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Six Sigma Green Belt Training

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SIX SIGMA GREEN BELT TRAINING Shailja Chaudhry National Institute of Technology, Kurukshetra
SIX SIGMA OVERVIEW AND EVOLUTION
What is Six Sigma A customer focused business improvement process Driven by teamwork, consensus & logical reasoning Structured methodology – DMAIC Focuses on making the process robust & reduce variations Applies to any Process
What is Six Sigma  Six sigma is a highly disciplined and quantitative strategic business improvement approach that seeks to increase both customer satisfaction and an organization’s financial health.  Six Sigma helps a company focus on developing and delivering near- perfect products (durable goods or services), to improve customer satisfaction and the bottom line.
What Six Sigma is NOT Six Sigma is NOT A Quality Program Cure for World Hunger Only for Technical People Just about Statistics Used when solution is known Used for Firefighting
Six-Sigma – A note from Originator of Six Sigma  “Six Sigma is not an improvement program. It is instead a business philosophy that employs a step by step approach to reducing variation, increasing quality, customer satisfaction, and in time, market share”
Overview of Six Sigma CULTURAL CHANGE TRANSFORM THE ORGANIZATION GROWTH REDUCE COSTS PAIN, URGENCY, SURVIVAL SIX SIGMA AS A PHILOSOPHY SIX SIGMA AS A PROCESS SIX SIGMA AS A STATISTICAL TOOL
What is Six Sigma?  Sigma is a measurement that indicates how a process is performing  Six sigma stands for Six Standard Deviations (Sigma is the Greek letter used to represent standard deviation in statistics) from mean. Six Sigma methodology provides the techniques and tools to improve the capability and reduce the defects in any process.  Six Sigma is structured application of tools and techniques applied on project basis to achieve sustained strategic results.
What is Six Sigma A Vision of a Six Sigma Company Organizational Issue • Problem Resolution • Behavior • Decision Making • Process Adjustment • Supplier Relationship • Planning • Design • Employee Training • Chain-of-command • Direction • Manpower Traditional approach • Fixing (symptoms) • Reactive • Experience-based • Tweaking • Cost (piece price) • Short-term • Performance • If Time Permits • Hierarchy • Seat-of-pants • Cost Six Sigma Approach • Preventing (causes) • Data-based • Controlling • Capability • Long-term • Producibility • Mandated • Empowered Teams • Benchmarking and metrics • Asset Traditional approach • Fixing (symptoms) • Reactive • Experience-based • Tweaking • Cost (piece price) • Short-term • Performance • If Time Permits • Hierarchy • Seat-of-pants • Cost Six Sigma Approach • Preventing (causes) • Data-based • Controlling • Capability • Long-term • Producibility • Mandated • Empowered Teams • Benchmarking and metrics • Asset
Character of 6s Traditional Quality / Six Sigma Quality Method ISSUE TRADITIONAL APPROACH SIX SIGMA APPROACH Index Data Target Range Method Action • % (Defect Rate) • Discrete Data • Satisfaction for Mfg. Process • Spec Outliner • Experience + Job • Bottom Up • σ • Discrete + Continuous Data • Customer Satisfaction • Variation improvement • Experience + Job + Statistical Ability • Top Down
Aligning The Focus Six Sigma Journey Started (Traditional) 1000 Unassigned Projects Six Sigma Project Strategic DirectionTactical Direction Individual Work Group (Lets do it) (Future)
What is Six Sigma Definition 2 Sigma 3 Sigma 4 Sigma 5 Sigma 6 Sigma 2σ 3σ 4σ 5σ 6σ Sigma Level Defects/ Million Opportunities% Yield 308,537 66,807 6,210 233 3.4 69.1 93.3 99.4 99.98 99.9997
Six Sigma : The Statistical Way LSL USL LSL USL LSL USL Process of Target Excessive Variation Reduce Variation % Center Process Customers feel the variation more than the mean Center Proces s Reduc e Spread Target Target Target
Six Sigma – Practical Meaning 99% Good (3.8 Sigma) 99.99966% Good (6 Sigma) • 20,000 lost articles of mail per hour • Unsafe drinking water for almost 15 minutes each day • 5,000 incorrect surgical operations per week • Two short or long landings at most major airports each day • 200,00 wrong drug prescriptions each year • No electricity for almost seven hours each month • Seven articles lost per hour • One unsafe minute every seven months • 1.7 incorrect operations per week • One short or long landing every five years • 68 wrong prescriptions per year • One hour without electricity every 34 years
Philosophy of Six Sigma  Know What’s Important to the Customer (CTQ)  Reduce Defects (DPMO)  Center Around Target (Mean)  Reduce Variation (σ)
Harvesting the fruit of Six Sigma
HISTORY OF SIX SIGMA
History of Six Sigma  Quality tools like SPC, Cost of Quality, Control Charts, Process capability etc. are known to industry for long time, much before birth of Six Sigma.  Quality Tools and Quality system implementation was not in conjunction with overall business Goals.  Traditional Quality Tools have limitations to orient the efforts on Quality Improvements to the Organizational direction basically due to approach.  Motorola was the first Company to initiate the Six Sigma breakthrough Strategy.
A Little Bit Of History  Six Sigma was developed by Bill Smith, QM at Motorola  It’s implementation began at Motorola n 1987  It allowed Motorola to win the first Baldrige Award in 1988  Motorola recorded more than $16 Billion savings as a result of Six Sigma  Several of the major companies in the world have adopted Six Sigma since then…. Texas Instruments, Asea Brown Boveri, AlliedSignal, General Electric, Bombardier, Nokia Mobile Phones, Lockheed Martin, Sony, Polaroid, Dupont, American Express, Ford Motor,….. The Six Sigma Breakthrough Strategy has become a Competitive Tool
Motorola Case Study In early 1980’ Motorola was facing a serious competitive challenge from Japanese Companies. Motorola was losing the market share and customer confidence. Motorola had not done any major changes to their products. The competitors from Japan were offering much better product at much lower price with no field failures.
Motorola Case Study….Continue  When Motorola studies the competitors products, it was revealed that the variation in key product characteristics is very low.  The competitors products were available at lower price.  The competitors products has very low warranty failure rate.  Motorola was not able to match the competitors price mainly due to high cost of Poor quality largely due to high reject rate, high rework / repair rate, high inspection cost, high warranty failure rate etc.  THE TECHNICAL TEAM CONCLUDED THAT THE COPETITORS ARE OFFERING BETTER PRODUCT AT LOWER COST.
Motorola Case Study Motorola requested to the competitors from Japan to permit the Team from Motorola to visit them fro Study. Motorola sent the team of managers to Japan to study the “Magic” of Japanese companies. What the team revealed?
Motorola Case Study  What Motorola learning was as follows:  Motorola was focusing too much on product Quality i.e. Inspection, rework, repair etc.  The internal defect rate was very high inside Motorola.  The reliability was slow since some of the defects were passing on to the customer as inspection lapses.  A dissatisfied customer was shouting loudly and was taking away min 10 potential customers.  As an effect of this, customers were lost to the competitors.
Motorola Case Study What was wrong?  Japanese were concentrating on ◦ Customers ◦ Processes ◦ People  Variation in product and process parameters was known and controlled  All people were well trained and highly motivated  All activities and processes were highly standardized i.e. no person dependence  Defect free lines and robust processes  Very less inspectors  Yet, very low defect rate, internal rejection and customer complaints  VERY HIGH LEVEL OF CUSTOMER SATISFACTION
Motorola Case Study WHAT WAS THE SECRET? THE SECRET WAS CONTROL OVER VARIATION Success factor: Proactive Vs. Reactive Quality
The Impact Of Added Inspection 3.4 ppm 100,000 ppm 6 ppm If the likelihood of detecting the defect is 70% and we have 10 consecutive inspectors with this level of capability, we would expect about 6 escaping defects out of every 1,000,000 products produced You can save yourself by producing quality not by
Motorola Case Study In order to address these issues, Motorola devised the Six Sigma methodology. Dr. Mikel Harry and Mr. Bill Smith were pioneers in Developing and implementing the Six Sigma methodology at Motorola. With implementation of Six Sigma, Motorola could achieve: 4σ level in one and half year time 5σ level in following year
Six Sigma Progress 1985 1987 1992 1995 2002 Johnson & Johnson, Ford, Nissan, HoneywellGeneral Electric Allied Signal Motorola Dr Mikel J Harry wrote a Paper relating early failures to quality
What can it do? Motorola:  5-Fold growth in Sales  Profits climbing by 20% pa  Cumulative savings of $14 billion over 11 years General Electric  $2 billion savings in just 3 years  The no. 1company in the USA Bechtel Corporation:  $200 million savings with investment of $30 million It is high time, that Indian Companies also start implementing Six Sigma for making breakthrough improvements and to remain
Quality and Value
Attempting to Define Quality Experts’ definitions of quality fall into two categories:  Level one quality is a simple matter of producing products or delivering services whose measurable characteristics satisfy a fixed set of specifications that are usually numerically defined.  Independent of any of their measurable characteristics, level two quantity products and services are simply those that satisfy customer expectations for their use or consumption. In short, level one quality means get it in the specs, and level two means satisfy the customer.
Quality Gap Quality Gap Understanding of Needs Customer Perception of Delivery Customer Expectations Design of Products Capability to Deliver Design Actual Delivery Design Gap Perception Gap Operations Gap Process Gap Understanding the Gap
Nine Dimensions of QUALITY According to modern management concepts, quality has nine dimensions: 1) Performance: main characteristics of the product/service 2) Aesthetics: appearance, feel, smell, taste 3) Special features: extra characteristics
Nine Dimensions of QUALITY 4) Conformance: how well the product/service conforms to customer’s expectations 5) Safety: risk of injury 6) Reliability: consistency of performance
Nine Dimensions of QUALITY 7) Durability: useful life of the product/service 8) Perceived Quality: indirect evaluation of quality (e.g. reputation) 9) Service after Sale: handling of Customer complaints and checking customer satisfaction.
Evolution of Quality Historically Contemporary Reactive Quality Quality Checks (QC) - Taking the defectives out of what is produced Proactive Quality “Create process that will produce less or no defects”
Old Concept Of Quality Past concepts of quality focused on “conformance to standards”. This definition assumed that as long as the company produced quality products and services, their performance standard was correct regardless of how those standards were met. Moreover, setting of standards and measurement of performance was mainly confined to the production areas and the commercial and other service functions were managed through command control.
