Understanding Six Sigma

Understanding Six Sigma

Understanding Six Sigma Manual

6σ

History of Six Sigma

Understanding Six Sigma

The U.S Defense system developed a system known as SQC to manage the complex weapon system & to handle the distributed defense contractors. SQC:- is a set of tools that originated in the Military Standards and the basis of SQC process was 3 sigma limits which yields a rate of 2700 defects per million. After World war 2 US companies returned to their Original strategy while the defeated countries were rebuilding their Industries. General Mc Arthur who was the Governor general of JAPAN at that time Imported some of the U.S. Pioneers of SQC to help train their counterparts in JAPAN. By 1970~ 1980 Japanese producers were renowned for their Quality & durability. U.S Companies slowly realized that to attain the desired Quality level two things are necessary One should be able to measure the quality level I.e it should be Quantifiable & Measurable. Motorola pioneered the Use of Six Sigma , Bill Smith VP &Senior QA Manager of Motorola is regarded as the Father of Six Sigma.

History of Six Sigma

Understanding Six Sigma

Quality Management & Six Sigma “ Six Sigma Management” --Lead by American Companies

• Small group of sections • Toward total Solution

Improve Product Quality

Quality ` Control

Six Sigma

Total Quality Management

Total Quality Control

“ Production line focussed Improvement “ --- Lead by Japanese Companies

Total Optimization of R&D, Production , Sales and Service is necessary

History of Six Sigma

Understanding Six Sigma Most of service called products were from reworked products at the factories

• Bill Smith Report

Hidden Factory and Rolled throughput Yield concept are Induced

• Actual Practice Strategy by Dr. Michael Harry • The Malcom Baldridge Award of 1988 of Motorola • After Motorola, Texas Instruments, ABB, Allied Signal , GE, LG Electronics, Polaroid, Nokia, Lockheed Martin, …. Sony started Six Sigma

Definitions Population Total Study Group

Understanding Six Sigma Sample Small Group taken from Population

The Group of Sample 100 people who possesses a Fan

Data The facts derived from the sample

Age and Number of decision makers purchasing a Fan

Definitions

Understanding Six Sigma

Data Point : The single entity in the sample. Data : The trend of data points in a sample I.e the facts derived from the sample. Information : The data presented in a form which conveys some result, Conclusion Sample : A Sample is a portion of the whole collection of Items (population) Population : The population consists of the set of all measurements in which the investigation is interested. It is also called Universe. Statistic : A numerical value such as standard deviation or mean , that characterizes the sample. Statistics : An application theory & method to reach appropriate & wise decisions in unknown circumstances.

Population Sample

The Characteristic of populaton : parameter

The Characteristic of Sample : Statistic

Understanding Six Sigma

Types of Data A.) Continuos Data : - The Data which can be measured and has unit associated with it is called continuous data. It can be in fractions. E.g Length of a Playground, Thickness of the paint coating, ect.

B) Attribute Data :- The data which can has only two options Yes/No, True/False is called Attribute data. E.g Quality of Food ( OK/NG),

C) Discreet Data : - The data which can be measured only in whole number and has no units associated with it is called Discreet data. It is the Count of the Number of Attributes E.g number of Heart Beats in one minute, Number of Type A defects.

Understanding Six Sigma

Characteristics of Normal Curve Normal Distribution Curve is known as Density Curve meaning the area under the curve is equal to one. A.) The Normal curve is a bell shaped curve and it has single peak (Mode ) at the center. B) The mean & median of the distribution are equal and are located at the peak. C) The Normal distribution curve is symmetrical USL about the mean. D) The curve is asymptotic I.e the curve gets close to X-Axis but it never touches it Standard Normal curve is one having Mean=0, and standard deviation =1.

x

LSL

Understanding Six Sigma

Measures of Central Tendency of Data A.) Mean : The mean of a set of Observations is their average. It is equal to the sum of all Observations divided by the number of Observations in the set. E.g Consider the data set given below

1 , 2 , 3 , 4 , 5 ; Mean = 1+ 2 + 3 + 4 + 5 = 15/5 = 3 Mean of Sample,

X = ΣX = n

i

x1 +x2 +x3 +x4+……..xn

I=1

n b.) Median : The Median term of the given data is given by n + 1 th term, where n is the number of Median = 2 Observations in the given data(arranged in increasing order).

E.g 3, 5 , 1 , 8 , 2 , 7 , 1 , 4 . No. of Terms = 8, Arrange data in Increasing Order = 1 , 1 , 2 , 3 , 4 , 5 , 7, 8

Median Term = 4.5th Term = 4th Term + 0.5 ( 5th - 4th Term ) = 3 + 0.5 ( 4- 3) = 3.5

Understanding Six Sigma

Measures of Central Tendency of Data C)

Mode

: The value Occurring maximum number of times

E.g 3, 4 , 3, 6, 5, 3, 7, 4 , Mode = 3

E.g Calculate the Median of the Following data : 2, 4, 6, 8, 12, 15, 10 Sol. Arrange the data in Increasing Order 2, 4, 6 , 8 , 10 , 12 , 15 No.of Observations = (7 + 1 ) / 2 = 4th Term = 8

Understanding Six Sigma

Measures of Spread/Dispersion of Data a.) Range : The Difference between Maximum & Minimum value. b) Standard Deviation ( σ ) : It gives tells us about the variation in data . c ) Variance ( σ 2 ) : It is defined as the square of the standard deviation to account for the total variation observed in the data. E.g : The Process Specification = 10 ± 2 Sol. USL = 12 , LSL =8 Range = USL - LSL = 12 - 8 = 4 Consider the data set 3, 2, 5, 1, 4 Mean = 3 Standard Deviation ( σ ) = (10/4 )1/2 = 1. 5 Variation ( σ 2 ) = 2.25

Data 1 2 3 4 5

Mean = 3 Variation Variation 2 -2 4 -1 1 0 0 1 1 2 4 0 10

Understanding Six Sigma

Measures of Spread/Dispersion of Data d) Quartile : It divides the total range into 4 equal parts (quarters ) and tells that in which Quartile a particular data point is lying.

Q1 = n + 1 4

th

, Q2 = 2 * n + 1

4

th

, Q3 = 3 * n + 1

th

4

where Q1,Q2 and Q3 are 1st 2nd & 3rd Quartiles resp. Interquartile Range ( IQR ) = Q3 Q1, contains 50 % of the Total data points

Upper Limit UL = Q3 + 1.5 IQR Lower Limit

LL = Q1 - 1.5 IQR

Eg. Calculate the First , second & Third Quartile for the data given below. Also calculate the IQR & Upper & Lower Limits. 10, 17, 14, 23, 18 , 11 , 9, 13 Sol . Arrange data in Increasing order = 9 , 10 , 11, 13, 14 , 17 , 18 , 23

Q1 = [ ( 8 +1) / 4 ] th = 2.25th Term = 2nd Term + 0.25 ( 3rd Term - 2nd Term ) = 10 + 0.25( 1) = 10.25 Q2= 2* 2.25th Term = 4.5th Term = 13 + 0.5 (14 - 13 ) = 13.5, Q3= 3* 2.25th Term = 6.75th Term = 17.75 IQR = Q3 -Q1 = 17.75 - 10.25 = 7.50 ; UL = 17.75 + 1.5( 7.50 ) =

29 ; UL = 10.25 - 1.5 (7.50) = -1

Understanding Six Sigma

Six Sigma : Introduction

What is Six Sigma ….???????

1) Statistical Measurement : We measure defect rates in all the Processes through an expanding statistical concept, and we use ‘ σ ’in measuring process capability. 2) Business Strategy : We gain a competitive edges in Quality, Cost, Customer Satisfaction. 3) Philosophy : We should work smarter, not harder

Understanding Six Sigma

Six Sigma : Introduction Z Level

Z

PPM

6

3.4

5

233

4

6,210

3

66,807

2

308,537

Process Capability

Defect Opportunity

Harvest Sweet Fruit Design for Manufacturablity 5 σ Wall, Improve Designs

Bulk of Fruit Process Characterization and Optimization 4 σ Wall, Improve Processes

Lower Hanging Fruit Seven Basic Tools 3 σ Wall, Work with suppliers

Ground Fruit Logic and Intuition

Understanding Six Sigma

Six Sigma : Introduction

The Percentage Acceptable Area under the Curve Increases as the Z Value ( the Number of Standard Deviations σ ) Increases .

1σ 2σ 3σ 4σ 5σ 6σ

USL

x

68.3 % 95.45 % 99.73 % 99.9936 % 99.99 99 4 % 99.99 99 99 8 %

LSL

The Area Under the Curve represents the Acceptance or Yield, whereas the Area outside the Curve represents the Rejection

Understanding Six Sigma

Six Sigma : D-M-A-I-C

Objective

Theme Selection

Theme Selection (Define)

Measurement Y

Capability OK ?

Redesign

Measurement

N

• Define problem • Define range • Measuring capability of CTQ • Clearfy measuring method

Analysis Analysis Redesign ?

Y

N Improvement

N

Improvement

Capability OK ? Y Control

• Clearfy factors

Control

• Find vital few • Optimize process

• Control vital few • Set up control system

3σ Vs 6σ Company

The 3 Sigma Company • Spends 15-25% of sales dollars on cost of failure

Understanding Six Sigma

The 6 Sigma Company • Spends 5% of sales dollars on cost of failure

• Produces 66,807 defects per million • Produces 3.4 defects per million opportunities opportunities • Relies on capable prcesses that • Relies on inspection to find defects don’t produce defects • Belives high quality is expensive • Benchmarks themselves against their competition

• Knows that the high quality producer is the low cost produer • Benchmarks themselves against the best in the world

• Believes 99% is good enough

• Believes 99% is Unacceptable

• Defines CTQs internally

• Defines CTQs from customers

Define-Application of Six Sigma

Understanding Six Sigma

Six Sigma is a tool that can be applied to all business systems, Design, Manufacturing, Sales & Service

Six Sigma

R&D

Mfg

SVC

Guarantee for design completion • Selecting CTQ to meet customer requirement • Deciding reasonable Tolerance • Guarantee CTQ’s through capability analysis

Quality Assurance in Manufacturing Stages • Improve serious Problems • Real Time Monitoring • CTQ control system

Maximizing Sales & Service • Improve cycle time & accuracy • Cost Improvement

Define

Understanding Six Sigma

Define Phase • Pareto Analysis • Process Mapping • Logic Tree • 3/5 Why Analysis • RTY • QFD • FMEA • Brainstorming

Define

Understanding Six Sigma

Pareto Analysis : The Origin of the Tool lies with the Italian Economist Vilfredo Pareto. Pareto Principle is also known as 80/20 .20% of items purchased by the company accounts for 80% of the value 1st Item in the figure below indicates the highest no of faults

Pareto Charts are a type of bar chart in which the horizontal axis represents categories of interest, rather than a continuous scale. The categories are often "defects." By ordering the bars from largest to smallest, a Pareto chart can help you determine which of the defects comprise the "vital few" and which are the "trivial many." A cumulative percentage line helps you judge the added contribution of each category. Pareto charts can help to focus improvement efforts on areas where the largest gains can be made. Pareto chart can draw one chart for all your data (the default), or separate charts for groups within your data.

