CAPABILITY ANALYSIS…

PROCESS CAPABILITY STUDY ASSUMPTIONS:

1. The performance measure data reflects statistical control when

plotted over a control chart (i.e.: X–Bar & Range Chart)

2. The performance measure data distributed normally.

Process Capability Index:

Cp — Measure of Potential Capability

6variationprocess

variationprocess LSLUSL

actual

allowable

Cp

68

Cp = 1

Cp < 1

Cp > 1

LSL USL Cp measures the

relationship between the

tolerance width and the

total range of process

variation.

Cp does not consider the

location of the mean and

therefore represents the

potential of the process to

produce characteristics

within specification.

PROCESS CAPABILITY ANALYSIS…

Process Capability Index:

Cpk — Measure of Actual Capability

69

“σ” is the standard deviation of the production process

Cpk considers both process variation () and process

location (X)

PROCESS CAPABILITY ANALYSIS…

Cpk takes into account any difference between the design target and the

actual process mean.

PROCESS CAPABILITY ANALYSIS:

EXAMPLE (Minitab)70

The length of a camshaft for an automobile

engine is specified at 600 + 2 mm. To avoid

scrap / rework, the control of the length of the

camshaft is critical.

The camshaft is provided by an external supplier.

Access the process capability for this supplier.

Filename: “camshaft.mtw”

Stat > quality tools > capability analysis

(normal)

Process Capability Indices & Sigma Quality Level

PROCESS CAPABILITY ANALYSIS:

EXERCISE (Minitab)–BOTH SUPPLIERS71

Histogram of camshaft length suggests mixed

populations.

Further investigation revealed that there are two

suppliers for the camshaft. Data was collected

over camshafts from both sources.

Are the two suppliers similar in performance?

If not, What are your recommendations?

FILENAME: “camshafts.mtw”

PROCESS CAPABILITY FOR

NON–NORMAL DATA72

To address non – normal data is to identify exact type of

distribution other than normal distribution

INDIVIDUAL IDENTIFICATION OF DISTRIBUTION

Use to evaluate the optimal distribution for your data

based on the probability plots and goodness-of-fit

tests prior to conducting a capability analysis study.

Choose from 14 distributions.

You can also use distribution identification to

transform your data to follow a normal distribution

using a Box–Cox transformation or a Johnson

transformation.

PROCESS CAPABILITY FOR

NON–NORMAL DATA73

EXAMPLE:

Suppose you work for a company that manufactures floor tiles and are concerned

about warping in the tiles. To ensure production quality, you measure warping in

10 tiles each working day for 10 days.

The distribution of the data is unknown. Individual Distribution Identification allows

you to fit these data with 14 parametric distributions and 2 transformations.

Open Worksheet: Tiles.mtw

PROCESS CAPABILITY FOR

NON–NORMAL DATA74

Distribution ID Plot for Warping

Box-Cox transformation: Lambda = 0.5

Johnson transformation function:

0.882908 + 0.987049 * Ln( ( X + 0.132606 ) / ( 9.31101 – X ) )

Goodness of Fit Test:

Distribution AD P LRT P

Normal 1.028 0.010

Box-Cox Transformation 0.301 0.574

Lognormal 1.477 <0.005

3-Parameter Lognormal 0.523 * 0.007

Exponential 5.982 <0.003

2-Parameter Exponential 3.892 <0.010 0.000

Weibull 0.248 >0.250

3-Parameter Weibull 0.359 0.467 0.225

Smallest Extreme Value 3.410 <0.010

Largest Extreme Value 0.504 0.213

Gamma 0.489 0.238

3-Parameter Gamma 0.547 * 0.763

Logistic 0.879 0.013

Loglogistic 1.239 <0.005

3-Parameter Loglogistic 0.692 * 0.085

Johnson Transformation 0.231 0.799

Best fit

distribution will

be having p–

value greater

than 0.05. But

The Best fit is

Johnson

Transformation

.

PROCESS CAPABILITY FOR

NON–NORMAL DATA75

PROCESS CAPABILITY FOR

NON–NORMAL DATA76

PROCESS CAPABILITY ANALYSIS:

EXERCISE (Minitab)–BOTH SUPPLIERS77

Histogram of camshaft length suggests mixed

populations.

Further investigation revealed that there are two

suppliers for the camshaft. Data was collected

over camshafts from both sources.

Are the two suppliers similar in performance?

If not, What are your recommendations?

FILENAME: “camshafts.mtw”

MODULE # 5

78 DMAIC METHODOLOGY:

ANALYSIS PHASE

Testing of Hypothesis for Variable data

Testing of Hypothesis for Attribute data

Scatter Plot

Linear Regression & Correlation

STATISTICAL METHODS

Statistical

Methods

Descriptive

Statistics

Inferential

Statistics

Estimation

Hypothesis

Testing

79

Inferential Statistics – To draw inferences about the process or

population being studied by modeling patterns of data in a way that

account for randomness and uncertainty in the observations. 2

2. Wikipedia.com

HYPOTHESIS TESTING

80

Population

I believe the

population mean

age is 50

(hypothesis).

Mean

X = 20

Reject

hypothesis! Not

close.

Random

sample

Hypothesis means A Belief

about a Population Parameter

NULL & ALTERNATIVE

HYPOTHESIS81

The hypotheses to be tested consists of two complementary statements:

1) The null hypothesis (denoted by H0) is a statement about the value of a

population parameter; it must contain the condition of equality.

2) The alternative hypothesis (denoted by H1) is the statement that must be true

if the null hypothesis is false. e.g.:

H0: μ = some value vs H1: μ ≠ some value

H0: μ ≤ some value vs H1: μ > some value

H0: μ ≥ some value vs H1: μ < some value

What Do You Do? If You have:

Different types of Materials. (Stainless, Carbon Steel & Aluminum)

Different types of oils. (Shell & Mobil)

Different type of Cleaning solutions. (Hydrocarbon & Water base)

You want to find which method of cleaning yield the best results for all

these materials?

SAMPLING RISK

82 α – Risk, also referred as Type I Error or Producer’s Risk:

Is the risk of rejecting H0 when H0 is true.

i.e. concluding that the process has drifted when it really has not.

β – Risk, also referred to Type II Error or Consumer’s Risk:

Is the risk of accepting H0 when H0 is false.

i.e. failing to detect the drift that has occurred in a process.

HYPOTHESIS STATEMENT:

H0 : μ = some value

H1 : μ ≠ some value

Criteria for “Accepting” & “Rejecting” a Null

Hypothesis: 1. For any fixed α, an increase in

the sample size will cause a

decrease in β.