Value Enrichment The term ‘Value Enrichment’ for the company means that they must strive to produce highest quality products at the lowest possible costs to be competitive in the global markets. For customers, the term ‘Value Enrichment’ means that they have the right to purchase high quality products/services at the lowest cost.
Concept Of Value Value to Customers Value = 𝐖𝐡𝐚𝐭 𝐘𝐨𝐮 𝐑𝐞𝐜𝐞𝐢𝐯𝐞 𝐖𝐡𝐚𝐭 𝐘𝐨𝐮 𝐏𝐚𝐲 Price + Inconvenienc e Real + Perceived
Definitions VALUE: THOSE ACTIVITIES THAT CONVERT MATERIALS OR IDEAS INTO GOODS OR SERVICES THAT GENERATE CASH
Definition ANYTHING THAT IS NOT VALUE IS WASTE
Six Sigma and Cost Of Quality Six Sigma has a very significant impact on the cost of quality. As the Sigma level moves up, the cost of quality comes down and vice versa. Traditionally recorded quality cost generally account for only 4 to 5 percent of sales which mainly comprise of scrap, re-work and warranty. There are additional costs of quality which are hidden and do not appear in the account books of the company, as they are intangible and difficult to measure.
Visible And Hidden Costs Visible Costs Hidden Costs • Scrap • Rework • Warranty Costs • Conversion efficiency of materials • Inadequate resources utilization • Excessive use of materials • Cost of re-design and re— inspection • Cost of resolving customer problems • Lost customers / Goodwill • High Inventory
Cost OF Quality At Various Levels Of Sigma 6 3.4 <10% 5 233 10-15% 4 6210 15-20% 3 66807 20-30% 2 308537 30-40% 1 6,90000 >40% Sigma Defect Rate (PPM) Cost Of Quality Competitive Level World Class Industry Average Non Competitive
What is The Cost Of Quality?  Cost of Quality: the cost of ensuring that the job is done right + the cost of not doing the job right. Cost of Conformance + Cost of Non- Conformance(Prevention and Appraisal) (Internal/External Defects)
Cost Of Quality Prevention Costs • Quality Planning • Process Evaluation / Improvement • Quality Improvement Meetings • Quality Training External Failure Costs • Complaint Handling • Rework / Correction • Re-Inspection Internal Failure Costs • Rework / Correction • Re-Inspection • Internal Reject • Loss of Business Appraisal Costs • Source Inspection • In / End-Process Inspection • Calibration • Specialist Cost Direct Costs Prevention Costs • Quality Planning • Process Evaluation / Improvement • Quality Improvement Meetings • Quality Training External Failure Costs • Complaint Handling • Rework / Correction • Re-Inspection Internal Failure Costs • Rework / Correction • Re-Inspection • Internal Reject • Loss of Business Appraisal Costs • Source Inspection • In / End-Process Inspection • Calibration • Specialist Cost
PHASES OF SIX SIGMA
Fundamental Steps There are 5 fundamental Steps involved in applying the breakthrough strategy for achieving Six Sigma. These steps are :-  Define  Measure  Analyze  Improve  Control
Define Phase This phase defines the project. It identifies critical customer requirements and links them to business needs. It also defines a project charter and the business processes to be undertaken for Six Sigma.
Define Define D CM A I Define Activities Identify Project, Champion and Project Owner Determine Customer Requirements and CTQs Define Problem, Objective, Goals and Benefits Define Stakeholders/Resource Analysis Map the Process Develop Project Plan Define Quality Tools Project Charter and Plan Effort/Impact Analysis Process Mapping Tree Diagram VOC Kano Model Pareto Analysis
Measurement Phase This phase involves selecting product characteristic, mapping respective process, making necessary measurements and recording the results of the process. This is essentially a data collection phase.
Measure – Operational Definition Measure M CD A I Measure Activities Determine operational Definitions Establish Performance Standards Develop Data Collection and Sampling Plan Validate the Measurements Measurement System Analysis Determine Process Capability and Baseline Measure Quality Tools Measurement Systems Analysis Check Sheet Process Capability Process FMEA
Analysis Phase In this phase an action plan is created to close the “gap” between how things currently work and how the organization would like them to work in order to meet the goals for a particular product or service. This phase also requires organizations to estimate their short term and long term capabilities.
Analyze Analyze A CMD I Analyze Activities Benchmark the Process or Product Analysis of the Process Map Brainstorm for likely causes Establish Causal Relationships Using Data Determine Root Cause(s) Using Data Analyze Quality Tools Cause and Effect or Event Diagram Graphical Analysis Statistical Analysis of Data Hypothesis Testing Correlation Regression DOE
Improvement Phase  This phase involves improving processes/product performance characteristics for achieving desired results and goals. This phase involves application of scientific tools and techniques for making tangible improvements in profitability and customer satisfaction.
Improve Improve I CMD A Improve Activities Develop Solution Alternatives Assess Risks and Benefits of Solution Alternatives Implement error-proofing solutions Validate Solution using a Pilot Implement Solution Determine Solution Effectiveness using Data Improve Quality Tools Brainstorming FMEA Risk Assessment Poka Yoke
Control Phase This Phase requires the process conditions to be properly documented and monitored through statistical process control methods. After a “setting in” period, the process capability should be reassessed. Depending upon the results of such a follow-up analysis, it may be sometimes necessary to revisit one or more of the preceding phases.
Control – Develop Standards Control CIMD A Control Activities Determine Needed Controls (measurement, design, etc.) Implement and Validate Controls Develop Transfer Plan Realize Benefits of Implementing Solution Institutional Changes Close Project and Communicate Results Control Quality Tools Statistical Process Control Process Map and FMEA Control Plans 5S Control Charts
Six Sigma Projects
Why Project Selection is Important?  High leverage projects lead to largest Savings  Large returns are expected by management to justify the investment in time and effort  Developing a Six Sigma culture depend upon successful projects having significant business impact
How To Focus Projects Process Cost Savings Focus Project Quality focus Product focus (Six Sigma Design) Problem Focus (Least Desirable Use)
Project Selection Align with company objectives and business plan (Annual Operating Plan) – Voice of Customer/CT’s Inputs – Quality (CTQ)/Cost (CTC)/ Delivery (CTD) – PPM / COPQ / RTY / Cycle Time Consistent with principles of Six Sigma – Eliminate process defects Concentrate on “Common” issues/opportunities …not “fir-fighting” Large enough to justify the investment
Project Desirability • Effort Required:- includes time required of team members and expenditure of money. • Probability of Success:- An assessment that takes into account various risk factors: + Time – uncertainty of the completion date + Effort – uncertainty of the investment required + Implementation – uncertainty of roadblocks
Project Desirability Matrix Hi Med Low HiMedLow Hi Med Low IMPACT EFFORT
Additional Project Considerations  Projects must serve as a learning experience for Green Belts to use the six Sigma tools  Projects scope should not be too large or take too long to implement  Projects scope should be manageable and take at least 255 of the potential Green Belt’s time.  Pareto Chart may be used to Scope the Project  Desirable to have a measurable variable for the primary project output/metrics
Additional Project Considerations • Projects must serve as a learning experience for Green Belts to use the six Sigma tools • Projects scope should not be too large or take too long to implement • Projects scope should be manageable and take at least 255 of the potential Green Belt’s time. • Pareto Chart may be used to Scope the Project • Desirable to have a measurable variable for the primary project output/metricsDO NOT try to Solve World Hunger
Strategy At Various Levels Almost every Organization can be divided into 3 basic levels:- 1. Business level 2. Operations level 3. Process level. It is extremely important that Six Sigma is understood and integrated at every level.
Strategies At Various Levels  Executives at the business level can use Six Sigma for improving market share, increasing profitability and organizations long term viability.  Managers at operations level can use Six Sigma to improve yield and reduce the labor and material cost.  At the process level engineers can use Six Sigma to reduce defects and variation and improve process capability leading to better customer satisfaction.
Factors To Control in Improvement Project  Resources Team availability The right tools  Schedule Be realistic Be aggressive Get buy-in  Scope of Work Watch for scope creep Stay focused Anticipate and mitigate risk Control any two areas, the third floats in response
Meetings – Make Them Effective  Defined goal for meeting  Notice and agenda  Decision makers prepare and participate  Action Items  Records  Balance Sheet – Focused on process, not topic – What helped us get to our goal – What could have been better – Take appropriate action
Skills Needed  People Leadership behaviors Communication  Process Time Management Schedule Coordination Problem Solving Risk analysis and mitigation Tactical Planning  Technical Six Sigma / Lean Tools Business Knowledge
Voice of Customer – CTP and CTQ
Establishing Customer Focus • Customer – Anyone internal or external to the organization who comes in contact with the product or output of work • Quality – performance to the standard expected by the Customer
Variation is the Enemy in Achieving Customer Satisfaction Variation •Uncertainty •Unknown •Disbelief •Risk •Defect Rate
What is Variation Variation is any deviation from the expected outcome.
Something more on Variation  Any process has variation  There are two kinds of variation Common cause variation Special cause variation  Variation is measured in terms of sigma or standard deviation.
Variation and Standard Deviation If a good deal of variation exists in a process activity, that activity will have a very large standard deviation. As a result, the distribution will be very wide and flat. Less Variation More Variation
Types of Variation  Special Cause: something different happening at a certain time or place Common Cause: always present to some degree in the process We tamper with the system if we treat all variations as if it were special cause
Dealing with Variation  Eliminate special cause variation by recognizing it and dealing with it outside of the process  Reduce common cause variation by improving the process
Whom would you Prefer? Operator - 1 Operator - 2
Critical To Quality (CTQ) are the key measurable characteristics of a product or process whose performance standards or specification limits must be met in order to satisfy the customer. They align improvement or design efforts with customer requirements.
Critical To Quality (CTQ) 1. To put it in layman’s terms, CTQs are what the customer experts of a product... 2. ...the spoken needs of the customer. 3. The customer may often express this in plain English, but it is up to us to convert them to measurable terms using tools such as QFD, DFMEA, etc.
Critical To Quality (CTQ) 1. List customer needs. 2. Identify the major drivers for these needs (major means those which will ensure that the need is addressed). 3. Break each driver into greater detail. 4. Stop the breakdown of each level when you have reached sufficient information that enables you to measure whether you meet the customer need or not.
Example – CTQ Tree Ease of Operation Ease of Maintenance Ease of Operation and Maintenance Operator Training Time (hrs.) Setup Time (minutes) Operation Accuracy (errors/1000 ops) Mean Time to Restore (MTTR) # Special Tools Required Maintenance Training Time (hrs.) Need CTQsDrivers SpecificGeneral Hard to Measure Easy to Measure
PROJECT CHARTER
Importance of Project Charter A project charter is a written document and works as an agreement between management and the team about what is expected. The charter:  Clarifies what is expected of the team.  Keeps the team focused.  Keeps the team aligned with organizational priorities.  Transfers the project from the champion(s) to the project team.