Example :The company you work for manufactures metal bookcases. During final inspection, a certain number of bookcases are rejected due to scratches, chips, bends, or dents. You want to make a Pareto chart to see which defect is causing most of your problems. First you count the number of times each defect occurred, then you enter the name of the defect each time it occurs into a worksheet column called Damage.

Define

Understanding Six Sigma

Damage

Counts

Scratch Scratch Bend Chip Dent Scratch Chip Scratch

274 59 19 43 4 8 6 10

• Choose Stat > Quality Tools > Pareto Chart. • Choose Chart defects data in and enter Damage. Click OK. Graph window output

Define

Understanding Six Sigma

Process Mapping • Process mapping is used to document process to examine part and information flow. • It is a key tool in identifying opportunities for improvement. The Process Mapping Method • Define the Process boundary. (General area or specific process you intend to improve) • Brainstorm and order process steps with your team. • Code activities using symbols for easy analysis. • Walk through the process to validate map. • Add key process metrics - yield, costs, rolled throughput yield, scrap, overtime $, capacity, %schedule, %OTD • Analyze map for key business issues -could be in the areas of : - Process loss or waste - Cycle time improvements - Quality improvements - Flow improvements

Define

Understanding Six Sigma

Process Mapping Car Shop arrival

Look around third car

[ex] Vehicle Purchasing Meeting Salesman Trial Driving?

Small Talk

No

Trial Driving

Look around second car

Shop around for another shop

Yes Receive key and Tag

Look around first car

No Whether you Yes buy or not?

Decide to buy

Operation

Review the sales manager

Decision

Decide Contract Value

Decide the Price

Whether you No purchase another or not? Yes

Delay measurement

Visit another car shop

Storage

No Review the sales manager

Decide the price

Get a loan

Credit check

Stand by for a loan

Yes Reasonable price?

Drive the new car

Make out final contract

w

Decide to buy

Transmission

Define

Understanding Six Sigma

Process Mapping

[ex] Refrigerator - R1 Line Rolled Throughput Yield Door Ass’y 89.7% D/Plate plate/paint 99.0%

D/Liner extrusion/mold 99.7%

99.6%

Door forming 93.4%

Door assembly 97.3%

81.0%

Case forming

I/Case extrusion/mold

Cycle

assembly 83.8%

97.7%

Front - CTQ, L painting

O/Case, B/Plate

99.2%

91.7%

Case Ass’y 73.4%

LQC & appearance 96.5%

Output

Rolled Though put Yield = 73.4% ▲ Case

× 89.7% ×97.7% × 83.8%×

▲ Door

▲ Cycle

▲ assembly

96.5% = 52.0%

▲ LQC& appearance

Define

Understanding Six Sigma

Logic Tree (Structure Tree) • Used to break down problem into manageable groups to identify root cause or area of focus. • Breakdown the problem on the base of MECE - MECE - Mutually Exclusive Collectively Exhaustive Mutually Exclusive : When a Problem is broken into further sub parts there should not be anything common among the factors. Collectively Exhaustive : Also there should be nothing left to represent the Main factor Why

Electromagnetic

Lamination Why

Rotor Endrings

RPM

Losses Inductance

Mechanical

OD

Area A

Core length

Stator Area B

Assembly 6σ is a kind of type which can improve the problem (RPM) by practicing improvement activity for the lower level displayed in the long run

Define

Understanding Six Sigma

5 Why Analysis : Five Why analysis is done to determine the root cause of the Problem . It is a kind of brainstorming to reach the root cause of the Problem. It is Observed that by the time you arrive at the 5th Why the solution of the Problem is with you. It is not essential to ask why 5 times, you can locate the root cause at the 3rd or 4th Why also..

Define

Understanding Six Sigma Rolled Throughput Yeild

RTY : Rolled throughput Yeild : It is the Probability that the product will pass through all the stages without any rejection / rework. RTY is calculated by calculating the YFT’s of Individual stages.

RTY = YFT1 * YFT2 * ……… YFTn

,

where YFT’s are the First time yields of the Individual Stages/Process connected in series.

YNA = Normalized Yield = ( RTY of the line )1/no. of stages in Line YNA gives the average yield of line . This is used to calculate when we have to compare the performance of two lines on the basis of RTY.

Define

Understanding Six Sigma

RTY is the probability of going through all the processes with zero-defect the first time . Also, it provides an indication of opportunities to reduce the waste. Goal

Input -Process 1: (Acceptancerate:99.0%)

Target

-Process 2 (92.0%) -Process 3 (97.0%) Final Inspn. (97.0%)

Un-controlled Loss

Total Process Defect-rate

Overall process’s defect, m/c trouble, No work,L/B,Model

Change Loss,Non-value added work

Tool

6 Sigma

Activity

TDR, 6 Sigma, NWT, One man one project

Final Product

- Process defect rate - Self & sequential inspection

RTY = 0.99×0.92×0.97×0.97= 85.7 % * RTY : Rolled Throughput Yield

To increase productivity through Quality Improvement

To improve all the hidden defects of all the processes

Define

Understanding Six Sigma Yield First Time

Painted Components

1000

820

Are the parts Good ?

Yes

No 100

Rework & Reprocess ( Rust, Chemical Wash)

Yes

Can the parts be repaired ? No

50

Scrap (Dent)

30

Yes

Rework & Pass-on ( Paint Touch up)

Define

Understanding Six Sigma

First Time Yield

YFT = S/U YFT = First Time Yield S = No. of units that pass the first time U = No. of units tested

YFT gives the probability of going through one process with zero defects.

Rolled Throughput Yield

YRT = YFT1 x YFT2 x YFT3 YRT

Normalised Yield

YNA = (YRT)**1/Opp

= Rolled Throughput yield YFT1,YFT2,… = First Time Yield of each process

YNA = Normalized Yield Opp = Number of opportunities

•YRT gives the probability of going through all the processes with zero defects in the first time. •YRT provides an indication of opportunities to reduce waste.

•YNA us average yield of processes. •YNA allows for calculation of Z value of processes. •YNA allows for comparison between processes.

Define QFD :

Understanding Six Sigma Quality Function Deployment ( What Customer Wants )

It is defined in two steps : a) Converting customer’s Voice into Engineers Voice b) Converting Engineers voice into Technical CTQ’s & CTP’s QFD is tool which is used to generate data in the form of taking feedback from the customer through quality matrix, converting those requirements into Tech changes in the Process through Quality Matrix. • Identify key consumer cues by reviewing market, reliability requirements, general requirements and current quality issues. • Rank cues by importance and translate them into technical specifications required to meet customer cues. Rank technical specifications by impact on customer cues and translate them into potential part characteristics(CTQ’S). • Rank part characteristics by impact on meeting technical specifications(CTQ’S) QFD translates the Voice of the Consumer into the Voice of the Engineer.

Define

Understanding Six Sigma

QFD : Sub Process 1 : To Convert Consumer’s Voice into Engineer’s Voice

Capacity of Motor

Blower /Scroll

System for Gas Charging

HE Coils Fins per Inch

Priroty Ranking

Type of Comp Customer's Requirement

Engineer's Voice

Less Price

9

3

1

1

1

10

Low Noise

3

9

9

1

1

6

Air Flow

1

1

9

3

3

5

Less Power

9

9

1

3

1

8

More Cooling

9

3

3

9

9

3

Rating

212

170

126

82

66

Define

Understanding Six Sigma

QFD : Sub Process 2 : To Convert Engineer’s Voice into Potential CTP’s & CTQ’s

Comp Specifications

KW Rating

EER Specifications

Gas Charging Qty Variation less than 5mg

HE Design Spec to be maintained

Engineer's Voice

Potential CTP's & CTQ's

Type of Comp

9

1

3

3

1

212

Capacity of Motor

1

9

3

1

1

170

Blower /Scroll Design

1

1

9

1

3

126

System for Gas Charging

1

1

1

9

3

82

HE Coils Fins per Inch

1

1

1

3

9

66

3096

2016

2428

1848

1600

Priroty Ranking

Define

Understanding Six Sigma

FMEA : Failure Mode Effect Analysis ( What Customer Doesn’t want ) It gives you possible reasons in which a given Process / Design of part of a Product can Fail. To every Failure Mode we associate RPN Number RPN : Risk Priority Number = Severity * Occurrence * detection Rating Scale Severity

(1 ~ 10)

(1 ~ 10)

(1 ~ 10)

1 : If the Problem is Less Severe 10 : If the Problem is Life Threatening. 1 : If the Problem has chances of less occurrence

Occurrence

10 : If the Problem has more chances of occurrence

Detection

1 : If the Problem is easily detectable. 10 : If it is difficult to locate/detect the Problem

FMEA is used to proactively identify and rank risks in a product design and assign appropriate actions to be taken to prevent the failure mode.

Define

Understanding Six Sigma

FMEA Process • Brainstorm potential failures of the product design. • Assign severity and probability (likelihood of occurrence) ratings to each potential failure mode. • Determine existing control measures being taken to eliminate significant failure modes. • Develop actions to be taken to eliminate or reduce risk on all remaining significant failure modes.