2. For any fixed sample size, a

decrease in α will cause an

increase in β. Conversely, an

increase in α will cause a

decrease in β.

3. To decrease both α and β,

increase the sample size.

What is P-Value?83

This is the probability that a value as extreme as X–Bar (i.e.

≥ X–Bar) is observed, given that H0 is true. We reject H0 if

the obtained P-Value is less than α.

Interpreting P-Value:

H0 : μ = 5

H1 : μ ≠ 5

α = 0.05

A low p-value for the statistical test points to

rejection of Null hypothesis because it indicates

how unlikely it is that a test statistic as extreme as

or more extreme than a observed from this

population if Null Hypothesis is true.

If a p-value = 0.005, this means that if the

population means were equal (as hypothesized),

there is only 5 in 1000 chance that a more

extreme test statistic would be obtain using data

from this population and there is significant

evidence to support the Alternative Hypothesis

(H1).

P-value ≥ α, Accept Ho

P-value < α, Reject Ho

ANALYZE PHASE

HYPOTHESIS TESTING FOR

CONTINUOUS DATA

84

STATISTICAL INFERENCE

85

Comparing Two

groups

Data Normally

Distributed

Equality of Variances

Equal Variances if P

≥ 0.05

Unequal Variances if

P<0.05

Indep. Samp. T Tests Indep. Samp. T Tests

(Weltch

Approximation)

Comparing one

group with a Target

One Sample

Measured once

One Sample

Measured Twice

Data Distribution

Normal Data (P≥0.05)

One sample T Test

Data Distribution

Normal (P≥0.05)

Paired Sample T-Test

Comparing More

than Two groups

Data Distribution

One Way Anova Test *Welch Test

Testing of Hypothesis

Decision Making

1.Data is Normal when p ≥ 0.05 ,Use Anderson test

2.The Variance of groups are equal when p ≥ 0.05 Use the Levenes Test

3.Accept the Null Hypothesis when P≥0.05 otherwise accept the alternative hypothesis

Levenes Test

Normal (P≥0.05)

Equality of Variances

Equal Variances if P

≥ 0.05

Unequal Variances if

P<0.05

Levenes Test

* Not Available in Minitab

Testing of Hypothesis for Variable Data

TEST OF MEANS (t-tests): 1 Sample t

Measurements were made on nine widgets. You know that the distribution of

widget measurements has historically been close to normal, but suppose that

you do not know Population Standard deviation. To test if the population mean

is 5 and to obtain a 90% confidence interval for the mean, you use a t-

procedure.

1. Open the worksheet EXH_STAT.MTW.

2. Check the Normality of the data using Normality Test “VALUE”.

3. Choose Stat > Basic Statistics > 1-Sample t.

4. In Samples in columns, enter Values.

5. Check Perform hypothesis test. In Hypothesized mean, enter 5.

6. Click Options. In Confidence level, enter 90. Click OK in each dialog

box.

86

Target

A 1-sample t-test is used to compare an

expected population Mean to a target.

μsample

TEST OF MEANS (t-tests): 2-Sample

(Independent) t Test

Practical Problem:

We have conducted a study in order to determine the effectiveness of a new heating

system. We have installed two different types of dampers in home ( Damper = 1 and

Damper = 2).

We want to compare the BTU.In data from the two types of dampers to determine if

there is any difference between the two products.

Open the MINITABTM worksheet: “Furnace.MTW”

Statistical Problem:

Ho:μ1 = μ2

Ha:μ1 ≠ μ2

2-Sample t-test (population Standard Deviations unknown).

α = 0.05

87

No, not that kind of damper!

2-Sample (Independent) t Test:

Follow the Roadmap…88

NORMALITY TEST

2-Sample (Independent) t Test:

Follow the Roadmap…

89

TEST OF EQUAL VARIANCE

Stat ANOVA Test for

Equal Variances…

Damper

95% Bonferroni Confidence Intervals for StDevs

2

1

4.03.53.02.52.0

Damper

BTU.In

2

1

2015105

F-Test

0.996

Test Statistic 1.19

P-Value 0.558

Levene’s Test

Test Statistic 0.00

P-Value

Test for Equal Variances for BTU.In

Sample 1

Sample 2

2-Sample (Independent) t Test:

Equal Variance90

There is no difference between

the dampers for BTU’s in.

2-Sample (Independent) t

Test: EXERCISE91

A bank with a branch located in a commercial district of a city has the business

objective of developing an improved process for serving customers during the

noon- to-1 P.M. lunch period. Management decides to first study the waiting

time in the current process. The waiting time is defined as the time that

elapses from when the customer enters the line until he or she reaches the

teller window. Data are collected from a random sample of 15 customers, and

the results (in minutes) are as follows (and stored in Bank-I):

4.21 5.55 3.02 5.13 4.77 2.34 3.54 3.20

4.50 6.10 0.38 5.12 6.46 6.19 3.79

Suppose that another branch, located in a residential area, is also concerned

with improving the process of serving customers in the noon-to-1 P.M. lunch

period. Data are collected from a random sample of 15 customers, and the

results are as follows (and stored in Bank-II):

9.66 5.90 8.02 5.79 8.73 3.82 8.01 8.35

10.49 6.68 5.64 4.08 6.17 9.91 5.47

Is there evidence of a difference in the mean waiting time between the two

branches? (Use level of significance = 0.05)

PARAMETRIC STATISTICAL

INFERENCE92

Comparing Two

groups

Data Normally

Distributed

Equality of Variances

Equal Variances if P

≥ 0.05

Unequal Variances if

P<0.05

Indep. Samp. T Tests Indep. Samp. T Tests

(Weltch

Approximation)

Comparing one

group with a Target

One Sample

Measured once

One Sample

Measured Twice

Data Distribution

Normal Data (P≥0.05)

One sample T Test

Data Distribution

Normal (P≥0.05)

Paired Sample T-Test

Comparing More

than Two groups

Data Distribution

One Way Anova Test *Welch Test

Testing of Hypothesis

Decision Making

1.Data is Normal when p ≥ 0.05 ,Use Anderson test

2.The Variance of groups are equal when p ≥ 0.05 Use the Levenes Test

3.Accept the Null Hypothesis when P≥0.05 otherwise accept the alternative hypothesis

Levenes Test

Normal (P≥0.05)

Equality of Variances

Equal Variances if P

≥ 0.05

Unequal Variances if

P<0.05

Levenes Test

* Not Available in Minitab

TEST OF MEANS (t-tests):

PAIRED T-TEST

Practical Problem:

We are interested in changing the sole material for a popular brand of

shoes for children.