Team Charter  Problem Statement – Currently we carry out reblows to the extent of about 11-15% resulting in lower converter life, lower productivity of converter and increased Ferro-alloy and oxygen consumption.  Scope – All batches and all converters in SMS 1.  Project Goal and Measures – Reblows should be less than 7.5% and 9%.  Expected Business Results – We hope to save Rs. Xxxxx lakhs per year due to this reduction in reblows.
Team Charter  Team Members –Supervisor, two operators, technical services, quality control  Support Required –Allow for weekly team meetings –Team budget for quick wins  Schedule –Measure (7wks), Analyze (4wks), Improve (6wks), Check (2wks), Control (1wk), Standardise/Close (1wk)
Usual elements of a Project Charter  Project Description – Business Case  Scope – Process/Product  Goals and Measures (Key Indicators)  Expected Business Results  Team Members  Support Required  Expected Customer Benefits  Schedule
MEASURE OVERVIEW
Measurement Objective The Measure phase aims to set a baseline in terms of process performance through the development of clear and meaningful measurement systems
The Measurement Process TOOLS AND TECHNIQUS OF MEASURE Develop Process Measures Collect Process Data Check Data Quality Understand Process Behavior Baseline Process Capability and Potential How do you measure the problem? When and where does the data come from? How does the process currently behave? What is the current performance of the process with respect to the customer Does the data represent what you think it does  Statistics  Operational Definitions  Data Worlds  Process Capability  Cp, Cpk  DPMO  Distributions  First pass yield  Short/long term variation  MSA  Gage R&R  Data Collection Methods  Data Collection Plans  Sampling
Statistical and Data World If the data is AttributeCountContinuous Relevant statistical model is … Binomial Distribution Defects per Unit (DPU) Always Poisson – if process is in control Poisson Distribution When does the statistical model apply Common statistics are… Always Binomial – if process is in control Percentage (Proportion) Average (mean), Standard Deviation (sigma) Not always – validity of normality needs to be checked Normal Distribution
Basic Statistics
Statistics  The science of: –Collecting, –Describing –Analyzing –Interpreting data... And Making Decisions
What are Statistics?  Descriptive Statistics – Summarize and describe a set of data – Mean, median, range, standard deviation, variance, ....  Analytical Statistical (or Statistics) – Techniques that help us make decisions in the face of uncertainty – Use concepts of descriptive statistics as a base – Hypothesis testing, means comparisons, variance comparisons, proportions comparisons, ...
Sample Versus Population  Using a small amount of data (Sample)... to make assumptions (inferences)... on a large amount of data (population).  Population: the total collection of observations or measurements that are if interest.  Sample: A subset of observations and measurements taken form the population.  Why do we use samples? Time Cost Destructive testing (need product left to sell !!) Other?
Measures of Central Tendency  What is the Median value of Distribution? – Median  What value represents the distribution? – Mode  What value represents the entire distribution? – Mean (x̄ )  What is the best measures of central tendency?
Data Distributions  Mean: Arithmetic average of a set of values – Reflects the influence of all values – Strongly influenced of all values  Median: Reflects the 50% rank – the center number after a set of numbers has been sorted from low to high. – Does not include all values in calculation – Is “robust” to extreme scores  Mode: The value or item occurring most frequently in a series of observations or statistical data.
Variable Data Location - MeanMonth # of Units Jan-2006 233 Feb-2006 281 Mar-2006 266 Apr-2006 237 May-2006 260 Jun-2006 250 Jul-2006 237 Aug-2006 275 Sep-2006 218 Oct-2006 279 Nov-2006 227 Dec-2006 246 Jan-2007 258 Feb-2007 272 Mar-2007 229 Apr-2007 240 May-2007 287 Jun-2007 260 Jul-2007 251 Aug-2007 288 Sep-2007 256 Oct-2007 219 Nov-2007 260 Dec-2007 249 n=24 = 𝟔𝟎𝟕𝟖 We have data on the monthly demand history of one of our key product lines. Let’s calculate the statistics for location.  Mean (𝑿)  Add all of the monthly numbers  Divide by the number of months in the sample.  N=24, = 𝟔𝟎𝟖𝟕 𝑿 = 𝟔𝟎𝟖𝟕 𝟐𝟒 = 𝟐𝟓𝟑. 𝟐𝟓 Our average monthly shipment is 253 units 𝝁 = 𝒊=𝟏 𝑵 𝑿𝒊 𝑵 𝒙 = 𝒊=𝟏 𝒏 𝑿𝒊 𝒏 Populatio n Sample
Variable Data Location - MedianMonth # of Units Jan-1999 233 Feb-1999 281 Mar-1999 266 Apr-1999 237 May-1999 260 Jun-1999 250 Jul-1999 237 Aug-1999 275 Sep-1999 218 Oct-1999 279 Nov-1999 227 Dec-1999 246 Jan-2000 258 Feb-2000 272 Mar-2000 229 Apr-2000 240 May-2000 287 Jun-2000 260 Jul-2000 251 Aug-2000 288 Sep-2000 256 Oct-2000 219 Nov-2000 260 Dec-2000 249 Month # of Units Sep-1999 218 Oct-2000 219 Nov-1999 227 Mar-2000 229 Jan-1999 233 Jul-1999 237 Apr-1999 237 Apr-2000 240 Dec-1999 246 Dec-2000 249 Jun-1999 250 Jul-2000 251 Sep-2000 256 Jan-2000 258 May-1999 260 Jun-2000 260 Nov-2000 260 Mar-1999 266 Feb-2000 272 Aug-1999 275 Oct-1999 279 Feb-1999 281 May-2000 287 Aug-2000 288 Statistics for location ~ Median (x) • Sort the data from lowest to highest • If there is an even number of observations, the median is the average of the two middle values (𝟐𝟓𝟏 + 𝟐𝟓𝟔) 𝟐 = 𝟐𝟓𝟑. 𝟓
Variable Data Location - MedianMonth # of Units Jan-1999 233 Feb-1999 281 Mar-1999 266 Apr-1999 237 May-1999 260 Jun-1999 250 Jul-1999 237 Aug-1999 275 Sep-1999 218 Oct-1999 279 Nov-1999 227 Dec-1999 246 Jan-2000 258 Feb-2000 272 Mar-2000 229 Apr-2000 240 May-2000 287 Jun-2000 260 Jul-2000 251 Aug-2000 288 Sep-2000 256 Oct-2000 219 Nov-2000 260 Dec-2000 249 Month # of Units Sep-1999 218 Oct-2000 219 Nov-1999 227 Mar-2000 229 Jan-1999 233 Jul-1999 237 Apr-1999 237 Apr-2000 240 Dec-1999 246 Dec-2000 249 Jun-1999 250 Jul-2000 251 Sep-2000 256 Jan-2000 258 May-1999 260 Jun-2000 260 Nov-2000 260 Mar-1999 266 Feb-2000 272 Aug-1999 275 Oct-1999 279 Feb-1999 281 May-2000 287 Aug-2000 288 Statistics for location ~  Mode • The most frequently occurring value is the mode 260 is the mode
Variable Data Location - Mode  Notes on mean –A measure of central tendency –Limitations:  Reflects the influence of all values  Strongly influenced by extreme values  Median (the centre number after sorting high to low) is robust to extreme values.
Variable Data Description – Range, Standard DeviationMonth # of Units Jan-2006 233 Feb-2006 281 Mar-2006 266 Apr-2006 237 May-2006 260 Jun-2006 250 Jul-2006 237 Aug-2006 275 Sep-2006 218 Oct-2006 279 Nov-2006 227 Dec-2006 246 Jan-2007 258 Feb-2007 272 Mar-2007 229 Apr-2007 240 May-2007 287 Jun-2007 260 Jul-2007 251 Aug-2007 288 Sep-2007 256 Oct-2007 219 Nov-2007 260 Dec-2007 249 • Let’s use this same data to calculate the statistics for dispersion  These statistics are Range and Standard Deviation
Example – commuting time Commute time (mins) 19.5 22.4 20.7 18.8 18.2 20.0 19.6 19.8 21.0 19.8 20.7 21.9 22.0 22.6 19.4 22.8 18.1 17.5 21.3 19.1 18.4 19.8 21.0 18.5 19.2 19.2 19.4 19.3 24.8 21.2 21.2 18.3 18.2 17.4 19.9 21.0 18.9 16.4 17.6 19.5 19.2 23.9 20.6 21.9 18.7 19.5 20.1 17.1 22.1 19.2 19.6 20.3 20.8 20.7 22.4 19.9 21.1 20.4 16.7 19.1 18.3 22.4 27.1 17.6 18.8 22.5 19.9 21.8 20.4 17.7 21.3 17.8 18.7 15.8 18.9 21.7 20.1 19.6 18.4 21.7 18.7 18.8 20.5 18.6 20.9 22.0 15.8 19.4 20.2 18.7 23.6 21.0 19.9 20.1 18.3 21.9 19.7 21.1 19.9 22.9 • Collect over a hundred occurrences. • Tabulate in chronological order. • Does the data show variation? • Can you make out anything with this arrangement of data? • Let us try and make some sense of this data…
Measure of variation – Standard Deviation and Range Category 1 15 2018 22 24 25 19.50 19.75 20.00 20.25 20.50 .. Summary for Commute time Anderson – Darling Normality Test A-Squared P-Value 0.42 0.312 What are the relative merits and demerits of standard deviation over range? Mean St.Dev Variance Skewness Kurtosis N Minimum 1st Quartile Median 3rd Quartile Maximum 95% Confidence Interval for Mean 95% Confidence Interval for Mean 95% Confidence Interval for Mean 19.632 20.006 1.884 3.550 0.54470 1.30256 100 15.754 18.714 19.819 21.186 27.054 20.380 19.448 20.263 1.654 2.189Mean Median One measure of variation (std. dev) Another Measure of variation (Range) Outlier *
Variable Data Dispersion – Standard Deviation “s” or “standard deviation”  What does it mean? –Standard deviation is a measure of dispersion (or how our data is spread out). –Range will tell us the difference between the highest and lowest values in a data set, but nothing about how the data are distributed. –We need deviation to statistically describe the distribution of values.