Define

Understanding Six Sigma

Brainstorming : It is Discussion among the Process Experts.The basic rule of brainstorming is no ideas are criticized. Brainstorming is of three Types : a) Freewheel b) Round Robin c) Card Method In Freewheeling type of brainstorming, everybody participates in the simultaneous discussion In Round Robin type of brainstorming each Individual in the group is given a chance to give his opinion In Card type method the Individuals write their Ideas on the Card

Measure

Understanding Six Sigma

Measure Phase • Gage R & R • Types of Sampling • Process Capability • Four Block Diagram

The Aim of the measure stage is to : a. To Establish the validity of the measuring system & operator b. It tells the present Level of the Process.

Measure

Understanding Six Sigma

Six Sigma is based on the measured data. There will be unfavorable consequences from analysis using statistical tool if we have a problem with measuring system. What’s more, the process gets worse, then experiment will end up in failure. Therefore, we do better secure correct measurement system before the project.

Overall Variation

Part to Part Variation

Measurement System Variation

Variation due to gage Repeatability

Variation due to Operator Reproducibility Operator

Operator by Part

Gage R & R - Introduction

Understanding Six Sigma

σ2Total = σ2Part-Part + σ2R&R Total variation

Variation due to differences among the parts.

Measurement error variation

σ2R&R = σ2Repeatibility + σ2Reproducibility Measurement error variation

Variation due to Gage

Variation due to Operator

Gage R & R - Introduction

Understanding Six Sigma

What is Gage R&R Gage R & R is Gage Repeatability & Reproducibility. ◆ Repeatability = EV(Equipment Variation) It is the variation observed in the system when one Operator measures the same part twice using the same gage. Repeatability is the variation due to equipment variation. Example : Consider one Operator who successively measures the Thickness of paint coating on Ref L/R part using the same gauge. Reading 1 = 12.5 , Reading 2 = 12.0 , Difference = 0.5 , variation due to Gage (Repeatability)

◆ Reproducibility = AV(Appraiser Variation) It is the variation observed in the system when two different operators measures the same part using the same gage. Reproducibility is the variation due to change in operator. Example : Consider two Operators who successively measures the Thickness of paint coating on same Ref L/R part using the same gauge. Operator 1 = 14.8 , Operator 2 = 14.2 , Difference = 0.6 , variation due to Operator (Reproducibility )

Total Gage R&R =

E.V2 +A.V2

Gage R &R - Long Study Method

Understanding Six Sigma Repeatability : “Getting consistent results”

Repeatability ? ☞ Variation observed with one measurement device when used several times by one operator while measuring the identical characteristic on the same parts.

Measure/Re-measure variation

Reproducibility ? Variation obtained from different operators using the same device when measuring the identical characteristic on the same parts.

Operator B Operator C

Operator A

Reproducibility

Gage R & R - Purpose

Understanding Six Sigma

Gage R&R • Gage R & R is used to ensure that the measured data used

for statistical tests is valid. • Selection of the most appropriate gage for the task. • When we want to compare the performance of each Gage. • When we want to exclude the Gage error from results. • Maintenance of measurement system ( Calibration ) • For measurement training for existing and New Staff.

Measure ~ Gage R & R

Understanding Six Sigma

Two types of Gage R&R Study ◆ Short study method • Requires minimum 2 operators and minimum 5 parts with each part measured at least once. • This method cant separate the total variation Observed through Gage R &R into repeatability & reproducibility • Permits speedy acceptance for adapting Gauge. ◆ Long study method • Requires minimum 2 Operators, minimum 10 parts with each part measured at least twice. • This method can divide the total variation observed in the system through Gage R & R into repeatability & reproducibility, so that we can get to know what we have to improve Operator or Gage

Gage R & R ~ Short Study

Understanding Six Sigma Short Study Method

Example : The height of a component has specifications given by 5.0 ± 0.5. (tolerance = 1.0 ) Solution : The Measurements taken by the two Operators for the Five Parts are listed below Part Operator 1 Operator 2 Ranges (1-2) 1 2 3 4 5

4.9 4.7 5.2 5 4.8

4.8 4.7 5.1 5.1 4.7 Range Sum

0.1 0 0.1 0.1 0.1 0.4

• Average Range ( R-bar ) = Σ R / n = 0.4 / 5 = 0.08 • Gauge Error

= ( 6.0 /d) ( R-bar) = (5.15 /1.19) (0.08 ) = 0.3464

5.15 indicates the Confidence Level of 99 % ; 6.0 indicates a Confidence Level of 99.73 %

•Gauge R & R as % of Tolerance = (0.3464 x 100) /1.0 = 34.64 %

Gauge Error is calculated by multiplying the average range by a constant d ( to be taken from the Table )

Gage R & R ~ Short Study

Understanding Six Sigma

d* values for distribution of the average range

Number of parts 1 2 3 4 5 6 7 8 9 10

2 1.41 1.28 1.23 1.21 1.19 1.18 1.17 1.17 1.16 1.16

Number of operators 3 4 1.91 2.24 1.81 2.15 1.77 2.12 1.75 2.11 1.74 2.10 1.73 2.09 1.73 2.09 1.72 2.08 1.72 2.08 1.72 2.08

5 2.48 2.40 2.38 2.37 2.36 2.35 2.35 2.35 2.34 2.34

Gage R & R ~ Guidelines

Understanding Six Sigma

Pre Requisites for Gage R & R ★

Blind Test :

• The Operator should not be aware that Gage R&R is going On. • The Previous Readings should not be conveyed while taking Next Reading. ★ Gage selection(Resolution)

•The Gage must have a resolution of less than or equal to 10% of the one sided specification or process variation. • Resolution is the smallest unit of measure the gage is able to read. •Ex) In case of part feature tolerance equals +/-0.020, Gage must have resolution 0.002 and Gage R&R ≤20% to be recommended. ★

Intentional Sampling :

• The samples must not be randomly selected, the sampling must be proceeded by a plan, so the total range of variation and specification are covered. • Most values should lie near the LSL/USL , because the chances of discrepancy are more near these limits

Gage R &R ~ Long Study Method

Understanding Six Sigma

An acceptable value for a Gage R&R Study (Continuos Data - ) ≤ 20%

% Study Variation & % Study Tolerance

20% to 29% ≥ 30%

: Acceptable : Conditional : Unacceptable

An improvement plan to lower the gauge R&R variation should be implemented. If there is no improvement , consideration should be made for the risks associated with high Gauge R &R

Number of Distinct Categories P Value of the Operator * Part

>4 > 0.25

Gage R &R - Long Study Method Long study method (using Minitab)

Select: ANOVA

Understanding Six Sigma Input : Parts, Operator & Measurement data

Gage R &R - Long Study Method

Understanding Six Sigma

Why ANOVA method is more accurate than X(bar) R Method…..???????? X - R Method

ANOVA

Part to Part Variation

Part to Part Variation

Repeatability

Repeatability

Reproducibility

Reproducibility

Operator

Operator by Part

ANOVA Method further breaks the Variation due to Operator(Reproducibility) into Operator & Operator by Part

Gage R &R - Long Study Method

Understanding Six Sigma

Long study method (using Minitab) If significant, P-value < 0.25 indicates that an operator is having a problem measuring some the parts. Hence Gage R&R is not acceptable.

Gage R&R Study - ANOVA Method Two-Way ANOVA Table With Interaction Source Parts Operator Parts * Operator Repeatability Total

DF 9 1 9 20 39

SS 81.6 0.1 0.4 1.0 83.1

MS 9.06667 0.10000 0.04444 0.05000

F 204.000 2.250 0.889

P 0.000 0.168 0.552

Two-Way ANOVA Table Without Interaction Source Parts Operator Repeatability Total

DF 9 1 29 39

SS 81.6 0.1 1.4 83.1

MS 9.06667 0.10000 0.04828

F 187.810 2.071

P 0.000 0.161

Gage R &R - Long Study Method

Understanding Six Sigma

Long study method (using Minitab) % Study Variation < 20 % and % Study Source Total Gage R&R Repeatability Reproducibility Operator Part-To-Part Total Variation

Source Total Gage R&R Repeatability Reproducibility Operator Part-To-Part Total Variation

VarComp 0.05086 0.04828 0.00259 0.00259 2.25460 2.30546

(of VarComp) 2.21 2.09 0.11 0.11 97.79 100.00

StdDev (SD) 0.22553 0.21972 0.05085 0.05085 1.50153 1.51837

Study Var (6 * SD) 1.35316 1.31831 0.30513 0.30513 9.00919 9.11024

Tolerance> 20%, Gage R&R is acceptable

%Study Var (%SV) 14.85 14.47 3.35 3.35 98.89 100.00

%Tolerance (SV/Toler) 33.83 32.96 7.63 7.63 225.23 227.76

For Gage R&R to be acceptable, number of distinct categories > 4 Number of Distinct Categories = 9

Gage R &R - Long Study Method

% Study Variation =

% Study Tolerance = 6.0 *

Understanding Six Sigma

σ Gage R&R

* 100

σ total variation σ Gage R&R

* 100

σ total variation

Gage R & R( Nested ) : Used for Destructive Testing Gage R & R (Nested ) is used when each part is measured once only as when measuring. Ex. The torque release of a bolt during QC sampling, we cannot measure again

Gage R &R - Long Study Method Accuracy ?

Understanding Six Sigma True (Reference) value

The degree of agreement of the measured value to the true magnitude (unbiased values). (Accuracy is typically expressed as 1-%Bias)

Stability ?

Accuracy

* Setting a true value is a one that is measured by the most accurate measuring device.

Observed average Time 1

Stability is the total variation in the measurements obtained with a measurement system on the same master or reference value when measuring the same characteristic over an extended time period. Stability Time 2

Bias : It is a measure of the distance between the average value the measurements and the "True" or "Actual" value of the sample or part.

Gage R &R - Long Study Method Linearity ?

Understanding Six Sigma

LSL

USL

Actual values

Linearity is the difference in the bias values throughout the expected operating range of the gage. (Gage is less accurate at the low end of specification or operating range than at the high end).