In order to account for variation in activity of children wearing the

shoes, each child will wear one shoe of each type of sole material. The

sole material will be randomly assigned to either the left or right shoe.

Statistical Problem:

Ho: μδ = 0

Ha: μδ ≠ 0

Paired t-test (comparing data that must remain paired).

α = 0.05

93

Just checking your souls,

er…soles!

EXH_STAT.MTW

TEST OF MEANS (t-tests):

PAIRED T-TEST94

NORMALITY TEST: “Delta”

Calc Calculator

AB Delta

Percent

1.51.00.50.0-0.5

99

95

90

80

70

60

50

40

30

20

10

5

1

Mean

0.622

0.41

StDev 0.3872

N 10

A D 0.261

P-Value

Probability Plot of AB Delta

Normal

TEST OF MEANS (t-tests):

PAIRED T-TEST

Analyze this data is to use the paired t-test command.

95

Stat Basic Statistics Paired T-test

Paired T-Test and CI: Mat-A, Mat-B

Paired T for Mat-A – Mat-B

N Mean StDev SE Mean

Mat-A 10 10.6300 2.4513 0.7752

Mat-B 10 11.0400 2.5185 0.7964

Difference 10 -0.410000 0.387155 0.122429

95% CI for mean difference: (-0.686954, -0.133046)

T-Test of mean difference = 0 (vs not = 0): T-Value = -3.35 P-Value = 0.009

The P-value of from this

Paired T-Test tells us the

difference in materials is

statistically significant.

EXERCISE: PAIRED T-TEST

96

Nine experts rated two brands of Colombian coffee in a taste-testing experiment. A

rating on a 7- point scale (1 = extremely unpleasing, 7 = extremely pleasing) is given

for each of four characteristics: taste, aroma, richness, and acidity. The following

data (stored in coffee) display the ratings accumulated over all four characteristics.

Brand

Expert A B

C.C. 24 26

S.E. 27 27

E.G. 19 22

B.L. 24 27

C.M. 22 25

C.N. 26 27

G.N. 27 26

R.M. 25 27

P.V. 22 23

At the 0.05 level of significance, is there evidence of a difference in the mean

ratings between the two brands?

ANOVA: EXAMPLE

We have three potential suppliers that claim to have equal levels of quality. Supplier

B provides a considerably lower purchase price than either of the other two vendors.

We would like to choose the lowest cost supplier but we must ensure that we do not

effect the quality of our raw material.

97

Supplier A Supplier B Supplier C

3.16 4.24 4.58

4.35 3.87 4.00

3.46 3.87 4.24

3.74 4.12 3.87

3.61 3.74 3.46

We would like test the data to determine whether there is a difference between the

three suppliers.

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

FOLLOW THE ROADMAP…TEST FOR NORMALITY

98

Supplier C

Percent

5.04.54.03.53.0

99

95

90

80

70

60

50

40

30

20

10

5

1

Mean

0.910

4.03

StDev 0.4177

N 5

AD 0.148

P-Value

Probability Plot of Supplier C

Normal

Supplier B

Percent

4.504.254.003.753.50

99

95

90

80

70

60

50

40

30

20

10

5

1

Mean

0.385

3.968

StDev 0.2051

N 5

AD 0.314

P-Value

Probability Plot of Supplier B

Normal

Supplier A

Percent

4.54.03.53.02.5

99

95

90

80

70

60

50

40

30

20

10

5

1

Mean

0.568

3.664

StDev 0.4401

N 5

AD 0.246

P-Value

Probability Plot of Supplier A

Normal All three suppliers samples are

Normally Distributed.

Supplier A P-value = 0.568

Supplier B P-value = 0.385

Supplier C P-value = 0.910

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

TEST FOR EQUAL VARIANCE

STACK DATA FIRST:

Data stack Columns…

99

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

TEST FOR EQUAL

VARIANCE…

100

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

ANOVA Using Minitab

101

Click on “Graphs…”,

Check “Boxplots of data”

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

Data

Supplier CSupplier BSupplier A

4.6

4.4

4.2

4.0

3.8

3.6

3.4

3.2

3.0

Boxplot of Supplier A, Supplier B, Supplier C

ANOVA: Session window

102

Test for Equal Variances: Suppliers vs ID

One-way ANOVA: Suppliers versus ID

Analysis of Variance for Supplier

Source DF SS MS F P

ID 2 0.384 0.192 1.40 0.284

Error 12 1.641 0.137

Total 14 2.025

Individual 95% CIs For Mean

Based on Pooled StDev

Level N Mean StDev ———-+———+———+——

Supplier 5 3.6640 0.4401 (———–*———–)

Supplier 5 3.9680 0.2051 (———–*———–)

Supplier 5 4.0300 0.4177 (———–*———–)

———-+———+———+——

Pooled StDev = 0.3698 3.60 3.90 4.20

Normal data P-value > .05

No Difference

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

ANOVA Assumptions

In one-way ANOVA, model adequacy can be checked by either of the following:

1. Check the data for Normality at each level and for homogeneity of variance across all

levels.

2. Examine the residuals (a residual is the difference in what the model predicts and the

true observation).

i. Normal plot of the residuals

ii. Residuals versus fits

iii. Residuals versus order

103

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

104

Residual

Frequency

0.60.40.20.0-0.2-0.4-0.6

5

4

3

2

1

0

Histogram of the Residuals

(responses are Supplier A, Supplier B, Supplier C)

The Histogram of

residuals should show a

bell shaped curve.

ANOVA Assumptions

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

Residual

Percent

1.00.50.0-0.5-1.0

99

95

90

80

70

60

50

40

30

20

10

5

1

Normal Probability Plot of the Residuals

(responses are Supplier A, Supplier B, Supplier C)

Normality plot of the

residuals should follow a

straight line.

Results of our example look

good.

The Normality assumption is

satisfied.

105

Fitted Value

Residual

4.054.003.953.903.853.803.753.703.65

0.75

0.50

0.25

0.00

-0.25

-0.50

Residuals Versus the Fitted Values

(responses are Supplier A, Supplier B, Supplier C)

The plot of residuals versus fits examines constant variance.

The plot should be structureless with no outliers present.

Our example does not indicate a problem.

ANOVA Assumptions

TEST FOR MORE THAN TWO MEANS

(F – Test): ANOVA…

ANOVA EXERCISE

EXERCISE OBJECTIVE: Utilize what you have learned to

conduct and analyze a one way ANOVA using MINITABTM.

You design an experiment to assess the durability of four experimental

carpet products. You place a sample of each of the carpet products in

four homes and you measure durability after 60 days. Because you

wish to test the equality of means and to assess the differences in

means, you use the one-way ANOVA procedure (data in stacked form)

with multiple comparisons. Generally, you would choose one multiple

comparison method as appropriate for your data.