Variable Data Dispersion – Standard Deviation How we calculate it…  A measure of how far each point deviates from the mean  We square each distance so that all the numbers are positive  The sum of the squares, divided by the sample size, is equal to the variance  The square root of the variance is the standard deviation – Variance can be added; standard deviations cannot 𝜎 = 𝑖=1 𝑛 (𝑥𝑖 − 𝜇)2 𝑁 s = 𝑖=1 𝑛 (𝑥𝑖 − 𝑥)2 𝑛 − 1Population Sample
Variable Data Dispersion – Standard Deviation CalculationMonth # of Units Jan-2006 233 Feb-2006 281 Mar-2006 266 Apr-2006 237 May-2006 260 Jun-2006 250 Jul-2006 237 Aug-2006 275 Sep-2006 218 Oct-2006 279 Nov-2006 227 Dec-2006 246 Jan-2007 258 Feb-2007 272 Mar-2007 229 Apr-2007 240 May-2007 287 Jun-2007 260 Jul-2007 251 Aug-2007 288 Sep-2007 256 Oct-2007 219 Nov-2007 260 Dec-2007 249 -20.25 27.75 12.75 -16.25 6.75 -3.25 -16.25 21.75 -35.25 25.75 -26.25 -7.25 4.75 18.75 -24.25 -13.25 33.75 6.75 -2.25 34.75 2.75 -34.25 6.75 -4.25 -20.25 27.75 12.75 -16.25 6.75 -3.25 -16.25 21.75 -35.25 25.75 -26.25 -7.25 4.75 18.75 -24.25 -13.25 33.75 6.75 -2.25 34.75 2.75 -34.25 6.75 -4.25 𝑿𝒊 − 𝑿 (𝑿𝒊−𝑿) 𝟐 𝑺 = 𝒊=𝟏 𝒏 (𝒙𝒊 − 𝒙) 𝟐 𝒏 − 𝟏 𝑿 = 𝟐𝟓𝟑. 𝟐𝟓 Calculate the Mean Count the Samples n = 24 Square each subtraction result Subtract the mean from each value Sum the Squares Calculate the Denominator Complete the Calculation 𝒊=𝟏 𝒏 (𝒙𝒊 − 𝒙) 𝟐= 𝟗. 𝟖𝟑𝟏 𝒏 − 𝟏 = 𝟐𝟒 − 𝟏 = 𝟐𝟑 𝒔 = 𝟗. 𝟖𝟑𝟏 𝟐𝟑 = 𝟐𝟎. 𝟕
Variable Data Dispersion – Standard Deviation 𝜎 = 𝑖=1 𝑛 (𝑥𝑖 − 𝜇)2 𝑁  Standard deviation of a population – If your data is from a population versus a sample from a population, use this formula to calculate standard deviation – The difference is the denominator “N” versus “n-1”
Fundamental Topic The Normal Curve ◦ Processes have natural variation ◦ Many processes behave “normally” ◦ Characterized by Bell Shaped Curve – Mean near peak – Curve is symmetric ◦ Mean ◦ Standard Deviation Histogram of Diameter, with Normal Curve Diameter Frequency
Measures of Variability •The Range is the distance between the extreme values of data set. (Highest – Lowest) •The Variance(S ) is the Average Squared Deviation of each data point from the Mean. •The Standard Deviation (s) is the Square Root of the Variance. •The range is more sensitive to outliners than the variance. •The most common and useful measure of variation is the Standard Deviation.
Sample of Statistics versus Population Parameters EstimateStatistics Parameters µ = Population Mean s = Sample Standard Deviation X = Sample Mean σ = Population Standard Deviation
Statistical Calculation (Sample) 𝑋 = 𝑖=1 𝑛 𝑋𝑖 𝑛 𝑆2 = 𝑖=1 𝑛 (𝑋𝑖−𝑋)2 𝑛 − 1 𝜎 = 𝑅/𝑑2 s = 𝑖=1 𝑛 (𝑋𝑖 − 𝑋)2 𝑛 − 1 Standard Deviation Standard Deviation VarianceMean n 2 3 4 6 𝒅 𝟐 1.128 1.693 2.059 2.326
Statistical Calculation (Population) 𝜇 ≈ 𝑋𝝈 𝟐 = 𝒊=𝟏 𝑵 (𝑿𝒊 − 𝝁) 𝟐 𝑵 𝝈 = 𝒊=𝟏 𝑵 (𝑿𝒊 − 𝝁) 𝟐 𝑵 Standard Deviation VarianceMean
Normal Distribution Description of a NORMAL DISTRIBUTION LOCATION: •The Central Tendency •It is usually expressed as the AVERAGE SPREAD: •The dispersion •It is usually expressed as standard deviation (Sigma) LOCATION SPREAD
Properties of Normal Distribution •Normal Distribution is Symmetric –Has equal number of points on both sides –Mean Median and Mode Coincide •Normal Distribution is Infinite –The chance of finding a point anywhere on the plus and minus side (around the mean) is not absolutely Zero.
Properties Of Normal Distribution Normal Curve & Probability Areas -3𝝈 -2𝝈 -1𝝈 0 1𝝈 2𝝈 3𝝈 68% 95% 99.73%
Let’s Summarize… We need data study, predict and improve the processes. Data may be Variable or Attribute. To understand a data distribution, we need to know its Center, Spread and Shape. Normal Distribution is the most common but not the only shape.
Standard Deviation - Graphically 0 1 2 3 4 5 Monthly Demand in Units Frequency Month # of Units Jan-1999 233 Feb-1999 281 Mar-1999 266 Apr-1999 237 May-1999 260 Jun-1999 250 Jul-1999 237 Aug-1999 275 Sep-1999 218 Oct-1999 279 Nov-1999 227 Dec-1999 246 Jan-2000 258 Feb-2000 272 Mar-2000 229 Apr-2000 240 May-2000 287 Jun-2000 260 Jul-2000 251 Aug-2000 288 Sep-2000 256 Oct-2000 219 Nov-2000 260 Dec-2000 249 Let’s take our demand data and develop a histogram 1. Set up the scale and limits per subdivision 2. Plot the count of values that fall within each subdivision on the scale
Standard Deviation - Graphically 0 1 2 3 4 5 Monthly Demand in Units Frequency If my data is normal… 𝑿 = 𝟐𝟓𝟑. 𝟐𝟓 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕
Standard Deviation - Graphically 0 1 2 3 4 5 Monthly Demand in Units Frequency If my data is normal… ±3𝝈 = 𝟗𝟗. 𝟕% 𝒐𝒇 𝒅𝒂𝒕𝒂 ±2𝝈 = 𝟗𝟓. 𝟒% 𝒐𝒇 𝒅𝒂𝒕𝒂 ±1𝝈 = 𝟔𝟖. 𝟑% 𝒐𝒇 𝒅𝒂𝒕𝒂
Standard Deviation – Simple ApplicationFrequency 800 600 400 200 I have a process with mean of 43 and a standard deviation of 3 1200 1400 1000 37 43 42 41 40 39 38 48 47 46 45 44 49 35 36 51 50 68.3% of the data lies between what points? 95.4% of the data lies between what points? 99.7% of the area lies between what points?
Standard Deviation – Simple ApplicationFrequency 800 600 400 200 I have a process with mean of 43 and a standard deviation of 3 1200 1400 1000 37 43 42 41 40 39 38 48 47 46 45 44 49 35 36 51 50 68.3% of the data lies between 40 and 45 𝑋 ± 1𝜎 = 43 ± 3 95.4% of the data lies between 38 and 47 𝑋 ± 2𝜎 = 43 ± 6 99.7% of the area lies between 36 and 49 𝑋 ± 3𝜎 = 43 ± 9 𝑿 = 𝟒𝟑 ±3𝝈 ±2𝝈 ±1𝝈
Standard Deviation – Class Exercise  What is the probability that a random sample taken from this process…  Will have a value between 40 and 45?  Will have a value between 36 and 48?  Will have a value between 33 and 51? 68.3% 95.4% 99.7%
Probability theory And Probability Distribution
Probability What is the role of Probability in Statistics?  Any conclusion we reach on a population, based on what we know about a sample, is subject to uncertainty.  This uncertainty is calculated and described using probability theory  Every output (response) from a process adds up to 100% of the
Probability Measure  Every event (=set of outcomes) is assigned a probability measure.  The probability of every set is between 0 and 1, inclusive.  The probability of the whole set outcomes is 1.  If A and B are two event with no common outcomes, then the probability of their union is the sum of their probabilities.
Probability Measure  Probability of an event A = P (A)  P (A) = 𝑪𝒉𝒂𝒏𝒄𝒆𝒔 𝒐𝒇 𝒇𝒂𝒗𝒐𝒓𝒊𝒏𝒈 𝒆𝒗𝒆𝒏𝒕 𝑻𝒐𝒕𝒂𝒍 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒆𝒗𝒆𝒏𝒕𝒔 Cards  Events: a red card (1/2); a jack (1/13)  Chances of calling correctly on toss of a coin is ½ i.e. 0.5
Probability Building an Understanding  We’ll start with a pair of dice  Our customer will only accept combinations that equal 3,4,5,6,7,8,9,10 and 11.  What is the probability of meeting his requirement?