Actual values (No Bias) Reference value

Reference values

Larger Bias

Small Bias

Measure - 4 Block Diagram Block Diagram

Understanding Six Sigma

Poor

2.5

Zshift (Process Control)

2.0

A

B

C

D

1.5 1.0 0.5

Good Poor

1

2

3 Z.st 4 5 (Process Technology )

A : Poor control, poor technology B : Must control the process better, technology is fine C : Process control is good, poor technology D : World Class

6

Good

Gage R & R - Discreet Data

Understanding Six Sigma

Pre-Requisites Gage R & R (discreet data) • The Minimum Number of Samples should be at least 20 • Minimum Number of Operators should be at least 2 • Each Operator must take at least two readings of each Part. Acceptability of Gage R & R (discreet data) • % Gage R & R should be less than 5 %

Gage R & R - Discreet Data

Understanding Six Sigma

Gage R & R (discreet data)

18 Samples

Visual Inspection Gage Study 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Appraiser "A" 1 2 G G G G NG G NG NG G G G G NG NG NG NG G G G G G G G G G NG G G G G G G G G G G

Appraiser "B" 1 2 G G G G G G NG NG G G G G NG NG G G G G G G G G G G G G G G G G G G G G G G

19 20

NG G

NG G

NG G

NG G

•The gage is acceptable if both the Appraisers (four per part) agree.. • % Gage R&R = No. of Disagreements/Total Opportunities X100 = 3 / 20 x 100% = 15% • If the results of checkers are different, the gage • must be improved and re-evaluated. • If the gage cannot be improved, it is unacceptable and an alternate measurement system should be found.

Gage R & R - Importance

Understanding Six Sigma

From the Gauge R & R Study we can determine the Following : •

Gage resolution is adequate.

• The Measurement System is stable over time. • The measurement system error is small enough and acceptable enough relevant to the process variation or Specification Gage R & R Indicates that whether the Measurement System is good enough for the collection of Data.

Measure - Capability Analysis

Understanding Six Sigma

Capability analysis is a set of calculations used to assess whether a system is statistically able to meet a set of specifications or requirements. Specifications or requirements are the numerical values within which the system is expected to operate, that is, the minimum and maximum acceptable values. Specifications are numerical requirements, goals, aims, or standards.

Measure

Understanding Six Sigma

Cp = Specification Width Process Capability

Cp

=

Cp = Product Specification Cpk Manufacturing Variability k =

T-µ

USL - LSL 2

CPU = USL - X

3σwithin

USL-- LSL 6σwithin = Cp (k-1) CPL = X - LSL

3σwithin

Process Capability ( Cp) is the Tolerance width in Relation to the process capability, expressed as the best short Term performance. Takes no account of the process centering.

Capability Index (Cpk) accounts for the process centering. Considers sample data variation & location simultaneously.

Measure - Central Limit Theorem

Understanding Six Sigma

Central limit theorem states that as the sample size increases, the sampling distribution of the mean will approach normality. Statisticians use the normal distribution as an approximation to the sampling distribution, whenever the sample size is at least 30. SEM : Standard Error of Mean

SEM *

Standard error of mean gives the difference between the standard deviation of Population & Standard deviation of sample.

SEM = σp (n)1/2

Standard deviation of Population Sample Size

n=8

n=30 Sample Size

Measure - Central Limit Theorem

Understanding Six Sigma

From the graph shown on the previous slide, it is evident that for sample size 30 the difference between the standard deviations of sample and population is very less. Even though this difference reduces further by increasing sample size, but this reduction is negligible. Hence while sampling, sample size of 30 is considered as the idle sample size.

Sample Size

Difference b/t Standard deviation of Population and Sample

• Less than 8

high variation

• Between 8 ~ 30

Moderate Variation

• 30 & above

Minimum Variation

As the sample size increases, above 8 samples the difference between standard deviation of Population & sample reduces drastically. At sample size 30 the difference is minimum and it remains constant & beyond 30 it remains constant., so the curve line representing the difference becomes parallel to X-axis.

Measure - Sampling

Understanding Six Sigma

Types of Sampling 1.) Random Sampling : In this type of Sampling each data point of the Population has an equal chance/ Probability of being selected. Example : During the draw of lottery tickets each & every lottery ticket number has an equal chance of winning the Prize. 2.) Stratified Sampling : In this type of Sampling , the sub group taken for sampling has data points of same type. Example : For determining the Quality of Food in the Canteen, if we take the Sample group in which all Supervisors/Operators/Managers are there, then the difference in the variation of taste within the sub group would be minimum but among the Sub groups would be maximum. 3) Clustered Sampling : In this type of Sampling,each & every type of data point present in the population would be covered in the Sample. Example : In the above example if take the sample in such a way that in the subgroup operator, supervisor and manager are taken so the difference in the taste would be maximum within the sub group and minimum among the subgroup.

Measure

Understanding Six Sigma

BLACK NOISE

PROCESS RESPONSE

(Signal)

RATIONAL SUBGROUPS

TIME

WHITE NOISE (Common Cause variation)

Measure

Understanding Six Sigma

White Noise • White noise represents the variation present in every process. Also known as common cause variation • It is not controllable variation within the existing technology. • Represents that best the process can be with the present technology(Inherent process capability).

Black Noise • Black Noise represents the outside influences on a process that cause average to shift and drift. Also known as Special Cause or assignable cause variation. • It is potentially controllable variation with the existing process technology. • It represents how the process is actually performing over time(Sustained process capability).

Measure

Understanding Six Sigma

From a statistics perspective, There are only two problems.

Problem with Spread

Problem with Centering Desired

Desired

Current situation

Current situation

LSL

T

USL

LSL

T

USL

Shift Accurate but not Precise

Precise but not Accurate

Measure

Understanding Six Sigma

Process Capability Ratios The greater the design margin, the lower the Total Defects Per Unit Design margin is measured by the Process Capability Index (Cp)

Cp =

Maxium Allowable Range of Characteristic Normal variation of Process

X -3σ

+3σ

Process Width

Zst = 3 Cp Zlt = 3Cpk

Design Width

Cp =

USL - LSL + 3σ

Measure

Understanding Six Sigma

Is it Control or Technology? Long Term Data

Short Term Data

. Data taken over a period of time . Data taken over a short enough period of time that there are no long enough that external factors external influences on the can influence the process. process Z st : Z lt

. Z lt (σlt ) .

Cpk

Z st (σst ) . Cp

Zlt is always less tahn Zst, because the long term value is reduced by the shift of the process

. Defined by technology and process . Defined by technology control . Process Capability (Entitlement - The best process . Process Performance can be)

6σ means Zlt=4.5 and Cpk=1.5

Technology: Control

6σ means Zst=6.0 and Cp=2.0

Analyze

Understanding Six Sigma

Analyze Phase • Cause & Effect Diagram • Hypothesis Testing • Mean Testing • Variance Testing • Regression Analysis

The Aim of the Analyze Phase is to : a. To List down all the Possible factors through Brainstorming. b. Pick out the Vital few Potential Factors out of Trivial Many. c. Statistical verification of Potential Factors by means of various type of Tests. d. Ascertaining whether we have considered all factors & nothing is Left Out.

Understanding Six Sigma

Analyze - Cause & Effect Diagram

Cause & Effect Diagram Cause Cause

Cause

Cause

Effect Cause Cause Cause

The Purpose of this tool is to Find out the start of the collection of Data and analysis. It list down all the Probable causes responsible for the main effect . Cause & Effect Diagram is also known as Fishbone Diagram / 4M Diagram. The Symptom or result is put under the Dark Box on the Right.. Lighter Boxes at the end of the Large Bones are main groups in which ideas are classified. The Lighter Boxes may consist of Five Ms - Man,Machine,Measurement & Method.(Money can be considered wherever relevant) . The Middle Bones indicates the direction of path from cause to effect

Understanding Six Sigma

Analyze - Cause & Effect Diagram

Machine

Man

Cylinder Failure

New casual Handling problem

M/C not clean

Die Setting OUT CASE DENT Dented sheet

Chips on sheet

Material

Sheet thickness

Piece check Method

Piece unloading

Analyze - Hypothesis

Understanding Six Sigma Hypothesis Testing

Hypothesis means something taken to be true for the Purpose of argument or Investigation , an assumption. Hypothesis Testing is defined as the comparison of two Populations (equality of mean/ variance ) by taking samples from those Populations. It is assumed in the beginning that the two Populations are equal (Null Hypothesis ;µ1 µ2 ; σ1 = σ2 ) or not equal (Alternate Hypothesis : µ1≠ µ2 ; σ1 ≠ σ2 ) . The equality is confirmed by actually conducting tests on the sample. There is always a risk associated with the Hypothesis , in case the sample taken for comparison from Population does not correctly represent the Population.. There are many types of hypothesis test. The test is selected depending on the type of data or the comparison required.

Continuous Data 1) F-test : Compares Variances • Levene’s Test • Bartlett’s Test 2) t-test : Compares means • 1 sample t-test • Paired t-test • 2 sample t-test

Discrete Data 3) Chi Square Test : Compares counts • Goodness of Fit • Contingency Table

Analyze - Hypothesis

Understanding Six Sigma Hypothesis Testing

Ho(Null Hypothesis) is assumed to be true .This is like the defendant being assumed to be innocent.

Ha(Alternative Hypothesis) is alternatives the Null Hypothesis.

Ha is the one

that must be proved. Population Ho

In this case as the samples does not correctly represent the Population

Correct

Ho Decision

Sample

In this case the samples correctly represent the Population so sample mean = Population mean.

so sample mean ≠ Population mean. Incorrect Decision

Ha

Type 1 Error α

Ha

Type 2 Error β

In this case as the samples does not correctly represent the Population so sample mean ≠ Population mean. Incorrect Decision In this case as the samples correctly represent the Population so sample mean = Population mean. Correct Decision

Analyze - Hypothesis

Understanding Six Sigma Important Terms

1.) Type 1 Error : This error gives us the probability of rejecting the Right Material . This happens when a weird sample gets selected for the comparison of mean/variance. It is also known as α − Error or Producer’s Risk. Generally It’s value lies around 5 %. 2. ) Type 2 Error : This error gives us the probability of accepting the wrong material. This also happens when a weird sample is selected for comparison. It is also known as β − Error or Consumer’s Risk. It’s value generally lies around 10 %.

3 .) 1-α = Confidence of the Test The probability that can be determined as a right thing when the Null Hypothesis is correct. 4) 1-β = Power of the test The rejecting probability when null Hypothesis you want to test is not right. It is not possible to simultaneously commit a Type 1 and Type 2 decision error.