1. Open the worksheet EXH_AOV.MTW.

2. Choose Stat > ANOVA > One-Way.

3. In Response, enter Durability. In Factor, enter Carpet.

4. Click OK in each dialog box.

106

HYPOTHESIS TESTING ROADMAP

ATTRIBUTE DATA

Attribute Data

One Factor Two Factors

One Sample

Proportion

Two Sample

Proportion

MINITABTM:

Stat – Basic Stats – 2 Proportions

If P-value < 0.05 the proportions

are different

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 the factors are not

independent

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 at least one

proportion is different

Two or More

Samples

Two

SamplesOne Sample

107

PROPORTION VERSUS A TARGET

This test is used to determine if the process proportion (p) equals some

desired value, p0.

The hypotheses:

H0: p = p 0

Ha: p ¹ p 0

The observed test statistic is calculated as follows: (normal approximation)

np1p

pp

Z

00

0

obs

ˆ

ONE SAMPLE PROPORTION (Z – Test)

108

PROPORTION VERSUS A TARGET

ONE SAMPLE PROPORTION (Z – Test)

A county district attorney would like to run for the office of state district attorney. She has

decided that she will give up her county office and run for state office if more than 65% of her

party constituents support her. You need to test H0: p = .65 versus H1: p > .65.

As her campaign manager, you collected data on 950 randomly selected party members and

find that 560 party members support the candidate. A test of proportion was performed to

determine whether or not the proportion of supporters was greater than the required

proportion of 0.65. In addition, a 95% confidence bound was constructed to determine the

lower bound for the proportion of supporters.

1. Choose Stat > Basic Statistics > 1 Proportion.

2. Choose Summarized data.

3. In Number of events, enter 560. In Number of trials, enter 950.

4. Check Perform hypothesis test. In Hypothesized proportion, enter 0.65.

5. Click Options. Under Alternative, choose greater than. Click OK in each dialog

box.

As P-Value > 0.05, mean accept H0; means

As her campaign manager, you would

advise her not to run for the office of state

district attorney.

109

EXAMPLE: You are the shipping manager and are in charge of improving

shipping accuracy. Your annual bonus depends on your ability to prove

that shipping accuracy is better than the target of 80%.

Out of 2000 shipments only 1680 were accurate.

• Do you get your annual bonus?

PROPORTION VERSUS A TARGET

ONE SAMPLE PROPORTION (Z – Test)

Choose Stat Basic Statistics 1 Proportion

Choose Summarized data.

110

HYPOTHESIS TESTING ROADMAP

ATTRIBUTE DATA

Attribute Data

One Factor Two Factors

One Sample

Proportion

Two Sample

Proportion

MINITABTM:

Stat – Basic Stats – 2 Proportions

If P-value < 0.05 the proportions

are different

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 the factors are not

independent

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 at least one

proportion is different

Two or More

Samples

Two

SamplesOne Sample

111

COMPARING TWO PROPORTIONS

This test is used to determine if the process defect rate (or

proportion, p) of one sample differs by a certain amount ‘D’

from that of another sample (e.g., before and after your

improvement actions)

The hypotheses:

H0: p1 – p2 = D

Ha: p1 – p2 = D

The test statistic is calculated as follows:

222111

21

obs

npˆ1pˆnpˆ1pˆ

Dpˆpˆ

Z

TWO SAMPLE PROPORTIONS (Z – Test)

112

Hypotheses:

H0: p1 – p2 = 0.0

Ha: p1 – p2 = 0.0

Two sample proportion test

Choose level of Significance = 5%

COMPARING TWO PROPORTIONS: EXAMPLE

As your corporation’s purchasing manager, you need to authorize the purchase of

twenty new photocopy machines. After comparing many brands in terms of price,

copy quality, warranty, and features, you have narrowed the choice to two: Brand

X and Brand Y. You decide that the determining factor will be the reliability of the

brands as defined by the proportion requiring service within one year of purchase.

Because your corporation already uses both of these brands, you were able to

obtain information on the service history of 50 randomly selected machines of

each brand. Records indicate that six Brand X machines and eight Brand Y

machines needed service. Use this information to guide your choice of brand for

purchase.

TWO SAMPLE PROPORTIONS (Z – Test)

113

COMPARING TWO PROPORTIONS

Choose Stat > Basic Statistics > 2 Proportions.

Choose Summarized data.

In First sample, under Events, enter 44. Under Trials, enter 50.

In Second sample, under Events, enter 42. Under Trials, enter 50. Click

OK.

TWO SAMPLE PROPORTIONS (Z – Test)

As Both P – Value > 0.05, Accepts H0; means the proportion of photocopy

machines that needed service in the first year did not differ depending on

brand. As the purchasing manager, you need to find a different criterion to

guide your decision on which brand to purchase.

114

HYPOTHESIS TESTING ROADMAP

ATTRIBUTE DATA

Attribute Data

One Factor Two Factors

One Sample

Proportion

Two Sample

Proportion

MINITABTM:

Stat – Basic Stats – 2 Proportions

If P-value < 0.05 the proportions

are different

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 the factors are not

independent

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 at least one

proportion is different

Two or More

Samples

Two

SamplesOne Sample

115

TWO OR MORE SAMPLE PROPORTIONS

(Chi–Square Test) “ONE FACTOR”….

Contingency Tables

The null hypothesis is that the population proportions of each group are the same.

H0: p1 = p2 = p3 = … = pn

Ha: at least one p is different

Statisticians have shown that the following statistic forms a chi-square distribution when

H0 is true:

Where “observed” is the sample frequency, “expected” is the calculated frequency based

on the null hypothesis, and the summation is over all cells in the table.

expected

expectedobserved

2

c

1j ij

2

ijij

r

1i

2

o

E

)E(O

χ

Chi-square Test:

Test Statistic Calculations

116

TWO OR MORE SAMPLE PROPORTIONS

(Chi–Square Test) “ONE FACTOR”….

Contingency Tables

117

The political affiliation of a certain city’s population is: Republicans 52%, Democrats

40%, and Independent 8%. A local university student wants to assess if the

political affiliation of the university students is similar to that of the population. The

student randomly selects 200 students and records their political affiliation.

1. Open the worksheet POLL.MTW.

2. Choose Stat Tables Chi-Square Goodness-of-Fit Test (One Variable).

As P-Value < 0.05; Reject H0; means the

political affiliation of the university

students is not the same as those of the

population.