Probability Building an Understanding The customer defines a response of 2 or 12 as a defect Die 1 Roll 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 1 0 5 6 7 8 9 1 0 11 6 7 8 9 1 0 11 1 2 Die2Roll Calculate all possible responses from the combinations of inputs How many total combinations exist? How many times is my response a 2? What is the probability of a response of 2? How many times is my response a 12? What is the probability of a response of 12? What is the probability of a defect? (2 or 12) 𝟏𝑹𝒆𝒔𝒑𝒐𝒏𝒔𝒆 𝒊𝒏 𝟑𝟔 = 𝟏 𝟑𝟔 = 𝟎. 𝟎𝟐𝟕𝟖 = 𝟐. 𝟕𝟖% 𝟔 𝒅𝒊𝒆 𝟏 × 𝟔 𝒅𝒊𝒆 𝟐 = 𝟑𝟔 𝑻𝒐𝒕𝒂𝒍 𝑪𝒐𝒎𝒃𝒊𝒏𝒂𝒕𝒊𝒐𝒏𝒔 𝟏𝑹𝒆𝒔𝒑𝒐𝒏𝒔𝒆 𝒊𝒏 𝟑𝟔 = 𝟏 𝟑𝟔 = 𝟎. 𝟎𝟐𝟕𝟖 = 𝟐. 𝟕𝟖% 𝟎. 𝟎𝟐𝟕𝟖 + 𝟎. 𝟎𝟐𝟕𝟖 = 𝟎. 𝟎𝟓𝟓𝟔 = 𝟓. 𝟓𝟔%
Probability Building an Understanding Die 1 Roll 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 1 0 5 6 7 8 9 1 0 11 6 7 8 9 1 0 11 1 2 Die2Roll • Another example  What is the probability of rolling a 7 using a fair pair of dice? Die 1 Die 2 Probability 1 6 0.0278 2 5 0.0278 3 4 0.0278 4 3 0.0278 5 2 0.0278 6 1 0.0278 Total 0.1668 The probability of each roll is included in each block 16.68% Probability
Probability Value (Response) Frequency Probability 2 1 0.0278 3 2 0.0556 4 3 0.0833 5 4 0.1111 6 5 0.1389 7 6 0.1667 8 5 0.1389 9 4 0.1111 10 3 0.0833 11 2 0.0556 12 1 0.0278 Total 1.0000 Probability of any given value on Die 1 𝑭𝒐𝒓 𝒓𝒐𝒍𝒍𝒊𝒏𝒈 𝒕𝒉𝒆 𝒅𝒊𝒄𝒆, 𝒘𝒉𝒂𝒕 𝒊𝒔 𝒕𝒉𝒆 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏? 𝟏 𝟔 = 𝟎. 𝟏𝟔𝟔𝟔 = 𝟎. 𝟏𝟔𝟔𝟕 Probability of any given value on Die 2 𝟏 𝟔 = 𝟎. 𝟏𝟔𝟔𝟔 = 𝟎. 𝟏𝟔𝟔𝟕 𝟏 𝟔 × 𝟏 𝟔 = 𝟏 𝟑𝟔 = 𝟎. 𝟎𝟐𝟕𝟕 = 𝟎. 𝟎𝟐𝟕𝟖 Probability of any given combination
Probability 0 2 4 6 8 10 12 14 16 18 2 3 4 5 6 7 8 9 10 11 12 This represents the response of our system (In Probability) Response (Dice Total) Probability 0.0278 0.0278 0.1389 0.1111 0.0833 0.0556 0.1667 0.0556 0.0833 0.1111 0.1389
Probability  Our customer will only accept combinations that equal 3,4,5,6,7,8,9,10,11  We have a 99.44% probability of meeting the customers specification  The curve of this distribution becomes it’s Probability Density Function 0 5 10 15 20 2 3 4 5 6 7 LSL USL Probability Response (Dice Total) Value (Response) Frequenc y Probabilit y 2 1 0.0278 3 2 0.0556 4 3 0.0833 5 4 0.1111 6 5 0.1389 7 6 0.1667 8 5 0.1389 9 4 0.1111 10 3 0.0833 11 2 0.0556 12 1 0.0278
Probability  Our customer will only accept combinations that equal 3,4,5,6,7,8,9,10,11  We have a 99.44% probability of meeting the customers specification  The curve of this distribution becomes it’s Probability Density Function 0 5 10 15 20 LSL USL Probability Response (Dice Total) Value (Response) Frequenc y Probabilit y 2 1 0.0278 3 2 0.0556 4 3 0.0833 5 4 0.1111 6 5 0.1389 7 6 0.1667 8 5 0.1389 9 4 0.1111 10 3 0.0833 11 2 0.0556 12 1 0.0278
Probability Theory What is a Probability Density Function? A Mathematical Function  It models the probability density reflected in a histogram With more observations  Class intervals become narrower and more numerous  The histogram of the variable takes on the appearance of a smooth curve The total area under the curve must equal 1. The probability that a random variable will assume a value between any two points is equal in value to the area under the random variable’s probability density function between these two points. h What does this mean to us?
Probability Theory 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Response Intervals 1 2 3 4 5 6 7 8 9 10 11 Frequency • This histogram has 24 points distributed over 12 intervals
Probability Theory Response Intervals Frequency 400 300 200 100 • As the number of data increase, the intervals get smaller When we do this, the curve outlining the data gets smoother
Probability Theory What do we know about Probability Distribution?  The area under the curve always equals 1  We can determine the probability that a value of a random variable will fall between 2 points on the curve by calculating the area under the curve between the two points Why would we want to do this? How do we do this?
Using Probability Distribution The Standard Normal Distribution  Let’s take a look at the most important PD…the standard normal distribution  We can transform each point on our normal curve into a standard normal curve value using the Z transform 𝒁 𝒑𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒁 𝒔𝒂𝒎𝒑𝒍𝒆
Using Probability Distribution • Standard Normal Curve Characteristics The Standard Normal Distribution 𝑿 = 𝟎 𝑿 = 𝟐𝟓𝟑 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 It has a standard deviation of 1.0 𝒁 𝒔𝒂𝒎𝒑𝒍𝒆 = 𝑿 − 𝑿 𝑺 It has a mean of 0.0 The area under the curve equals 1 The curve is symmetrical After the Z Transform The Original Distribution
Using Probability Distribution The Standard Normal Distribution The “How” Find the points on the Standard Normal Distribution that correspond to your values Determine the area under the standard normal curve that is between the points you have found If our data is normal, we can use the Standard Normal Distribution This saves us from having to do the calculation for each specific situation!
Using Probability Distribution The Standard Normal Distribution A “Why” Example:  The unit sales of Product A follows a normal distribution and has a monthly average of 253 units with a standard deviation of 21 units = 253 S = 21 What is the probability that next months sales will be greater than 300 units? 𝑿
Using Probability Distribution The Standard Normal Distribution  What is the probability that next month’s unit sales will be greater than 300? 1. Find the point on the Standard Normal Distribution that corresponds to 300 =25 S = 21 𝒁 𝒔𝒂𝒎𝒑𝒍𝒆 = 𝑋 − 𝑋 𝑆 = 300 − 253 21 = 2.24 𝑿 This is telling us that 300 is 2.24 standard deviations from the mean
Using Probability Distribution The Standard Normal Distribution 2) Determine the area under the standard normal curve that is to the right of 2.24 – How? – Use the Table of the Standard Normal Distribution 2.24
StandardNormalTable Z was 2.24
Using Probability Distribution The Standard Normal Distribution  This table shows the area between 0 (the mean of a standard normal table) and Z  Because the curve is symmetric… The area of each ½ is 0.500 The area to the right of a positive value is 0.500 minus the area between 0 and the Z value  For Z = 2.24 (the equivalent of 300) Locate the row labeled .04 The area is 0.4875  Subtract this area from 0.500 0.500 – 0.4875 = 0.0125 I have a 1.25% probability that my unit sales next month will be greater than 300 units 2.24
Normal Distribution If you know your average value ( ) and your standard deviation (s) then for a given specification limit, it is possible to predict rejections (if any), that will occur even if you keep your process in control. Example: = 2.85, s = 0.02 (The dimensions relate to a punched part). Lat us find the percentage rejection if the specified value is 2.85±0.04 i.e. the part is acceptable between 2.81-2.89
Normal Distribution Applicable in real life: Acceptable RangeRejections Rejections 2.81 2.85 2.89
Normal Distribution Let A, B and C represent the areas under the curve for the following conditions: A – rejections for undersize B – acceptable range C – rejections for oversize Total Area = A+B+C Total Rejections = A+C BA C 2.85 2.892.81
Normal Distribution We will introduce a concept called Z which we can use with a one-sided distribution to determine the area under A, B and C and thus the percentage rejections and acceptable components. BA C 2.85 2.892.81
Normal Distribution  The area from Normal table corresponding to 2 is 0.02275  Hence Rejection for Over size (Area C) = 2.275%  Similarly one can find the rejection for undersize
Discrete Probability Distributions
Binomial Distribution When applicable: When the variable is in terms of attribute data and in binary alternatives such as good or bad, defective or non-defective, success or failure etc. Conditions:  The experiment consists of ‘n’ identical trials  There are only two possible outcomes on each trial. We denote as Success(S) and Failure(F).  The probability of ‘S’ remains the same from trial to trial and is denoted by ‘p’ and the probability of ‘F’ is ‘q’.  p+q = 1  The trials are independent
Binomial Distribution For a random experiment of sample size n where there are two categories of events, the probability of success of the condition x in one category (where there is n-x in the other category) is 𝑃(𝑋 = 𝑥) = 𝑛 𝑥 𝑝 𝑥 (𝑞) 𝑛−𝑥 , 𝑥 = 0,1,2, , 𝑛 Where (𝒒 = 𝟏 − 𝒑) is the probability that the vent will not occur. Where 𝑛 𝑥 = 𝑛! 𝑥! 𝑛−𝑥 !
Binomial Distribution Consider now that the probability of having the number “2” appear exactly three times in seven rolls of a six die is 𝑷 𝑿 = 𝟑 = 𝒏 𝒙 𝒑 𝒙(𝟏 − 𝒑) 𝒏−𝒙 = 𝟑𝟓 𝟎. 𝟏𝟔𝟕 𝟑 𝟏 − 𝟎. 𝟏𝟔𝟕 𝟕−𝟑 = 𝟎. 𝟎𝟕𝟖𝟒
Poisson Distribution When applicable:  No. of accidents in a specified period of time  No. of errors per 100 invoices  No. of telephone calls in a specified period of time  No. of surface defects in a casting  No. of faults of insulation in a specified length of cable  No. of visual defects in a bolt of cloth  No. of spare parts required over a specified period of time  The no. of absenteeism in a specified no. of time  The number of death claims in a hospital per day  The number of breakdowns of a computer per month  The PPM of Toxicant found in water or air emission from a manufacturing plant
Poisson Distribution Two Properties of a Poisson Experiment 1) The Probability of an occurrence is he same for any two intervals of equal length. 2) The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.
Poisson Distribution Conditions:  The experimental consists of counting the number of times a particular event occurs during a given unit of time or in a given area or volume or weight or distance etc.  The probability that an event occurs in a given unit of time is same for all the units.  The no. of events that occur in one unit of time is independent of the number that occur in other units.  The mean no. of events in each unit will be denoted by .