Analyze - Tests used for Comparison

Understanding Six Sigma

Analyze

Variance Testing

Mean Testing Continuous Data

Discrete Data

1 Sample Z Test

1 Proportion Test

1 Sample t Test

2 Proportion Test

2 Sample t Test

Chi-Square Test

ANOVA Testing

Test for Equal Variance

2 Variance Test

Analyze

Understanding Six Sigma

1 Sample Z Test :computes a confidence interval or performs a hypothesis test of the mean when the population standard deviation, σ is known. This procedure is based upon the normal distribution. This test compares the mean of the sample with some test Population with known standard deviation. Example : Measurements were made on nine widgets. You know that the distribution of measurements has historically been close to normal with s = 0.2. Because you know s, and you wish to test if the population mean is 5 and obtain a 90% confidence interval for the mean, you use the Z-procedure. Solution : 1 Open the worksheet enter the values.. 2 Choose Stat > Basic Statistics > 1-Sample Z. 3 In Samples in Columns, enter Values. 4 In Standard deviation, enter 0.2. 5 In Test mean, enter 5. 6 Click Options. In Confidence level, enter 90. Click OK. 7 Click Graphs. Check Individual value plot. Click OK in each dialog box.

Values 4.9 5.1 4.6 5 5.1 4.7 4.4 4.7 4.6

Analyze

Understanding Six Sigma

One-Sample Z: Test of

Test Mean doesn’t lie within the confidence Interval

Values

mu = 5 vs not = 5

The assumed standard deviation = 0.2 Variable

N Mean

StDev

Values

9 4.78889 0.24721

SE Mean

90% CI

0.06667 (4.67923, 4.89855)

Z -3.17

p value < 0.05, Hence Ha, alternate Hypothesis P 0.002

Analyze

Understanding Six Sigma Interpreting the results

The test statistic, Z, for testing if the population mean equals 5 is -3.17. The p-value, or the probability of rejecting the null hypothesis when it is true, is 0.002. This is called the attained significance level, p-value, or attained α of the test. Because the p-value of 0.002 is smaller than commonly chosen α-levels, there is significant evidence that m is not equal to 5, so you can reject H0 in favor of m not being 5. A hypothesis test at α = 0.1 could also be performed by viewing the individual value plot. The hypothesized value falls outside the 90% confidence interval for the population mean (4.67923, 4.89855), and so you can reject the null hypothesis.

1 Sample t test : computes a confidence interval or performs a hypothesis test of the mean when Population standard , σ is unknown. This procedure is based upon the tdistribution, which is derived from a normal distribution with unknown σ.

Example : 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 σ. To test if the population mean is 5 and to obtain a 90% confidence interval for the mean, you use a t-procedure.

Analyze

Understanding Six Sigma

Solution : 1 Open the worksheet enter the data. 2 Choose Stat > Basic Statistics > 1-Sample t. 3 In Samples in columns, enter Values. 4 In Test mean, enter 5. 5 Click Options. In Confidence level enter 90. Click OK in each dialog box

Values 4.9 5.1 4.6 5 5.1 4.7 4.4 4.7 4.6

One-Sample T: Values Test of mu = 5 vs not = 5 Variable N Mean Values

StDev

SE Mean

9 4.78889 0.24721 0.08240

90% CI

T

(4.63566, 4.94212) -2.56

P 0.034

Result Interpretation : The p-value < 0.05 , also “ 0 “ does not lie within the Confidence Interval so Null Hypothesis is rejected and Alternate Hypothesis is accepted. It confirms that the sample mean is not equal to Population Mean ).

Analyze

Understanding Six Sigma

2 Sample t test : computes a confidence interval and performs a hypothesis test of the difference between two population means when σ 's are unknown and samples are drawn independently from each other. This procedure is based upon the t-distribution, and for small samples it works best if data were drawn from distributions that are normal or close to normal. You can have increasing confidence in the results as the sample sizes increase.

Example : A study was performed in order to evaluate the effectiveness of two devices for improving the efficiency of gas home-heating systems. Energy consumption in houses was measured after one of the two devices was installed. The two devices were an electric vent damper (Damper=1) and a thermally activated vent damper (Damper=2). The energy consumption data (BTU.In) are stacked in one column with a grouping column (Damper) containing identifiers or subscripts to denote the population. Suppose that you performed a variance test and found no evidence for variances being unequal .Now you want to compare the effectiveness of these two devices by determining whether or not there is any evidence that the difference between the devices is different from zero.

Analyze Solution : 1 Open the worksheet , enter the data. 2 Choose Stat > Basic Statistics > 2-Sample T. 3 Choose Samples in one column. 4 In Samples, enter 'BTU.In'. 5 In Subscripts, enter Damper. 6 Check Assume equal variances. Click OK.

Understanding Six Sigma BTU.In 7.87 9.43 7.16 8.67 12.31 9.84 16.9 10.04 12.62 7.62 11.12 13.43 9.07 6.94 10.28 9.37 7.93 13.96 6.8 4 8.58 8 5.98 15.24 8.54 11.09 11.7 12.71 6.78 9.82 12.91 10.35 9.6 9.58 9.83 9.52 18.26 10.64 6.62 5.2 12.28 7.23 2.97 8.81 9.27 11.29 8.29 9.96 10.3 16.06 14.24 11.43 10.28 13.6 5.94 10.36 6.85 6.72 10.21 8.61

Damper 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Analyze

Understanding Six Sigma

Minitab Output : Two-Sample T-Test and CI: BTU.In, Damper Two-sample T for BTU.In Damper N

Mean StDev SE Mean

1

40

9.91

3.02

0.48

2

50 10.14

2.77

0.39

Difference = mu (1) - mu (2) Estimate for difference: -0.235250 95% CI for difference: (-1.450131, 0.979631) T-Test of difference = 0 (vs not =): T-Value = -0.38 P-Value = 0.701 DF = 88 Both use Pooled StDev = 2.8818

Analyze

Understanding Six Sigma

Result Interpretation : Minitab displays a table of the sample sizes, sample means, standard deviations, and standard errors for the two samples. Since we previously found no evidence for variances being unequal, we chose to use the pooled standard deviation by choosing Assume equal variances. The pooled standard deviation, 2.8818, is used to calculate the test statistic and the confidence intervals. A second table gives a confidence interval for the difference in population means. For this example, a 95% confidence interval is (-1.45, 0.98) which includes zero, thus suggesting that there is no difference. Next is the hypothesis test result. The test statistic is -0.38, with pvalue of 0.701, and 88 degrees of freedom. Since the p-value is greater than commonly chosen a-levels, there is no evidence for a difference in energy use when using an electric vent damper versus a thermally activated vent damper.

Analyze

Understanding Six Sigma

ANOVA : is a tool with which we can compare several means. It is a tool used to search for the significant X factors that have an influence on the response variable Y. In effect, analysis of variance extends the two-sample t-test for testing the equality of two population means to a more general null hypothesis of comparing the equality of more than two means, versus them not all being equal.

Example : 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. However, two methods are selected here to demonstrate Minitab's capabilities.

Analyze

Understanding Six Sigma

Solution : 1 Open the worksheet enter the data 2 Choose Stat > ANOVA > One-Way. 3 In Response, enter Durability. In Factor, enter Carpet. 4 Click Comparisons. Check Tukey's, family error rate. Check Hsu's MCB, family error rate Durability Carpet and enter 10. 5 Click OK in each dialog box.

18.95 12.62 11.94 14.42 10.06 7.19 7.03 14.66 10.92 13.28 14.52 12.51 10.46 21.4 18.1 22.5

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

Analyze Results for: EXH_AOV.MTW One-way ANOVA: Durability versus Carpet Source DF SS MS F P Carpet 3 146.4 48.8 3.58 0.047 Error 12 163.5 13.6 Total 15 309.9 S = 3.691 R-Sq = 47.24% R-Sq(adj) = 34.05% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---------+---------+---------+---------+ 1 4 14.483 3.157 (-------*-------) 2 4 9.735 3.566 (-------*--------) 3 4 12.808 1.506 (-------*-------) 4 4 18.115 5.435 (-------*-------) ---------+---------+---------+---------+ 10.0 15.0 20.0 25.0

Understanding Six Sigma

Analyze

Understanding Six Sigma

Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Carpet Individual confidence level = 98.83% Carpet = 1 subtracted from: Carpet Lower Center Upper ------+---------+---------+---------+--2 -12.498 -4.748 3.003 (------*-------) 3 -9.426 -1.675 6.076 (------*-------) 4 -4.118 3.632 11.383 (-------*------) ------+---------+---------+---------+---10 0 10 20 Carp = 2 subtracted from: Carpet Lower Center Upper ------+---------+---------+---------+--3 -4.678 3.073 10.823 (-------*-------) 4 0.629 8.380 16.131 (------*-------) ------+---------+---------+---------+---10 0 10 20 Carp = 3 subtracted from: Carpet 4

Lower -2.443

Center 5.308

Upper 13.058

------+---------+---------+---------+--(------*-------) ------+---------+---------+---------+---10 0 10 20

Analyze

Understanding Six Sigma Tukey's comparisons

Tukey's test provides 3 sets of multiple comparison confidence intervals: Carpet 1 mean subtracted from the carpet 2, 3, and 4 means: The first interval in the first set of the Tukey's output (-12.498, -4.748, 3.003) gives the confidence interval for the carpet 1 mean subtracted from the carpet 2 mean. You can easily find confidence intervals for entries not included in the output by reversing both the order and the sign of the interval values. For example, the confidence interval for the mean of carpet 1 minus the mean of carpet 2 is (-3.003, 4.748, 12.498). For this set of comparisons, none of the means are statistically different because all of the confidence intervals include 0. Carpet 2 mean subtracted from the carpet 3 and 4 means: The means for carpets 2 and 4 are statistically different because the confidence interval for this combination of means (0.629, 8.380, 16.131) excludes zero. Carpet 3 mean subtracted from the carpet 4 mean: Carpets 3 and 4 are not statistically different because the confidence interval includes 0. By not conditioning upon the F-test, differences in treatment means appear to have occurred at family error rates of 0.10. If Hsu's MCB method is a good choice for these data, carpets 2 and 3 might be eliminated as a choice for the best. When you use Tukey's method, the mean durability for carpets 2 and 4 appears to be different.