HYPOTHESIS TESTING ROADMAP

ATTRIBUTE DATA

Attribute Data

One Factor Two Factors

One Sample

Proportion

Two Sample

Proportion

MINITABTM:

Stat – Basic Stats – 2 Proportions

If P-value < 0.05 the proportions

are different

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 the factors are not

independent

Chi Square Test

(Contingency Table)

MINITABTM:

Stat – Tables – Chi-Square Test

If P-value < 0.05 at least one

proportion is different

Two or More

Samples

Two

SamplesOne Sample

118

Test for association (or dependency) between two

classifications

(Chi–Square Test) “TWO FACTORS”….

Contingency Tables

119

Exercise objective: To practice solving problem presented using

the appropriate Hypothesis Test.

You are the quotations manager and your team thinks that the reason

you don’t get a contract depends on its complexity.

You determine a way to measure complexity and classify lost contracts

as follows:

1. Write the null and alternative hypothesis.

2. Does complexity have an effect?

Low Med High

Price 8 10 12

Lead Time 10 11 9

Technology 5 9 16

Test for association (or dependency) between

two classifications

(Chi–Square Test) “TWO FACTORS”….

Contingency Tables

120

First we need to create a table in

MINITABTM

Secondly, in MINITABTM perform a

Chi-Square Test

Test for association (or dependency) between

two classifications

(Chi–Square Test) “TWO FACTORS”….

Contingency Tables

121

Are the factors independent of

each other?

Yes; Both factors are independent

ANALYZE PHASE…

SCATTER PLOT, CORRELATION

AND

SIMPLE & MULTIPLE

REGRESSION ANALYSIS

122

SCATTER PLOT

WHAT IS A SCATTER PLOT?

Is a graphical presentation of any possible relationship between

two sets of variables by a simple X-Y plot, which may or may

not be dependent.

123

EXAMPLE: You are interested in how well your company’s camera batteries are

meeting customers’ needs. Market research shows that customers become annoyed

if they have to wait longer than 5.25 seconds between flashes.

You collect a sample of batteries that have been in use for varying amounts of time

and measure the voltage remaining in each battery immediately after a flash

(VoltsAfter), as well as the length of time required for the battery to be able to flash

again (flash recovery time, FlashRecov). Create a scatter plot to examine the

results. Include a reference line at the critical flash recovery time of 5.25 seconds.

Open the worksheet BATTERIES.MTW

SCATTER PLOT

EXAMPLE… :

124

SCATTER PLOT

INTERPRETING THE RESULTS:

As expected, the lower the voltage in a battery after a

flash, the longer the flash recovery time tends to be.

The reference line helps to illustrate that there were many

flash recovery times greater than 5.25 seconds.

125

CORRELATION

Correlation analysis is a method that is used to measure the strength of

the linear relationship between two or more continuous variables.

126

When ‘r’ is close to +1, there is a strong positive correlation.

When ‘r’ is close to Zero, there is very little or no correlation.

When ‘r’ is close to –1, there is a strong negative correlation.

H0: There is No Correlation

H1: There is Correlation

SCATTER PLOT & CORRELATION

EXAMPLE: You are interested in how well your company’s

camera batteries are meeting customers’ needs. Market

research shows that customers become annoyed if they have to

wait longer than 5.25 seconds between flashes.

You collect a sample of batteries that have been in use for

varying amounts of time and measure the voltage remaining in

each battery immediately after a flash (VoltsAfter), as well as

the length of time required for the battery to be able to flash

again (flash recovery time, FlashRecov). Create a scatter plot to

examine the results. Include a reference line at the critical flash

recovery time of 5.25 seconds.

Open the worksheet BATTERIES.MTW

127

SCATTER PLOT & CORRELATION

EXAMPLE…:

Correlations: FlashRecov, VoltsAfter

Pearson correlation of FlashRecov and VoltsAfter = -0.478

P-Value = 0.002

GENERAL GUIDELINE:

+/-0.8 imply a good

correlation

Hypothesized statement:

H0 : No correlation between the 2 variables

H1 : Significant correlation between the 2 variable

128

SCATTER PLOT &

CORRELATION…

EXERCISE#2:

The following information taken

from annual report of a company

shows net sales (NS) & working

capital (WC).

a) Plot the variable NS and WC in

scatter plot. Format, what, if

any kind of relationship

appears to exist between

them.

b) Compute the correlation

coefficient between NS & WC.

YEAR NET SALES WORKING

CAPITAL

1989 234463 67168

1990 281462 69788

1991 294030 75306

1992 286495 84740

1993 318930 97343

1994 356595 108601

1995 418152 118550

1996 473103 145069

1997 502875 146975

1998 557840 141268

129

WHAT IS REGRESSION?

Method of determining the statistical relationship between a response

(or output) and one or more predictor (or input) variables.

Y = ƒ (X1, X2, . . . . Xn)

Where ‘Y’ is the RESPONSE and X1 to Xn are the PREDICTORS

130

Simple Linear Regression…

Is when the dependent variable is linearly proportional to just ONE

independent variable.

Multiple Regression…

May be viewed as an extension of simple regression analysis (where only

one predictor is involved) to the situation where there is more than ONE

predictor to be considered.

TYPES OF REGRESSION

SIMPLE LINEAR

REGRESSION

EXAMPLE:

A study was conducted with

vegetarians to see whether the

number of grams of protein

each ate per day was related to

diastolic blood pressure. The

data are given here. If there is

a significant relationship,

predict the diastolic pressure of

a vegetarian who consumers 8

grams of protein per day.

131

SIMPLE LINEAR

REGRESSIONEXAMPLE…

Regression Analysis: Pressure versus Grams

The regression equation is

Pressure = 64.9 + 2.66 Grams

Predictor Coef SE Coef T P

Constant 64.936 3.401 19.09 0.000

Grams 2.6623 0.4408 6.04 0.001

S = 2.84522 R-Sq = 83.9% R-Sq(adj) = 81.6%

Analysis of Variance

Source DF SS MS F P

Regression 1 295.33 295.33 36.48 0.001

Residual Error 7 56.67 8.10

Total 8 352.00

REGRESSION EQUATION

Coefficient of Determination-”R-Sq” is

the measure of the fit of the Regression

to the data. It suggest a very good fit

when R-Sq approach 100%

The F-test is a test of the hypothesis……

H0: All Regression coefficients, except b0 are Zero

H1: The Regression is Statistically significant

132

SIMPLE LINEAR

REGRESSION

EXERCISE:

A study is conducted to

determine the relationship

between a driver’s age and

the number of accidents he

or she has over a one-year

period. If there is a

significant relationship,

predict the number of

accidents of a driver who is

28.