Poisson Distribution The Poisson Random Variable ‘X’ is the number of events that occur in specified period of time. 𝑃 𝑋 = 𝑥 = 𝑒−𝝺 𝞴 𝑥 𝑥! 𝑥 = 0,1,2,3 … A company observed that over several years they had a mean manufacturing line shutdown rate of 0.10 per day. Assuming a Poisson distribution, determine the probability of two shutdowns occurring on the same day. For the Poisson distribution, 𝝺 = 𝟎. 𝟏𝟎 occurrence/day and 𝐱 = 𝟐 results in the probability 𝑃 𝑋 = 2 = 𝑒−𝝺 𝞴 𝑥 𝑥! = 𝑒−0.10.12 2! = 0.004524
Poisson Distribution Suppose the number of breakdowns of machines in a day follows Poisson Distribution with an average number of breakdowns is 3. Find the probability that there will be no breakdowns tomorrow. 𝞴 = 3 𝐏(𝐗 = 𝟎) = 𝒆−𝟑 𝟑 𝟎 𝟎! = 𝒆−𝟑 = 𝟎. 𝟎𝟒𝟕𝟗𝟕
Poisson Distribution Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? Example: Mercy Hospital
Control Charts
Process Accuracy And Precision  We have curves that describe our process  Some questions we may ask… Is my process accurate? Is my process precise?
Process Accuracy And Precision  Accuracy describes centering  Is my process mean at my target mean? LSL USL Target
Process Accuracy And Precision LSL Target USL •Precision describes spread •How does the spread of my process compare to the customer’s specification limits?
Inaccurate and Imprecise
Accurate and Imprecise
Precise But Inaccurate
Accurate And Precise
Capability In Statistic Terms… LSL USLLSL USL LSL USL LSL USL Mean is not centered in Specification Mean is centered in Specification SmallStandard Deviation LargeStandard Deviation
SPC PROCESS The combination of people, equipment, materials, methods, measurement and environment that produce output – a given product or service. Process is transformation of given inputs into outputs
SPC VARIATION The inevitable differences among individual outputs of a process. The sources of variation can be grouped into two major classes, Common Causes & Special Causes
SPC
SPC
SPC COMMON CAUSE A source of variation that affects all the individual values of the process output being studied This is the source of the inherent process variation.
SPC Common Causes: 1. Plenty in Numbers 2. Results in less Variation 3. Part of the Process 4. Results in constant Variation 5. Predictable 6. Management Controllable 7. Statistics shall apply
SPC Examples of Common Causes, MAN MACHINE MATERIAL Differences in Competency (setting, operating & inspection) of Employees working in shifts. Difference in Quality of Product when Production of same Part is being carried out as per plan. UPS provided for Electricity Supply Difference in Mechanical & Chemical Properties in 2 different lots of Material of same grade received from suppliers (Raw Material Manufacturers)
SPC SPECIAL CAUSE: A source of variation that affects only some of the output of the process; it is often intermittent and unpredictable. A special cause is some times called assignable cause. It is signaled by one or more points beyond the control limits or a non-random pattern of points within the control limits.
SPC Special Causes: 1. Few in numbers 2. Results in large variation 3. Visitors to the process 4. Variation due to external factors 5. Fluctuating Variation 6. Unpredictable 7. Controllable by Operating personnel 8. Statistics shall not apply Recognize and deal with special causes outside the (Six Sigma) process Implement Corrective and Preventive Action (CAPC)
SPC Examples of Special Causes, MAN MACHINE MATERIAL METHOD MEASUREMENT Untrained Employee working on the Machine Production of Product on Conventional Lathe machine where Product Run out requirement is 2 microns. Major & frequent breakdowns of Machine. Frequent Power Failures. Use of different grade of raw material Setting of process Parameters which are not proven. Tool breakage Use of Micrometer having range of 0-25 mm to check O.D. of 25 mm ± 0.1 mm.
Types of Control Charts VARIABLE 𝑿, R 𝑿, s 𝑿, mR CUSUM ATTRIBUTE p np c u
Control Charts Overview The first step for control charting is to identify the CTQ’s of the process which is required to be brought under control Types of Control Charts Depends on the nature of the variable needed to control:  Variable Control Charts  Attribute Control Charts
Variable Control Chart CONTROL CHARTS
Variable Control Chart Xbar – Rbar When to use: When studying the behavior of a single measurable characteristic produced in relatively high volumes. How: By plotting sample averages (X-bar) and ranges (R) on separate charts. This allows for independent monitoring of the process average and the variation about that average. Conditions:  Constant sample size.  One characteristic per chart.  Should have no less than 20 samples before calculating control limits.
Variable Control Chart Xbar – Rbar 1. Most common type of control chart for analyzing continuous variables. 2. The xbar part of the chart notes the variation between the averages of consecutive sub-groups of data points. 3. The R part of the chart notes the changes of variation within each of the consecutive sub-groups.
Variable Control Chart RATIONAL SUBGROUP CONCEPT  Subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized, while the chance for differences due to these assignable causes within a subgroup will be minimized.  Time order is frequently a good basis for forming subgroups because it allows to detect assignable causes that occur over time.  Two general approaches for constructing rational subgroups: ◦ Construction units of production ◦ Random sample of all process output over the sampling interval
Control Chart  Reviewing plots & Analysis of trends: Ensure that all points of both X and R charts within control limits. If any point touching to any of the control limits, review process related remark corresponding to particular sub-group. This is assignable cause. Study particular trends if any ◦ Case study: ◦ Consider process of side member sub- assembly where critical dimensional characteristics i.e. concentricity of mounting holes is controlled.
Control Chart TRENDS ANALYSIS IN SPC CHARTS ALL POINTS WITHIN CONTROL LIMIT S.NO Trend Type Meaning Precautions for better process control 1. All points within control limits with zigzag pattern Process under control, variation due to random causes. Zigzag pattern changing with each point over judgment Let process continue. Try to make it a natural process 2. 7 more consecutive points on one side of center line Process Centre shifted towards one of the specification limit Do changes to bring process to Centre 3. Cyclic trends Assignable cause happening periodically Study assignable cause and reason. Study to prevent 4. Continuous inclination towards one of the control limits Assignable cause for process drift. If not prevented, product may go out of control Study assignable cause, set process to prevent drifting
Control Chart TRENDS ANALYSIS IN SPC CHARTS ALL POINTS WITHIN CONTROL LIMIT S.NO Trend Type Meaning Precautions for better process control 1. All points suddenly going out of control limits Assignable cause present, study specific process event associated with period of specific point Study probable causes for assignable cause taking place try to resolve the same 2. Any point going out of control limits with definite trend Process going out of control due to assignable cause Study the trend type & establish controls to prevent the assignable cause occurring
Typical Out-Of-Control Patterns  Point outside control limits  Sudden shift in process average  Cycles  Trends  Hugging the center line  Hugging the control limits  Instability
Shift in Process Average
Cycles
Trend
Control Charts PURPOSE OF CONDUCTING SPC STUDIES:  To study and analyze process variation  To find out trends in processes  To identify random & sporadic causes  To manufacture products of consistent quality  To prevent wastage of material
Process Capability For Continuous Data
Capability vs Stability  Capability has a meaning only when a process is stable.  If a process is out of control, first we need to stabilize the process.  Improvement in the inherent variation can be made only when the process is stable.  Control Charts are used to study stability.  The first job of Six Sigma practitioner is to identify and remove Special Causes of Variation.  Once the process is made predictable, the next job is to identify the causes of inherent variation and remove them.
Calculating Capability 𝐶𝑎𝑝𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑎𝑛 𝑏𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑇𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑃𝑟𝑜𝑐𝑒𝑠𝑠 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 LSL USL𝑿 ± 𝟏𝝈 𝑿 ± 𝟑𝝈 𝑿 ± 𝟐𝝈 𝑪 𝒑 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝝈 𝑪 𝒑 = 𝑻 𝟔𝝈
Calculating Capability 𝑴𝒂𝒓𝒈𝒊𝒏𝒂𝒍 𝑪𝒂𝒑𝒂𝒃𝒊𝒍𝒊𝒕𝒚 LSL USL𝑿 ± 𝟏𝝈 𝑿 ± 𝟑𝝈 𝑿 ± 𝟐𝝈 𝑪 𝒑 = 𝟔𝝈 𝟔𝝈 𝑪 𝒑 = 𝑻 𝟔𝝈 𝑪 𝒑 = 𝟏
Calculating Capability LSL USL 𝑿 ± 𝟔𝝈 𝑪 𝒑 = 𝟏𝟐𝝈 𝟔𝝈 𝑪 𝒑 = 𝑻 𝟔𝝈 𝑪 𝒑 = 𝟐 • Six Sigma Capability 𝑿 ± 𝟑𝝈 𝟑𝝈 𝟑𝝈
Calculating Capability 0 2 4 6 8 10 12 14 16 18 20 𝑪 𝒑𝒌 = 𝑴𝒊𝒏 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 , 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑪 𝒑𝑳 = 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 𝑪 𝒑𝑼 = 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑿 − 𝑳𝑺𝑳 𝑼𝑺𝑳 − 𝑿 • Calculate 𝑪 𝒑 from Upper and Lower side
Calculating Performance 0 2 4 6 8 10 12 14 16 18 20 𝑷 𝑷𝒌 = 𝑴𝒊𝒏 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 , 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑷 𝑷𝑳 = 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 𝑷 𝑷𝑼 = 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑿 − 𝑳𝑺𝑳 𝑼𝑺𝑳 − 𝑿 • Calculate 𝑪 𝒑 from Upper and Lower side 𝑷 𝑷 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝝈
Calculating Performance 𝑪 𝑷 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝝈 𝑪 𝒑𝒌 = 𝑴𝒊𝒏 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 , 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 • If the formulae are same, what is the difference? • The difference is in Sigma Calculation! • Sigma in Capability covers Short Term Variation. • Sigma in performance covers Long term Variation. • How is the Data Collection Different?