Analyze

Understanding Six Sigma

1 Proportion Test : Performs a test of one binomial proportion. Use 1 Proportion to compute a confidence interval and perform a hypothesis test of the proportion. For example, an automotive parts manufacturer claims that his spark plugs are less than 2% defective. You could take a random sample of spark plugs and determine whether or not the actual proportion defective is consistent with the claim. For a two-tailed test of a proportion: H0: p = p0 versus H1: p ≠ p0 where p is the population proportion and p0 is the hypothesized value.

Analyze

Understanding Six Sigma

Example : 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 trials, enter 950. In Number of events, enter 560. 4 Click Options. In Test proportion, enter 0.65. 5 From Alternative, choose greater than. Click OK in each dialog box.

Analyze

Understanding Six Sigma

Session window output Test and CI for One Proportion Test of p = 0.65 vs p > 0.65 95% Lower Exact Sample X N Sample p Bound P-Value 1 560 950 0.589474 0.562515 1.000 Interpreting the results The p-value of 1.0 suggests that the data are consistent with the null hypothesis (H0: p = 0.65), that is, the proportion of party members that support the candidate is not greater than the required proportion of 0.65. As her campaign manager, you would advise her not to run for the office of state district attorney.

Analyze

Understanding Six Sigma

2 Proportion Test : Performs a test of two binomial proportions. Use the 2 Proportions command to compute a confidence interval and perform a hypothesis test of the difference between two proportions. For example, suppose you wanted to know whether the proportion of consumers who return a survey could be increased by providing an incentive such as a product sample. You might include the product sample with half of your mailings and see if you have more responses from the group that received the sample than from those who did not. For a two-tailed test of two proportions: H0: p1 - p2 = p0 versus H1: p1 - p2 ≠ p0 where p1 and p2 are the proportions of success in populations 1 and 2, respectively, and p0 is the hypothesized difference between the two proportions.

Analyze

Understanding Six Sigma

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. 1 Choose Stat > Basic Statistics > 2 Proportions. 2 Choose Summarized data. 3 In First sample, under Trials, enter 50. Under Events, enter 44. 4 In Second sample, under Trials, enter 50. Under Events, enter 42. Click OK.

Analyze

Understanding Six Sigma

Session window output

Test and CI for Two Proportions Sample

X

N

Sample

1

44

50

0.880000

2

42

50

0.840000

Difference = p (1) - p (2) Estimate for difference: 0.04 95% CI for difference: (-0.0957903, 0.175790) Test for difference = 0 (vs not = 0): Z = 0.58 P-Value = 0.564

p

Analyze

Understanding Six Sigma Interpreting the results

Since the p-value of 0.564 is larger than commonly chosen a levels, the data are consistent with the null hypothesis (H0: p1 - p2 = 0). That is, 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. You can make the same decision using the 95% confidence interval. Because zero falls in the confidence interval of (-0.096 to 0.176) you can conclude that the data are consistent with the null hypothesis. If you think that the confidence interval is too wide and does not provide precise information as to the value of p1 - p2, you may want to collect more data in order to obtain a better estimate of the difference.

Analyze

Understanding Six Sigma

Chi Square Test : It is a measure of the Observed & expected frequencies. Chi Square test is a statistical test which consists of three different type of Analysis. 1) Goodness of Fit 2) Test for Homogeneity 3) Test for Independence The test for Goodness of fit determines if the sample under analysis was drawn from a population that follows some specified distribution . Test for Homogeneity answers the proposition that several populations are homogenous with respect to some characteristic. Test for Independence is for testing Null hypothesis that two criteria of Classification

Analyze

Understanding Six Sigma

Example : You are interested in the relationship between gender and political party affiliation. You query 100 people about their political affiliation and record the number of males (row 1) and females (row 2) for each political party. The worksheet data appears as follows: Column 1

Column 2

Democrat

Republican

Column 3 Other

28

18

4

22

27

1

1 Open the worksheet EXH_TABL.MTW. 2 Choose Stat > Tables > Chi-Square Test (Table in Worksheet). 3 In Columns containing the table, enter Democrat, Republican and Other. Click OK.

Analyze

Understanding Six Sigma

Session window output Chi-Square Test: Democrat, Republican, Other Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts Democrat 1

2

Total

Other

Total

18

4

50

25.00

22.50

2.50

0.360

0.900

0.900

22

27

1

25.00

22.50

2.50

0.360

0.900

0.900

50

45

28

Republican

Chi-Sq = 4.320, DF = 2, P-Value = 0.115 2 cells with expected counts less than 5.

5

50

100

Analyze

Understanding Six Sigma

Session window output Chi-Square Test: Democrat, Republican, Other Expected counts are printed below observed counts Chi-Square contributions are printed below expected counts Other

Total

Observed Values

18

4

50

Expected Values

25.00

22.50

2.50

0.360

0.900

0.900

22

27

1

25.00

22.50

2.50

0.360

0.900

0.900

50

45

Democrat 1

2

Total

28

Republican

Chi-Sq = 4.320, DF = 2, P-Value = 0.115 2 cells with expected counts less than 5.

5

Chi Square Values 50 Row Totals

100

Grand Total

Column Totals

Analyze

Understanding Six Sigma

Interpreting the results No evidence exists for association (p = 0.115) between gender and political party affiliation. Of the 6 cells, 2 have expected counts less than five (33%). Therefore, even if you had a significant p-value for these data, you should interpret the results with caution. To be more confident of the results, repeat the test, omitting the Other category.

Formulae’s :

a) Expected Value = Row Total * Column Total Grand Total b) Chi Square Value = ( Observed Value - Expected Value )2 Expected Value c) Degrees of Freedom(DF) = (No. of rows - 1) * ( No. of Columns - 1)

Analyze

Understanding Six Sigma

2 Variance Test : Use to perform hypothesis tests for equality, or homogeneity, of variance among two populations using an F-test and Levene's test. Many statistical procedures, including the two sample t-test procedures, assume that the two samples are from populations with equal variance. The variance test procedure will test the validity of this assumption.

Analyze

Understanding Six Sigma

Example : A study was performed in order to evaluate the effectiveness of two devices for improving the efficiency of gas home-heating systems. Energy consumption in houses was measured after one of the two devices was installed. The two devices were an electric vent damper (Damper = 1) and a thermally activated vent damper (Damper = 2). The energy consumption data (BTU.In) are stacked in one column with a grouping column (Damper) containing identifiers or subscripts to denote the population. You are interested in comparing the variances of the two populations so that you can construct a two-sample t-test and confidence interval to compare the two dampers. 1 Choose Stat > Basic Statistics > 2 Variances. 2 Choose Samples in one column. 3 In Samples, select the column which contain the values 4 In Subscripts, enter Damper. Click OK.

Analyze

Understanding Six Sigma Test for Equal Variances for BTU.I n F-Test Test Statistic P-Value

Damper

1

1.19 0.558

Levene's Test Test Statistic P-Value

2

2.0

2.5 3.0 3.5 95% Bonferroni Confidence I ntervals for StDevs

4.0

Damper

1

2

5

10

15 BTU.I n

20

0.00 0.996

Analyze

Understanding Six Sigma

Test for Equal Variances: BTU.In versus Damper 95% Bonferroni confidence intervals for standard deviations Damper 1 2

N 40 50

Lower 2.40655 2.25447

StDev 3.01987 2.76702

Upper 4.02726 3.56416

F-Test (normal distribution) Test statistic = 1.19, p-value = 0.558 Levene's Test (any continuous distribution) Test statistic = 0.00, p-value = 0.996

Analyze

Understanding Six Sigma

Result Interpretation The variance test generates a plot that displays Bonferroni 95% confidence intervals for the population standard deviation at both factor levels. The graph also displays the side-by-side boxplots of the raw data for the two samples. Finally, the results of the F-test and Levene's test are given in both the Session window and the graph. Note that the 95% confidence level applies to the family of intervals and the asymmetry of the intervals is due to the skewness of the chi-square distribution.For the energy consumption example, the p-values of 0.558 and 0.996 are greater than reasonable choices of a, so you fail to reject the null hypothesis of the variances being equal. That is, these data do not provide enough evidence to claim that the two populations have unequal variances. Thus, it is reasonable to assume equal variances when using a two-sample t-procedure.

Analyze

Understanding Six Sigma

Test for Equal Variance is used when comparing the variance of two or more than two populations Example : You study conditions conducive to potato rot by injecting potatoes with bacteria that cause rotting and subjecting them to different temperature and oxygen regimes. Before performing analysis of variance, you check the equal variance assumption using the test for equal variances.

1Open the worksheet . 2 Choose Stat > ANOVA > Test for Equal Variances. 3 In Response, enter Rot. 4 In Factors, enter Temp Oxygen. Click OK.

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Understanding Six Sigma

Test for Equal Variances for Rot Temp

Oxygen Bartlett's Test

2 10

Test Statistic P-Value

6

Levene's Test Test Statistic P-Value

10

2 16

2.71 0.744

6 10 0 20 40 60 80 100 120 140 95% Bonferroni Confidence Intervals for StDevs

0.37 0.858

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Understanding Six Sigma

Test for Equal Variances: Rot versus Temp, Oxygen 95% Bonferroni confidence intervals for standard deviations Temp 10 10 10 16 16 16

Oxygen 2 6 10 2 6 10

N 3 3 3 3 3 3

Lower 2.26029 1.28146 2.80104 1.54013 1.50012 3.55677

StDev 5.29150 3.00000 6.55744 3.60555 3.51188 8.32666

Upper 81.890 46.427 101.481 55.799 54.349 128.862

Bartlett's Test (normal distribution) Test statistic = 2.71, p-value = 0.744 Levene's Test (any continuous distribution) Test statistic = 0.37, p-value = 0.858 Test for Equal Variances: Rot versus Temp, Oxygen

Analyze

Understanding Six Sigma

Interpreting the results The test for equal variances generates a plot that displays Bonferroni 95% confidence intervals for the response standard deviation at each level. Bartlett's and Levene's test results are displayed in both the Session window and in the graph. Note that the 95% confidence level applies to the family of intervals and the asymmetry of the intervals is due to the skewness of the chi-square distribution. For the potato rot example, the p-values of 0.744 and 0.858 are greater than reasonable choices of a, so you fail to reject the null hypothesis of the variances being equal. That is, these data do not provide enough evidence to claim that the populations have unequal variances.