Driver’s

Age

No. of

accidents

16 3

24 2

18 5

17 2

23 0

27 1

32 1

133

MULTIPLE LINEAR

REGRESSION134

(MLR) model: Y = β0 + β1 X1 + β2 X2 …….

Where X’S is the predictor (independent) variables

Y is the response (dependent) variable

β0 is the intercept

β1, β2… are the slopes for the respective predictors

EXAMPLE: As part of a test of solar thermal energy, we need to measure

the total heat flux from homes. We wish to examine whether total heat flux

(HeatFlux) can be predicted by the position of the focal points in the east,

south, and north directions.

We will evaluate the three-predictor (three input variables; east, south and

north) model using multiple regression.

1. Open the worksheet EXH_REGR.MTW

2. Choose Stat > Regression > Regression.

3. In Response, enter HeatFlux.

4. In Predictors, enter East South North.

5. Click OK in each dialog box.

MULTIPLE LINEAR REGRESSION

135 EXAMPLE…:

Results for: Exh_regr.MTW

Regression Analysis: HeatFlux versus East, South, North

The regression equation is

HeatFlux = 389 + 2.12 East + 5.32 South – 24.1 North

Predictor Coef SE Coef T P

Constant 389.17 66.09 5.89 0.000

East 2.125 1.214 1.75 0.092

South 5.3185 0.9629 5.52 0.000

North -24.132 1.869 -12.92 0.000

S = 8.59782 R-Sq = 87.4% R-Sq(adj) = 85.9%

The p-values for the estimated

coefficients of North and South are

both 0.000, indicating that they are

significantly related to HeatFlux.

The R-Square value indicates

87.4% of the variance in HeatFlux

is due to the predictors

MODULE # 6

136 DMAIC METHODOLOGY:

IMPROVE PHASE

Design of Experiment: An Introduction

2K Factorial Design

PROJECT STATUS REVIEW

137

1. Understand our problem and it’s impact on the business.

(DEFINE)

2. Established firm objectives / goals for improvement.

(DEFINE)

3. Quantified our output characteristic. (DEFINE)

4. Validated the measurement system for our output

characteristic. (MEASURE)

5. Identified the process input variables in our process.

(Measure)

6. Narrowed our input variables to the potential “X’s” through

statistical Analysis. (ANALYZE)

7. Selected the Vital few X’s to optimize the output response(s).

(IMPROVE)

8. Quantified the relationship of the Y’s to the X’s with Y = f(x).

(IMPROVE)

WHAT IS EXPERIMENT?

138

In statistics, an experiment refers to any process that generates a set of

data.

An experiment involves a test or series of test in which purposeful

changes are made to the input variables of a process or system so that

changes in the output responses can be observed and identified.

Noise Factors

TERMINOLOGIES

139

Terms used in Design of Experiments (DOE) need to defined, these are:

RESPONSE:

A measurable outcome of interest, e.g.: yield, strength, etc.

FACTORS:

Controllable variables that are deliberately manipulated to determine

their individual and joint effects on the response(s), OR Factors are

those quantities that affect the outcome of an experiment, e.g.:

temperature, time, etc.

LEVELS:

Levels refer to the values of factors for which the data is gathered,

“values that factor will take in an experiment”, e.g.:

Level–1 for time = 2hours

Level–2 for time = 3 hours

TREATEMENT:

A set of specified factor levels for an experimental run, e.g.:

Treatment–1: time = 2hrs and temperature = 1750 C

Treatment–2: time = 3hrs and temperature = 2250 C

EXAMPLES

140

EXAMPLE–1:

In a MACHING PROCESS

RESPONSE: Surface Finish “Y”

FACTORS: Speed of machine “XA” & Depth of Cut “XB”

LEVELS: High & Low

EXAMPLE–2:

In a POPCORN MAKING PROCESS

RESPONSE: Volume (ml) Yield of Popcorn “Y”

FACTORS: Type of Popper “XA” & Grade of corn used

“XB”

LEVELS: Air, and Oil & Budget, Regular and luxury

TYPES OF EXPERIMENTS

141

EXPERIMENTS

ONE-FACTOR AT A TIME

EXPRIMENTS

BEST GUESS

EXPERIMENTS

FACTORIAL

EXPERIMENTS

2K FACTORIAL

142

2K Factorial Designs are experiments where all

FACTORS have only TWO LEVELS

The number of combinations (Runs) for Full

Factorial Design is denoted as n = 2k (where

k=number of Factors)

2K

Factors

Levels

22 FACTORIAL

EXPERIMENTAL DESIGN143

EXAMPLE: Consider the manufacture of a product, for use

in the making of paint, in a batch process. Fixed amounts of raw

material are heated under pressure in rector-1 for a fixed period

of time and the product is then recovered. Currently the process

is operated at temperature 225o C and pressure 4.5 bar. As part

of Six Sigma project, aimed at increasing product yield, a 22

factorial experiment with two replications was planned. Yields

are typically around 90 Kg. It was decided after discussion

amongst the project team to use the levels 200o C and 250o C

for temperature and level 4.0 bar and 5.0 bar for pressure.

RESPONSE: Product Yield “Y”

FACTORS: Temperature “XA” & Pressure “XB”

LEVELS: 200o C and 250o C & 4.0 bar and 5.0

bar

22 FACTORIAL

EXPERIMENTAL DESIGN144

EXAMPLE…:

22 FACTORIAL

EXPERIMENTAL DESIGN145

EXAMPLE…:

Stat > DOE > Factorial > Factorial Plots

22 FACTORIAL

EXPERIMENTAL DESIGN146

EXAMPLE…:

The Main Effect Plot indicate that:

On average, increasing temperature

from 200o C to 250o C increases yield

of product by 8 kg.

On average, increasing pressure from 4

bar to 5 bar decreases yield of product

by 6Kg.

The parallel lines indicate no temperature–

Pressure interaction here.

22 FACTORIAL

EXPERIMENTAL DESIGN147

EXAMPLE…:

Stat > DOE > Factorial > Analyze Factorial Design…

Factorial Fit: Yield versus Temperature, Pressure

Estimated Effects and Coefficients for Yield (coded units)

Term Effect Coef SE Coef T P

Constant 92.000 0.9354 98.35 0.000

Temperature 8.000 4.000 0.9354 4.28 0.013

Pressure -6.000 -3.000 0.9354 -3.21 0.033

Temperature*Pressure 0.000 -0.000 0.9354 -0.00 1.000

S = 2.64575 PRESS = 112

R-Sq = 87.72% R-Sq(pred) = 50.88% R-Sq(adj) = 78.51%

The P–Value indicate

that both temperature &

pressure have a real

effect on Yield.