Process Capability Ratios ContinuousImprovement LSL USL 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 < 𝟏. 𝟎 LSL USL IncreasedNumberofDefects Process Capability Real Capability 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑𝒌 = 𝟐. 𝟎 𝑪 𝒑𝒌 < 𝟐. 𝟎 𝑪 𝒑𝒌 = 𝟏. 𝟎 𝑪 𝒑𝒌 = 𝟎. 𝟎 𝑪 𝒑𝒌 < 𝟎. 𝟎 𝑪 𝒑𝒌 < −𝟏. 𝟎 Understanding 𝑪 𝑷 and 𝑪 𝑷𝑲 𝑪 𝑷 only works for a process that is centered on the target 𝑪 𝑷𝑲 is a better measure for tracking performance
Capability Indices 𝑪 𝒑 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝑺 𝑪 𝒑𝒍 = 𝑿 − 𝑳𝑺𝑳 𝟑𝑺 𝑪 𝒑𝒖 = 𝑼𝑺𝑳 − 𝑿 𝟑𝑺 𝑪 𝒑𝒌 = 𝐦𝐢𝐧(𝑪 𝒑𝒖 , 𝑪 𝒑𝒍) 𝑪 𝒑 = 𝟎. 𝟑𝟏𝟕 − 𝟎. 𝟑𝟎𝟕 𝟔 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟏 . 𝟎𝟎𝟕𝟐 = 𝟏. 𝟑𝟖𝟗 𝑪 𝒑𝒖 = 𝟎. 𝟑𝟏𝟕 − 𝟎. 𝟑𝟏𝟓𝟓 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟎𝟏𝟓 . 𝟎𝟎𝟑𝟔 = 𝟎. 𝟒𝟏𝟕 𝑪 𝒑𝒍 = 𝟎. 𝟑𝟏𝟓𝟓 − 𝟎. 𝟑𝟎𝟕 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟎𝟖𝟓 . 𝟎𝟎𝟑𝟔 = 𝟐. 𝟑𝟔𝟏 𝑪 𝒑𝒌 = 𝐦𝐢 𝐧 𝟎. 𝟒𝟏𝟕, 𝟐. 𝟑𝟔𝟏 = 𝟎. 𝟒𝟏𝟕 Exampl e 𝑿 𝒎𝒆𝒂𝒏 = 𝟎. 𝟑𝟏𝟓𝟓 𝒔 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 = 𝟎. 𝟎𝟎𝟏𝟐 𝑳𝑺𝑳 𝒍𝒐𝒘𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟎𝟕 𝑼𝑺𝑳 𝒖𝒑𝒑𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟏𝟕 0.306 0.308 0.310 0.312 0.314 0.316 0.318 0.320 LSL USL
Capability Indices 𝑪 𝒑 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝑺 𝑪 𝒑𝒍 = 𝑿 − 𝑳𝑺𝑳 𝟑𝑺 𝑪 𝒑𝒖 = 𝑼𝑺𝑳 − 𝑿 𝟑𝑺 𝑪 𝒑𝒌 = 𝐦𝐢𝐧(𝑪 𝒑𝒖 , 𝑪 𝒑𝒍) 𝑪 𝒑 = 𝟎. 𝟑𝟐𝟐 − 𝟎. 𝟑𝟎𝟐 𝟔 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟐 . 𝟎𝟎𝟕𝟐 = 𝟐. 𝟕𝟕𝟖 𝑪 𝒑𝒖 = 𝟎. 𝟑𝟐𝟐 − 𝟎. 𝟑𝟏𝟒𝟑 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟎𝟕𝟕 . 𝟎𝟎𝟑𝟔 = 𝟐. 𝟏𝟑𝟗 𝑪 𝒑𝒍 = 𝟎. 𝟑𝟏𝟒𝟑 − 𝟎. 𝟑𝟎𝟐 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟏𝟐𝟑 . 𝟎𝟎𝟑𝟔 = 𝟑. 𝟒𝟏𝟕 𝑪 𝒑𝒌 = 𝐦𝐢 𝐧 𝟐. 𝟑𝟏𝟗, 𝟑. 𝟒𝟏𝟕 = 𝟐. 𝟑𝟏𝟗 Exampl e 𝑿 𝒎𝒆𝒂𝒏 = 𝟎. 𝟑𝟏𝟒𝟑 𝒔 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 = 𝟎. 𝟎𝟎𝟏𝟐 𝑳𝑺𝑳 𝒍𝒐𝒘𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟎𝟐 𝑼𝑺𝑳 𝒖𝒑𝒑𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟐𝟐 0.300 0.305 0.310 0.315 0.320 LSL USL
Let’s Summarize  A process cannot be improved till it is Stabilized.  Capability data should be utilized for stable processes  Subgroups should contain consecutive data, not random data.  Performance calculations should be done based on large amount of data representing Long Term Variation.
Process Capability For Attribute Data
Discrete Data Capability A discrete defect is an attribute, which can be counted. Such as: Scratches, Spots, Dent Marks, Cracks etc.  In these cases ½ does not make sense.  A defect is non conformance to the standards.  A defective unit can have more than one defect.  A sample of 100, may have 2 defectives but 5 defects.
Discrete Data Capability Defect Opportunities:  Defect opportunities are various types of defects, that may occur.  These creates dissatisfaction to the customers.  This is different than defects that occur.  Example : 12 type of defects that can occur on painted part. However, on a part produced, we may observe 0 to up to 12 defects. Thus a part may be defect free or may have1to 12 defects.
Discrete Data Capability Example: A sample of 100 nos have been taken. Following are the results of inspection: No of Defectives – 3 No of defects – 10 No of Opportunities - 12
Discrete Data Capability Example: The capability can be calculated as follows: No of units = U =100 Defects = D =10 No of Opportunities = O = 12 Total defect opportunities = UxO = 100x12 =1200 DPO = Defects per opportunity = 10/1200 =1/2 = 0.0083
Discrete Data Capability Example: Defect per million opportunities (DPMO) =DPO x 1,000,000 =0.0083 x 1,000,000 =8300 DPMO From the tables, the corresponding sigma level is 3.9.
Discrete Data Capability The same formula also can be expressed as DPMO = 𝑵𝑶. 𝑶𝑭 𝑫𝑬𝑭𝑬𝑪𝑻𝑺×𝟏𝟎 𝟔 𝑵𝑶 𝑶𝒇 𝑼𝑵𝑰𝑻𝑺×𝑶𝑷𝑷./𝑼𝑵𝑰𝑻
Discrete Data Capability – Example of DPMO Suppose we observe 200 letters delivered incorrectly to the wrong addresses in a small city during a single day when a total of 200,000 letters were delivered. What is the DPMO in this situation? DPMO = 𝟐𝟎𝟎 × 𝟏𝟎 𝟔 𝟐𝟎𝟎,𝟎𝟎𝟎 ×𝟏 = 𝟏, 𝟎𝟎𝟎 So, for every only million letters delivered this city’s postal managers can expect to have 1,000 letters incorrectly sent to the wrong address. What is the Six Sigma Level for this Process?
DPMO Example  IRS tax form advice  Survey of responses indicates predicted error rate  If 40% then: DPO = 0.40 DPMO = 0.40 defects/opportunity * 1,000,000 opportunities/million opportunities 400,000 DPMO = 1.75 Sigma
DPMO Example Example of Rolled throughput yield  If there are five processes with following yields:  Rolled throughput yield for this process is = 0.9 × 0.99 × 0.95 × 0.96 × 1 = 0.7279 = 0.73 = 73% Process No. Yield in % 1 90 2 99 3 95 4 96 5 100
DPMO - Exercise  You have 100 documents You take a sample of 10 documents There are 10 opportunities for defect on each document. 5 defects were found. What is DPMO Attendance Policy June 23, 2000 Crane Operational Excellence Program All Operational Excellence Leaders should be aware.
Complexity and Capability Payroll and Labor Tracking Process Does complexity have an important impact on process capability and quality? There are many opportunities for defects… Step 1 97.4% Read and record daily start and stop time 𝒀 𝑹𝑻 Output 79.1% Step 6 99.9% Create payroll checks Step 5 95.5% Transfer hour totals to payroll generatio n system Step 4 91.8% Total weekly work hours and job accounts . Submit time card Step 3 98.0% Total daily work hours Step 2 94.6% Read and record daily start and stop time Rolled Throughput Yield Example =
Complexity and Capability Payroll and Labor Tracking Process Our goal, reduce the total number of opportunities and increase the capability of remaining opportunities Step 1 97.4% Output 79.1% Step 6 99.9% Step 5 95.5% Step 4 91.8% Step 3 98.0% Step 2 94.6% Rolled Throughput Yield Example = 𝒀 𝑹𝑻 Output 98.9% Step 3 99.9% Print payroll checks from computer generated database Step 2 99.4% Scan employee badge and job card for labor start and stop time Step 1 99.6% Scan employee badge for start and stop time =
Complexity and Capability Notice any Difference? Step 1 93.32% Output 81.26% Step 3 93.32% Step 2 93.32% Step 2 99.999997% Rolled Throughput Yield Example = = Output 79.1% Step 2 99.999997% Step 2 99.999997%x x xx A Three Sigma Process A Six Sigma Process
Sigma Levels SIGMA Defect per Million Opportunities (DPMO) 1 690,000 2 308,537 3 66,807 4 6,210 5 233 6 3.4
Introduction To Hypothesis Testing
Hypothesis Testing Concept  Hypothesis testing is one of the most scientific ways of decision making.  It works very much like a court case.  We have a suspect, we have to take decision whether He / She is innocent or guilty.  Suppose there is person charged with murder, and both sides (defense and prosecution) do not have any evidence, what would be decision?  Innocent unless proven guilty?  Guilty unless proven Innocent? Null Hypothesis
Null Hypothesis  Null hypothesis is represented by Ho  It is statement of Innocence.  It is something that has to be assumed if you cannot prove otherwise.  It is statement of No Change or No Difference.
Null Hypothesis – A Court Case  Just Like a court case, we first assume the accused (X) is innocent and then try to prove it otherwise based on evidence (Data).  If evidence (Data) does not show sufficient difference, we cannot reject the innocence(Ho)  But if Evidence (Data) is strong enough, we reject the Innocence (Ho) and pronounce the suspect Guilty (Ha).  The statement that will be considered valid if null hypothesis is rejected is called Alternate Hypothesis (Ha)
Null hypothesis – A Concept  Hypothesis testing is a philosophy that real life situations.  You cannot prove two things equal.  You cannot prove two things different by proving only one difference  If you cannot prove 2 things different, you have to assume that they are equal.  But if you cannot prove them Different, are they really Equal?  What is the RISK involved?
Hypothesis Testing Concept In Truth, the Defendant is: Correct Decision Innocent individual goes Free Incorrect Decision Guilty Individual Goes Free Incorrect Decision Innocent Individual Is Disciplined Correct Decision Guilty Individual Is Disciplined 𝑯 𝑨: Guilty𝑯 𝒐: Innocent Verdict Innocent Guilty
Hypothesis Testing Concept Correct Decision Incorrect Decision Type II Error Probability = 𝛽 Incorrect Decision Type I Error Probability = 𝛼 Correct Decision 𝑯 𝑨is True𝑯 𝒐is True 𝑯 𝒐is True 𝑯 𝑨is True Decision True, But Unknown State of the World
Hypothesis Testing Concept Hypothesis testing Justice System  State the Opposing Conjectures, Ho and HA.  Determine the amount of evidence required, n, and the risk of committing a “type error”,  What sort of evaluation of the evidence is required and what is the justification for this? (type of test)  What are the conditions which proclaim guilt and those which proclaim innocence/ (Decision Rule)  Gather & Evaluate the evidence.  What is the verdict? (Ho or HA?)  Determine “Zone of Belief” : Confidence Interval.  What is appropriate justice? – Conclusions
Hypothesis Testing 1. Null Hypothesis (Ho) – statement of no change or difference. The statement is assumed true until sufficient evidence is presented to reject it. 2. Alternate Hypothesis (Ha) – statement of change or difference. This statement is considered true if Ho is rejected. 3. True I Error – the error in rejecting Ho when it is in true fact, there is no difference. 4. Alpha Risk – the maximum risk or probability of making a Type I Error. This Probability is always greater then zero, and is usually established at 5%. The researcher makes the decision to the greatest level of risk that is acceptable for a rejection of Ho. Also known as significant level. 5. Type II Error – The error in failing to reject Ho when it in fact false, or saying there is no difference when there really is a differerence.