Analyze - Regression

Understanding Six Sigma

One Variable Regression with Minitab Example: You are trying to optimize the performance of an paint cure oven. One theory says that blower fan velocity affects evaporation of solvent in the paint. You are trying to prove that such a relationship exists by analyzing the data below.

Understanding Six Sigma

Analyze - Regression

The Concept of Regression A mathematical equation of describing a relationship between the ”Y” and “X’s” →Creating a Model of process where b0 = constant

There appears to be a linear relationship between floor space and annual sales… That is, Is the Annual sales reducing or increasing according to change floor space

b1 = slope

350

Annual Sales

Y = b0 + b1x + error

300

250

200

50

100

Floor Space

150

Analyze - Regression

Understanding Six Sigma

Regression analysis is used to investigate and model the relationship between a response variable and one or more predictors. Use least squares procedures when your response variable is continuous. Use partial least squares regression when your predictors are highly correlated or outnumber your observations.· Use logistic regression when your response variable is categorical.Both least squares and logistic regression methods estimate parameters in the model so that the fit of the model is optimized. Least squares regression minimizes the sum of squared errors to obtain parameter estimates,

Analyze - Regression

Understanding Six Sigma

Example : You are a manufacturer who wants to obtain a quality measure on a product, but the procedure to obtain the measure is expensive. There is an indirect approach, which uses a different product score (Score 1) in place of the actual quality measure (Score 2). This approach is less costly but also is less precise. You can use regression to see if Score 1 explains a significant amount of variance in Score 2 to determine if Score 1 is an acceptable substitute for Score 2.

Score1 Score2 4.1 2.1 2.2 1.5 2.7 1.7 6.0 2.5 8.5 3.0 4.1 2.1 9.0 3.2 8.0 2.8 7.5 2.5

Analyze - Regression

Understanding Six Sigma

1 Choose Stat > Regression > Regression. 2 In Response, enter Score2. 3 In Predictors, enter Score1. 4 Click OK.

Regression Analysis: Score2 versus Score1 The regression equation is Score2 = 1.12 + 0.218 Score1 Predictor Coef SE Coef T P Constant 1.1177 0.1093 10.23 0.000 Score1 0.21767 0.01740 12.51 0.000 S = 0.127419 R-Sq = 95.7% R-Sq(adj) = 95.1% The R2 value shows that Score 1 explains 95.7% of the variance in Score 2, indicating that the model fits the data extremely well.

Analyze

Understanding Six Sigma

Analyze Phase • 4M Diagram • Hypothesis Testing • Mean Testing • Variance Testing • Regression Analysis

The Aim of the Analyze Phase is to : a. To List down all the Possible factors through Brainstorming. b. Pick out the Vital few Potential Factors out of Trivial Many. c. Statistical verification of Potential Factors by means of various type of Tests. d. Ascertaining whether we have considered all factors & nothing is Left Out.

Understanding Six Sigma

Analyze - Cause & Effect Diagram 4 M Diagram MAN

MACHINE Cause

Cause

Cause

Cause

Effect Cause Cause Cause METHOD

MATERIAL

4 M ( Man, Method, Machine & Material ) Diagram is used to list down all the Probable factors (causes ) responsible for the Major Problem ( Effect ). After brainstorming the Significant Factors are selected for further comparison ( Hypothesis Testing ) The Symptom or result is put under the Dark Box on the Right.. Lighter Boxes at the end of the Large Bones are main groups in which ideas are classified. The Lighter Boxes consist of Four Ms - Man,Method, Machine & Material. The Middle Bones indicates the direction of path from cause to effect.

Analyze

Understanding Six Sigma

Interpreting the results The test for equal variances generates a plot that displays Bonferroni 95% confidence intervals for the response standard deviation at each level. Bartlett's and Levene's test results are displayed in both the Session window and in the graph. Note that the 95% confidence level applies to the family of intervals and the asymmetry of the intervals is due to the skewness of the chi-square distribution. For the potato rot example, the p-values of 0.744 and 0.858 are greater than reasonable choices of a, so you fail to reject the null hypothesis of the variances being equal. That is, these data do not provide enough evidence to claim that the populations have unequal variances.

Understanding Six Sigma

Analyze - Regression

The Concept of Regression A mathematical equation of describing a relationship between the ”Y” and “X’s” →Creating a Model of process where b0 = constant

There appears to be a linear relationship between floor space and annual sales… That is, Is the Annual sales reducing or increasing according to change floor space

b1 = slope

350

Annual Sales

Y = b0 + b1x + error

300

250

200

50

100

Floor Space

150

Analyze - Regression

Understanding Six Sigma

Regression analysis is used to investigate and model the relationship between a response variable and one or more predictors. Use least squares procedures when your response variable is continuous. Use partial least squares regression when your predictors are highly correlated or outnumber your observations.. Use logistic regression when your response variable is categorical. Both least squares and logistic regression methods estimate parameters in the model so that the fit of the model is optimized. Least squares regression minimizes the sum of squared errors to obtain parameter estimates,

Understanding Six Sigma

Analyze - Regression

Example : Do regression and residual analysis for yield as shown in the table.Interpret the output results. Please note that A,B,C are factors & yield is response. S.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A 2 2 8 6 5 8 5 3 2 1 9 5 3 2 1 4 2 1 2 5

B 3 1 3 4 5 3 1 2 2 8 7 6 5 6 7 2 4 6 5 6

C 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Yield 85 71 3129 1384 875 3159 823 254 150 298 4631 978 367 296 303 556 266 294 313 1058

Solution : 1.) Enter the columns A, B, C (Factors ) and Yield ( response ) in minitab Excel sheet.

Analyze - Regression

Understanding Six Sigma 2) Go to stat> Regression > Regression.

Analyze - Regression 3) Select Yield as Response and A,B,C as Predictors by double clicking on all.

Understanding Six Sigma

Understanding Six Sigma

Analyze - Regression 4) Click OK. Regression Analysis: Yield versus A, B, C The regression equation is Yield = - 1277 + 458 A + 136 B - 1.54 C Predictor Constant A B C

Coef -1277.3 457.70 135.72 -1.544

SE Coef 360.6 47.64 61.28 4.579

T -3.54 9.61 2.21 -0.34

R2 & R2 (adj) > 64 % indicating a strong corelation between the Factors & the Response (Yield)

P 0.003 0.000 0.042 0.740

S = 487.537 R-Sq = 86.9% R-Sq(adj) = 84.5% Analysis of Variance Source DF SS MS F P Regression 3 25308562 8436187 35.49 0.000 Residual Error 16 3803071 237692 Total 19 29111633 Source A B C

DF 1 1 1

Seq SS 23966672 1314871 27018

Unusual Observations Obs 11

A 9.00

Yield 4631

Fit 3707

SE Fit 308

Residual 924

St Resid 2.44R

R denotes an observation with a large standardized residual.

Understanding Six Sigma

Minitab fits the regression Line using the Least square method. As shown in the diagram the least square method minimizes the sum of the squared distances between the points and the fitted Line.`

Response

Analyze - Regression

Predictors Fitted Value : The predicted y or ; the mean response value for the given predictor values using the estimated regression equation. Residuals :The difference (ei) between the observed values and predicted or fitted values (data minus fits). This part of the observation is not explained by the fitted model. The formula for the residual of an observation is: ei = (yi - i)

YIELD = RESIDUAL + FITS

Understanding Six Sigma

Analyze - Regression

Three Regressions Models Linear Y = bo + b1X

Quadratic Y = bo + b11X2

Cubic Y = bo + b1X + b11X2

Y is the response; X is the predictor; bo is the intercept; and b1, b11, and b111 are the coefficients

Control

Understanding Six Sigma

Control Phase • Statistical Process Control • Control Chart AIM of Control Phase Control Charts are used to track process statistics over Time and to detect the presence of Special Causes. • Provides structured closure of projects and re-allocation of resources • Provides systematic changes to ensure the process continues in a new path of optimization. It Transfers sustainability of the improvement to the appropriate members of the Advocacy Team •

Provides communication of new procedures and systems to process owners

• Ensures that the new process conditions are documented and monitored

Control

Understanding Six Sigma

Statistical Process Control (SPC)? (SPC) Statistical Statistical methods are used to monitor and analyze process variation from sample data Process Any repetitive (manual or automatic) task or steps Control Provides an early warning signal that a process has changed. The warning allows you to make decisions about the process while there is still time to correct the problem before it can be seen in the final output. Six Sigma Quality focuses on moving control up stream in a process to leverage the input characteristics for the Y response. If we can measure and control the vital few X’s, control of the Y should be assured. Statistical Process Control Enables us to control our process using statistical methods to signal when process adjustments are needed.

Understanding Six Sigma

Control The Logic of SPC

Process Capability

Desired Output

●

Upper Control Limit

X

Controller

●

Lower Control Limit

Samples

Input

Process

Output

A B C D E L M N O P Controllable factors Uncontrollable factors - Assignable causes - Common causes - Adjustable - Noise - Special - Inherent causes

SPC has traditionally been used to monitor and control the output of processes. Six Sigma Quality focuses on moving control upstream to the leverage input characteristic for Y. If we can measure and control the vital few X’s, control of Y should be assured.

Understanding Six Sigma

Control

Control Charts A Control chart is a graphical display of measurements ( usaually aggregated in the form of means or other statistics) of an Industrial Process through time. By carefully scrutinizing the chart, a quality engineer can identify any potential problem with the Production process . The idea is that when a process is in control , the variable being measured - the mean of every four Observations, for example - should remain stable through the time. the Mean should stay somewhere around the middle line ( the grand mean for the process ) and not wander off by more than the fixed standard deviations of the process . The required number standard deviations is chosen so that there will be a small probability of exceeding them when the process is in control . Addition and subtraction of the required number of standard deviations ( three ) give us the Upper Control Limit ( UCL ) and the Lower Control Limit ( LCL) of the control chart. When the bounds are breached , the process is deemed out of control. A control chart is a time plot of a statistic, such as a sample mean, range, standard deviation , or proportion, with a centerline and upper and lower control limits. The limits give the desired range of values for the statistic. When the statistic is outside the bounds, or when its time plot reveals certain patterns, the process may be out of control.

Understanding Six Sigma

Control Procedure of Control Chart Selection Characteristic definition of control chart Variable Data Type?