22 FACTORIAL

EXPERIMENTAL DESIGN148

EXAMPLE…:

22 FACTORIAL

EXPERIMENTAL DESIGN149

EXERCISE:

An Engineer desire to study which is

the 2 Factors determined that affect

the Defect Rate in his production

line.

FACTORS:

Temperature & Pressure

LEVELS:

Temperature – 60 & 70o C &

Pressure – 3.0 & 5.5 Bar

REPLICATES: 3

DEFECT

3.93183

2.30259

0.0000

2.07944

4.33073

3.33220

2.39790

0.69315

2.19722

2.83321

1.38629

1.38629

23 FACTORIAL

EXPERIMENTAL DESIGN150

EXAMPLE: A plastic manufacturing company had formed a

work improvement company had formed a work

improvement team consisting of engineers from different

department. The team objective is to strive to improve the

yield of a coating process. After a series of brainstorming

session, the team determined that the following are the

deciding factors and levels:

A: Temperature: 400o F and 450o F

B: Catalyst Con.: 10% and 20%

C: Processing Ramp time: 45 seconds and 90

seconds

The design is a 23 factorial and each run (treatment) is

replicated 3 times and total is 24 randomized trial.

23 FACTORIAL

EXPERIMENTAL DESIGN151

EXAMPLE (Cont…):

23 FACTORIAL

EXPERIMENTAL DESIGN152

EXAMPLE (Cont…):

Stat > DOE > Factorial > Factorial Plots

23 FACTORIAL

EXPERIMENTAL DESIGN153

EXAMPLE (Cont…):

23 FACTORIAL

EXPERIMENTAL DESIGN154

EXAMPLE (Cont…):

23 FACTORIAL

EXPERIMENTAL DESIGN155

EXAMPLE (Cont…):

Stat > DOE > Factorial > Analyze Factorial Design…

23 FACTORIAL

EXPERIMENTAL DESIGN156

EXAMPLE (Cont…):

MODULE # 7

157 DMAIC METHODOLOGY:

CONTROL PHASE

SPC: An Introduction

Attribute Control Charts

Variable Control Charts

Control Plan

INTRODUCTION TO SPC

In 1924, Shewhart applied the terms of “assignable-cause” and “chance-cause”

variation and introduced the “control chart” as a tool for distinguishing between

the two.

Central to an SPC program are the following:

Understand the causes of variability:

Shewhart found two basic causes of variability:

Chance causes of variability

Assignable causes of variability

158

OBJECTIVES OF SPC CHARTS

All control charts have one primary purpose!

To detect assignable causes of variation that cause significant process shift, so that:

investigation and corrective action may be undertaken to rid the process of the

assignable causes of variation before too many nonconforming units are

produced. In other words, to keep the process in statistical control.

The following are secondary objectives or direct benefits of the primary objective:

To reduce variability in a process.

To Help the process perform consistently & predictably.

To help estimate the parameters of a process and establish its process capability.

CONTROL CHART ANATOMY

159

Common Cause

Variation

Process is “In

Control”

Special Cause

Variation

Process is “Out

of Control”

Special Cause

Variation

Process is “Out

of Control”

Run Chart of

data points

Process Sequence/Time Scale

Lower Control

Limit

Mean

+/-3sigma

Upper Control

Limit

INTERPRETING CONTROL CHART

160

TYPES & SELECTION OF

CONTROL CHART161

What type of

data do I

have?

Variable Attribute

Counting defects

or defectives?

X-bar &

S Chart

I & MR

Chart

X-bar &

R Chart

n > 10 1 < n < 10 n = 1

Defectives Defects

What subgroup

size is available?

Constant

Sample Size?

Constant

Opportunity?

yes yesno no

P or np

Chart

u Chartp Chart c or u

Chart

Note: A defective unit can

have more than one defect.

Calculate the parameters of the “P”

Control Charts with the following:162

Where:

p: Average proportion defective (0.0 – 1.0)

ni: Number inspected in each subgroup

LCLp: Lower Control Limit on P Chart

UCLp: Upper Control Limit on P Chart

inspecteditemsofnumberTotal

itemsdefectiveofnumberTotal

p

in

pp )1(

3pUCLp

Center Line Control Limits

in

pp )1(

3pLCLp

Since the Control Limits are a function of sample

size, they will vary for each sample.

CONTROL CHARTS FOR ATTRIBUTE DATA

163

P Chart With constant sample size: EXAMPLE

Frozen orange juice concentrate is packed in 6- oz cardboard cans. A metal

bottom panel is attached to the cardboard body. The cans are inspected for

possible leak. 20 samplings of 50 cans/sampling were obtained. Verify if the

process is in control.

Choose Stat > Control Charts >Attributes

Charts > P

CONTROL CHARTS FOR ATTRIBUTE DATA…

164

P Chart With Variable sample size: EXAMPLE

Suppose you work in a plant that manufactures picture

tubes for televisions. For each lot, you pull some of the

tubes and do a visual inspection. If a tube has scratches on

the inside, you reject it. If a lot has too many rejects, you

do a 100% inspection on that lot. A P chart can define

when you need to inspect the whole lot.

1. Open the worksheet EXH_QC.MTW.

2. Choose Stat > Control Charts >Attributes Charts >

P.

3. In Variables, enter Rejects.

4. In Subgroup sizes, enter Sampled. Click OK.

Calculate the parameters of the “np”

Control Charts with the following:

165

Center Line Control Limits

Since the Control Limits AND Center Line are a function

of sample size, they will vary for each sample.

subgroupsofnumberTotal

itemsdefectiveofnumberTotal

pn )1(3pnUCL inp ppni

p)-p(1n3pnLCL iinp

Where:

np: Average number defective items per subgroup

ni: Number inspected in each subgroup

LCLnp: Lower Control Limit on nP chart

UCLnp: Upper Control Limit on nP chart

ATTRIBUTE CONTROL CHARTS …

166

NP Chart: EXAMPLE

You work in a toy manufacturing company and your job is to

inspect the number of defective bicycle tires. You inspect

200 samples in each lot and then decide to create an NP

chart to monitor the number of defectives. To make the NP

chart easier to present at the next staff meeting, you decide

to split the chart by every 10 inspection lots.

1. Open the worksheet TOYS.MTW.

2. Choose Stat > Control Charts > Attributes Charts > NP.

3. In Variables, enter Rejects.

4. In Subgroup sizes, enter Inspected.

5. Click NP Chart Options, then click the Display tab.

6. Under Split chart into a series of segments for display

purposes, choose Number of subgroups in each segment and

enter10.