No

No

No

Defect ratio

Faults of parts Yes No

n =constant Yes

Yes

Yes

c Control chart

u Control chart

No

Subgroup sampling?

Average calculation

Yes

Yes

X Control chart

No

Yes

Yes No

n =constant Yes

pn control chart

p Control chart

Yes

Xbar-R Control chart

n＞ 8

Easy to calculate Subgroup

Median Control chart

No

Xbar-R Control chart

Xbar-R Control chart

This procedure is on the condition that data can be collected after Gage R&R.

Control - Types of Control Charts

Understanding Six Sigma

T y p e s o f C o n tr o l C h a r ts V a r ia b le s C h a r ts f o r m o n i t o r in g c o n t in u o u s X 's

A ttr ib u te s C h a r ts f o r m o n it o r i n g d is c r e t e X 's

A v e ra g e & R a n g e Xbar & R n < 10, t y p ic a ll y 3 - 5

F r a c tio n D e f e c tiv e p C h a rt t y p i c a l ly n ≥ 5 0 tra c k s d p u /d p o

A v e ra g e & S t d D e v ia t i o n Xbar & σ n ≥ 10

N u m b e r D e fe c tiv e n p C h a rt n ≥ 5 0 ( c o n s ta n t) tra c k s # d e f

M e d ia n & R a n g e X & R n<10, t y p ic a ll y 3 - 5

N u m b e r o f D e fe c ts c C h a rt c>5

I n d iv id u a l & M o v in g R a n g e Xm R n=1

N u m b e r o f D e f / U n it u C h a rt n v a r ia b le

• There are basically two types of control charts: –Variables charts - these charts are used for monitoring X variables that are continuous, such as, a diameter or consumer satisfaction rating. –Attribute charts - these charts are used for monitoring discrete X variables, such as, good/bad counts, or inventory levels. • Refer to the diagram for a summary list of the specific control chart types

In order to select the appropriate control chart for monitoring your process, first determine if your key process variables (X’s) are continuous or discrete. There are specific control charts for both continuous data and discrete data.

Understanding Six Sigma

Control

Variable control chart : Customer Satisfaction Index Example : A consumer services organization wants to monitor consumer satisfaction for their company. Each week, a survey from each of the company’s ten regional service centers is evaluated and the scores are tabulated. The following is an example of how an Xbar/R control chart could be used to monitor consumer satisfaction. In this example, higher is better: The vital information for creating an Xbar/R control chart : Total subgroups = 25 Subgroup size, n = 10 Process average, X = 4.096 and R = 0.4504

Control

File Open : S4 > Xbar - R

Calc > Random Data > Normal Distributions - Generate : 10 - Store in column(s) : c1-c25 - Mean : 4.0 - Standard deviation : 0.6 ( Only c7 Mean : 2.8, Standard deviation : 1.6 Manip > Stack > Stack Columns - Stack the following columns : c1-c25 - Store the stacked data in : c26

Understanding Six Sigma

Understanding Six Sigma

Control Control Limit Formulas:

UCLX = X + A2 × R LCLX = X − A2 × R

While it is acceptable to compute temporary control limits after 5 to 10 subgroups, permanent limits require at least 25 subgroups of data points that are In-control for both the average and range charts.

UCLR = D 4 × R LCLR = D 3 × R Actual Control Limiit Calculations for the Data UCL = 4.096 + 0.308 x 0.4504 = 4.235 UCL = 4.096 - 0.308 x 0.4504 = 3.957 UCLR = 1.777 x 0.4504 = 0.8003 UCLR = 0.223 x 0.4504 = 0.1005

Understanding Six Sigma

Control - Control chart

Constant used for Control chart

n 1 2 3 4 5 6 7 8 9 10

A2 2.660 1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308

A3 3.760 2.659 1.954 1.628 1.427 1.287 1.182 1.099 1.032 0.975

D3 0 0 0 0 0 0.076 0.136 0.184 0.223

D4 3.267 2.575 2.282 2.115 2.004 1.924 1.864 1.816 1.777

B3 0 0 0 0 0.03 0.118 0.185 0.239 0.284

B4 3.267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716

d2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078

c4 0.7979 0.8862 0.9213 0.9400 0.9515 0.9594 0.9650 0.9693 0.9727

Understanding Six Sigma

Control

Example of a variable control chart : Customer Satisfaction Index

Sample Mean

Customer Satisfaction Index

X= 3.9 3 8 3.5

2.5 S u b g ro u p 0 4 Sample Range

3.0 S L= 4.5 4 6

4.5

3 2

-3.0 S L = 3.3 2 9 1

5

10

15

20

25 3.0 S L = 3.5 0 9 R = 1.9 7 5

1 0

-3.0 S L = 0.4 4 0 6

• The weekly evaluation averages 7 and 16 fell below 3.957. • This change in “consumer satisfaction” score was driven by some assignable cause (either system-related or region initiated). • The appropriate action would be to investigate, identify and fix the assignable source of the variation. • The variation among the regional centers for week 7 is larger than expected.

An “Out-of-control” indication can come from either chart, independently.

Understanding Six Sigma

Control ~ Attribute data

Example : A local dental group wanted to know why a lot of their patients fail to keep their appointments. A problem solving team was assembled and decided to use a p Chart to track the percentages of “no shows”. The dental clinic began logging monthly percentages of “no shows” for each month. Of the total appointments for each month, % “no shows” plus % “shows” equal 100%. Since a “no show” is a defective appointment, the average total fraction defective is called p. Year Month % Failed Year Month % Failed Month %Failed

1996 Jul 40

Aug 36

Sep 36

Jan 20 Jul 16

Feb 26 Aug 10

Mar 25 Sep 12

Oct 42

Nov 42

Dec 40

Apr 19 Oct 12

May 20

Jun 18

1997

p Chart Formulas:

np n np p= n

p = 236/600 = 0.39333, where np = 40+36+36+42+42+40 = 236 the fraction is based on 600, total possible for 6 months

p=

UCL = p + 3

(

p × 1− p n

)

and

LCL = p − 3

(

p × 1− p n

)

UCL = .39333+3(.39333*.60667)/100)? = 0.539 LCL = .39333- 3(.39333*.60667)/100)? = 0.246

Control

Understanding Six Sigma

Definition of Stability •

A process output is considered stable when it consists of only common-cause variation.

•

Stability also means all subgroup averages and ranges are between their respective control limits and display no evidence of assignablesource (special -cause) variation.

•

If nonrandom patterns of data appear on the control chart, or when a point is beyond the control limits, then this is a strong signal that assignable-source (special-cause) variation is present in your process.

A stable process will rarely produce an output that lies outside of the +/- three sigma stable process variation region.

Understanding Six Sigma

Control Determination control limit of control chart •

The Empirical Rules emphasized that when a subgroup average falls outside of the 3σ limits, it is a pretty rare event. Process stability is defined in terms of these three sigma limits.

•

Another way of visualizing how control charts work: Think about a sequential or time-ordered hypothesis test for each new subgroup. α/2

H o : µι = µ Ha: µι ≠ µ

α/2

The hypothesis test provides the criteria for determining if a difference exists between the subgroup mean and the process average

The control limits are variation limits, not acceptance limits! Specification limits do Not appear on SPC charts!

Understanding Six Sigma

Control Example : The following samples were taken: S1 22 28 15 17 16

S2 26 22 21 22 27

S3 20 21 20 24 19

S4 18 16 20 19 20

S5 19 22 21 24 22

Calculate the LCL & UCL for the X-R chart using Minitab. Comment on the results. Find out whether process is in control or not. Solution : Step 1 ) Copy the data in Minitab Worksheet.

Control 2 ) Stack the data into one column and subscript into other. Go to data>stack> columns 3) Select columns from S1-S5. Stack columns values into c7 and their subscript in c6.

Understanding Six Sigma

Understanding Six Sigma

Control 4) As the sample size is less than 10 so Xbar-R would be used Go to stat > control charts >variable charts for sub groups > X bar - R

5) Select the Option all observations in one column and select c7 in which all the data is stacked. Go to Xbar-Options

Control 6 ) Go to Estimate and choose Rbar 7 ) Go to S Limits and enter 1 2 3 in the dialog box

8 ) Go to Tests and select all the eight standard tests for special causes.

Understanding Six Sigma

Understanding Six Sigma

Control

Test for Special Causes • 1 point more than 3 standard deviations from center line • 9 points in a row on same side of center line • 6 points in a row, all increasing or all decreasing •14 points in a row, alternating up and down •2 out of 3 points > 2 standard deviations from center line (same side) • 4 out of 5 points > 1 standard deviation from center line (same side) • 15 points in a row within 1 standard deviation of center line (either side) • 8 points in a row > 1 standard deviation from center line (either side) The Tests for special causes detects a special pattern in the data plotted on the Chart. The occurrence of the pattern suggests a special cause for the variation.

Understanding Six Sigma

Control 9 ) Click OK. Xbar-R Chart of C7 40

Sample M ean

+12SL=36.07 30 +3SL=24.65 __ X=20.84 -3SL=17.03

20

10 -12SL=5.61 1

2

3 Sample

4

5

40

Sam ple R ange

+12SL=36.02 30 20 +3SL=13.96 _ R=6.6

10 0

-12SL=0 -3SL=0 1

2

3 Sample

4

5

10 ) All the Points are lying within +/- 3σ Limits, so the process is within the control limits.

Control

Understanding Six Sigma

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 Rejects Sampled when you need to inspect the whole lot. Solution: 1) Open the worksheet and enter data. 2) Choose Stat > Control Charts >Attributes Charts > P. 3) In Variables, enter Rejects. 4) In Subgroup sizes, enter Sampled. Click OK.

20 18 14 16 13 29 21 14 6 6 7 7 9 5 8 9 9 10 9 10

98 104 97 99 97 102 104 101 55 48 50 53 56 49 56 53 52 51 52 47

Understanding Six Sigma

Control Minitab Output :

P Chart of Rejects 0.35

UCL=0.3324

0.30

1

0.25 Proportion

Sample 6 is outside the upper control limit. Consider inspecting the lot.

0.20

_ P=0.1685

0.15 0.10 0.05

LCL=0.0047

0.00 2

4

6

8

10 12 Sample

Tests performed with unequal sample sizes

14

16

18

20