7. Click OK in each dialog box.

Calculate the parameters of the “c”

Control Charts with the following:

167

Center Line Control Limits

subgroupsofnumberTotal

defectsofnumberTotal

c c3cUCLc

c3cLCLc

Where:

c: Total number of defects divided by the total number of subgroups.

LCLc: Lower Control Limit on C Chart.

UCLc: Upper Control Limit on C Chart.

ATTRIBUTE CONTROL CHARTS …

168

C Chart: EXAMPLE

Suppose you work for a linen manufacturer. Each 100 square yards of

fabric can contain a certain number of blemishes before it is rejected. For

quality purposes, you want to track the number of blemishes per 100

square yards over a period of several days, to see if your process is

behaving predictably.

1. Open the worksheet EXH_QC.MTW.

2. Choose Stat > Control Charts > Attributes Charts > C.

3. In Variables, enter Blemish.

Calculate the parameters of the “u” Control

Charts with the following:

169

Center Line Control Limits

InspectedUnitsofnumberTotal

IdentifieddefectsofnumberTotal

u

in

u

3uUCLu

in

u

3uLCLu

Where:

u: Total number of defects divided by the total number of units inspected.

ni: Number inspected in each subgroup

LCLu: Lower Control Limit on U Chart.

UCLu: Upper Control Limit on U Chart.

Since the Control Limits are a function of

sample size, they will vary for each sample.

ATTRIBUTE CONTROL CHARTS

(Cont…)

170 U Chart: EXAMPLE

As production manager of a toy manufacturing company, you

want to monitor the number of defects per unit of motorized

toy cars. You inspect 20 units of toys and create a U chart to

examine the number of defects in each unit of toys. You

want the U chart to feature straight control limits, so you fix

a subgroup size of 102 (the average number of toys per

unit).

1. Open the worksheet TOYS.MTW.

2. Choose Stat > Control Charts > Attributes Charts > U.

3. In Variables, enter Defects.

4. In Subgroup sizes, enter Sample.

5. Click U Chart Options, then click the S Limits tab.

6. Under When subgroup sizes are unequal, calculate control

limits, choose Assuming all subgroups have size then enter

102.

7. Click OK in each dialog box.

Calculate the parameters of the X–Bar and R

Control Charts with the following:

171

Center Line Control Limits

k

x

X

k

1i

i

k

R

R

k

i

i

RAXUCL 2x

RAXLCL 2x

RDUCL 4R

RDLCL 3R

Where:

Xi: Average of the subgroup averages, it becomes the Center Line of the Control Chart

Xi: Average of each subgroup

k: Number of subgroups

Ri : Range of each subgroup (Maximum observation – Minimum observation)

Rbar: The average range of the subgroups, the Center Line on the Range Chart

UCLX: Upper Control Limit on Average Chart

LCLX: Lower Control Limit on Average Chart

UCLR: Upper Control Limit on Range Chart

LCLR : Lower Control Limit Range Chart

A2, D3, D4: Constants that vary according to the subgroup sample size

Rbar (computed above)

d2 (table of constants for subgroup size n) (st. dev. Estimate) =

Calculate the parameters of the X–Bar and S

Control Charts with the following:

172

Center Line Control Limits

k

x

X

k

1i

i

k

s

S

k

1i

i

SAXUCL 3x

SAXLCL 3x

SBUCL 4S

SBLCL 3S

Where:

Xi: Average of the subgroup averages, it becomes the Center Line of the Control Chart

Xi: Average of each subgroup

k: Number of subgroups

si : Standard Deviation of each subgroup

Sbar: The average S. D. of the subgroups, the Center Line on the S chart

UCLX: Upper Control Limit on Average Chart

LCLX: Lower Control Limit on Average Chart

UCLS: Upper Control Limit on S Chart

LCLS : Lower Control Limit S Chart

A3, B3, B4: Constants that vary according to the subgroup sample size

Sbar (computed above)

c4 (table of constants for subgroup size n) (st. dev. Estimate) =

VARIABLE CONTROL CHARTS

(Cont…)173

X–Bar & S Charts: EXAMPLE

You are conducting a study on the blood glucose levels of 9 patients

who are on strict diets and exercise routines. To monitor the mean and

standard deviation of the blood glucose levels of your patients, create

an X-Bar and S chart. You take a blood glucose reading every day for

each patient for 20 days.

1. Open the worksheet

BLOODSUGAR.MTW.

2. Choose Stat > Control Charts

> Variables Charts for

Subgroups > Xbar-S.

3. Choose All observations for a

chart are in one column,

then enter Glucoselevel.

4. In Subgroup sizes, enter 9.

Click OK.

Calculate the parameters of the Individual and

MR Control Charts with the following:

174

Center Line Control Limits

k

x

X

k

1i

i

k

R

RM

k

i

i

RMEXUCL 2x

RMEXLCL 2x

RMDUCL 4MR

RMDLCL 3MR

Where:

Xbar: Average of the individuals, becomes the Center Line on the Individuals Chart

Xi: Individual data points

k: Number of individual data points

Ri : Moving range between individuals, generally calculated using the difference

between each successive pair of readings

MRbar: The average moving range, the Center Line on the Range Chart

UCLX: Upper Control Limit on Individuals Chart

LCLX: Lower Control Limit on Individuals Chart

UCLMR: Upper Control Limit on moving range

LCLMR : Lower Control Limit on moving range

E2, D3, D4: Constants that vary according to the sample size used in obtaining the moving

range

MRbar (computed above)

d2 (table of constants for subgroup size n) (st. dev. Estimate) =

VARIABLE CONTROL CHARTS

(Cont…)175 I & MR Charts: EXAMPLE

As the distribution manager at a limestone quarry, you

want to monitor the weight (in pounds) and variation in the

45 batches of limestone that are shipped weekly to an

important client. Each batch should weight approximately

930 pounds. you want to examine the same data using an

individuals and moving range chart.

1. Open the worksheet EXH_QC.MTW

2. Choose Stat > Control Charts > Variables Charts for

Individuals > I-MR.

3. In Variables, enter Weight.

4. Click I-MR Options, then click the Tests tab.

5. Choose Perform all tests for special causes, then click OK in

each dialog box.

176

WHAT IS CONTROL

PLAN?

The Control Plan describes the actions

that are required at each phase of the

process to ensure that all process

outputs will be in state of control.

Control plan is a living document,

reflecting the current methods of

control, and measurement systems

used.

Accessible at work station

17

7

CONTROL PLAN: EXAMPLE

What to Check? How important it is …?

How to Check?

How many & When to Check?

What to do when some thing is wrong?

178

CONTROL PLAN

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