Hypothesis Testing Concept 6) Beta Risk – The risk probability or making a Type II Error, or overlooking an effective treatment or solution to the problem. 7) Significant Difference – The term where a difference is too large to be reasonably attributed to chance.
𝛼 𝑎𝑛𝑑 𝛽 Risks  𝛼 Risk is also called producer’s risk.  𝛽 Risk is also called consumer’s risk.  Can we commit both type I and type II error at the same time?  As it necessary that we will have both 𝛼 and 𝛽risks?  Are𝛼 and 𝛽 risks equal?  Is 𝛼 and 𝛽 = 1?  Is there any relationship between 𝛼 and 𝛽?  Which risk is more important?
𝛼 𝑎𝑛𝑑 𝛽 Risks  An 𝛼 Risk of 5% is generally accepted.  An 𝛽 Risk of 10% is generally Accepted.  Since Ha cannot be proved, our attempt is to try and reject it.  What risk do we get in trying to reject the Ho.  Minitab represents 𝛼 risk by p-panel!
Steps in Hypothesis Testing  Define Ho  Define Ha  Select Appropriate Test.  Decide Significance Level (𝛼 and 𝛽)  Decide Sample Size  Collect Data  Conduct Test  Interpret!
Define Ho/Ha For following Cases To find if a distribution is normal or not.  Ho =>  Ha =? To find if the defects from three machines are same or different  Ho =>  Ha => To find if 2 groups of students from different streams have differing IQ  Ho =>  Ha =>
Basic Concepts Statistical Error
Statistical Error Definitions  Null Hypothesis Ho:  “Status quo”  “Nothing is different”  Equality  We fail to reject Ho based on statistical evidence  Alternate Hypothesis Ha:  “Something is different”  Statement about the population that requires strong evidence to prove  If we reject Ho, we in practice accept Ha.  Alpha Risk (𝛼)  Also called type I Error  Hypothesis the null hypothesis when it is fact true.  Beta Risk (𝛽)  Also called a Type II Error  Accepting the null Hypothesis when it is in fact false.
Statistical Error Typical 𝛼 𝑎𝑛𝑑 𝛽Risks  Typically, the 𝛼 level is set at 0.05 and the 𝛽 level is set at 0.10  They can be set at any level depending on what you want to know The risk is also called the “p-value” 1-𝛼 = confidence that an observed outcome in the sample is “real” We typically look for a p-value of 0.05 because:  1-0.05 = 0.95 (or 95% confidence)
The Central Limit Theorem Normally  Why are distributions normal? When all factors are random Some measurements are actually averages over time of “micro-measurements” In other words, what we see as a measurement is actually an average The Central Limit Theorem explains why a distribution of averages tends to be normal
Confidence Sample statistics estimate the mean or standard deviation of a population The “True” population mean and standard are unknown Confidence limits, levels, and intervals are used to determine the population statistics For means… We use t distribution to calculate limits, levels, and intervals For Standard Deviations… We use the 𝑐2 distribution to calculate limits, levels, and intervals
Definition Confidence Level:  The level of risk we are willing to take  How sure we want to be that the population mean or standard deviation falls between the confidence level is typical  95% confidence level is typical.  95% chance that the population mean or standard deviation falls between the limits.  5% chance (alpha risk) that the population mean or standard deviation isn’t contained within the calculated limits.  Risk (𝛼/2) Risk (𝛼/2) Risk (𝛼)
Definition Confidence Limit Upper and Lower limits that bracket the “true” mean or standard deviation of a population  Calculation from the sample data and the appropriate test statistic.  Test statistic is dependent on the risk we accept that our results will be wrong.
Definition Confidence Interval The interval defined by the upper and lower confidence limits. A range of values based on  Sample mean or sample standard deviation  Sample size  Confidence level  Appropriate test statistic Contains  Population mean or  Population standard deviation
Basic Concepts Confidence Limits For Means
Confidence Limit Formulas Means Lower Confidence Limit Upper Confidence Limit 𝑋 - 𝑡( 𝛼 2 ,𝑛−1) 𝑠 √𝑛 𝑋 + 𝑡( 𝛼 2 ,𝑛−1) 𝑠 √𝑛 𝑋 = 𝑚𝑒𝑎𝑛 𝑠 = 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑛 = 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒 𝑡 = 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑠𝑡𝑢𝑑𝑒𝑛𝑡′ 𝑠 𝑡 − 𝑡𝑎𝑏𝑙𝑒 𝑓𝑜𝑟 𝛼 2 𝑎𝑛𝑑 𝑛 − 1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚 (𝑓𝑜𝑟 𝑎 95% 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙, 𝛼 2 = 0.025) 𝛼 = 𝐴𝑙𝑝ℎ𝑎 𝑅𝑖𝑠𝑘
Confidence Limit Formulas Means Confidence Interval Lower Confidence Limit ≤ Mean ≤ Upper Confidence Limit ≤ 𝝁 ≤𝑋 - 𝑡( 𝛼 2 ,𝑛−1) ( 𝑠 𝑛 ) 𝑋 + 𝑡( 𝛼 2 ,𝑛−1) ( 𝑠 √𝑛 )
Confidence Limit - Example The tensioning device (rubber band) used on the Silobuster has come under scrutiny  Two sets of tensioners are measured and descriptive statistics are run. What is the 95% confidence interval for the variation? ≤ 𝝁 ≤ Set 1 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 25 Set 2 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 100 We are 95% confident that the interval 0.2479 to 0.2521 brackets the true process standard deviation (0.0042 width) We are 95% confident that the interval 0.2490 to 0.2510 brackets the true process standard deviation (0.0020 width) 𝑿 - 𝒕( 𝜶 𝟐 ,𝒏−𝟏)( 𝒔 𝒏 ) 𝑿 + 𝒕( 𝜶 𝟐 ,𝒏−𝟏) ( 𝒔 √𝒏 ) . 𝟐𝟓𝟎 ± 𝟐. 𝟎𝟔𝟑𝟗 . 𝟎𝟎𝟓 𝟐𝟓 = . 𝟐𝟒𝟕𝟗 𝒕𝒐 . 𝟐𝟓𝟐𝟏 . 𝟐𝟓𝟎 ± 𝟏. 𝟗𝟖𝟒𝟐 . 𝟎𝟎𝟓 𝟏𝟎𝟎 = . 𝟐𝟒𝟗𝟎 𝒕𝒐 . 𝟐𝟓𝟏𝟎
Basic Concepts Confidence Limits For Standard Deviation
Confidence Limit Formulas Variation Confidence Interval - Lower Confidence Limit ≤ Standard Deviation ≤ Upper Confidence Limit 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒍𝒐𝒘𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 ≤ 𝝈 ≤ 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒖𝒑𝒑𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 Population Standard Deviation
Confidence Limit Formulas Variation The tensioning device (rubber band) used on the Silobuster has come under scrutiny  Two sets of tensioners are measured and descriptive statistics are run. What is the 95% confidence interval for the variation? 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒍𝒐𝒘𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 ≤ 𝝈 ≤ 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒖𝒑𝒑𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 𝟎. 𝟎𝟎𝟓 𝟐𝟒 𝟑𝟗.𝟑𝟔 = 0.0039 and𝟎. 𝟎𝟎𝟓 𝟐𝟒 𝟏𝟐.𝟒𝟎 = .0070 𝟎. 𝟎𝟎𝟓 𝟗𝟗 𝟏𝟐𝟖𝟒𝟐 = 0.0044 and 𝟎. 𝟎𝟎𝟓 𝟗𝟗 𝟕𝟑.𝟑𝟔 = .0058 Set 1 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 25 Set 2 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 100 We are 95% confident that the interval 0.0039 to 0.0070 brackets the true process standard deviation (0.0031 width) We are 95% confident that the interval 0.0044 to 0.0058 brackets the true process standard deviation (0.0014 width)
TEST OF HYPOTHESIS - roadmap You want to compare the averages/ medians of samples of data to decide if they are statistically different Are samples normally distributed Compare median values instead if average How many samples do you wan to compare Kruskall Wallis Test For samples that do not have any outliners One-way ANOVA For comparing averages of three or more samples against one another 1 Sample t-test Comparing av. of one sample against target Paired t-test For comparing averages of two samples that contain data that is linked in pairs Two Sample t-test For comparing averages of two samples against each other Mood’s Median Test For samples that have some outliners Transform Data Yes 2 1 or 3 or more No No or
Design of Experiments
EXERCISE  Represent the following data in graphical form: Temperature 100 100 120 120 Response 275 285 270 325 Pressure 250 300 250 300
EXERCISE - continued a) Determine what parameter settings yield the largest response. b) Determine what parameter settings of pressure would be bets if it were important to reduce the variability of the responses that results from frequent temperature variations between two extremes.
EXERCISE - continued 0 100 200 300 400 500 600 700 100 120 Pressure = 300 Pressure = 250 Response Temperature
Design Of Experiments Design of Experiments (DOE) is a valuable tool to optimize product and process designs, to accelerate the development cycle, to reduce development costs, to improve the transition of products from research and development to manufacturing and to effectively trouble shoot manufacturing problems. Today, Design of Experiments is viewed as a quality technology to achieve product excellence at lowest possible overall cost.
Design of Experiments General Comments  Keep your experiments simple  Don’t try to answer all the questions in one study  Use 2 level designs to start  Try potential business results to the project  The best time to design an experiment is after the previous one is finished  Always verify results in a follow-up study ( verification)  Be ready for changes  A final report is a must to share the knowledge  Avoid DoE infatuation…do your homework first!  Measure & Analyze to reduce potential variables  Use Graphical Analysis  Use the basic tools of Operational Execllence
Design of Experiments Be Proactive  DOE is a proactive tool  If DOE output is inconclusive: You may be working with the wrong variables Your measurement system may not be capable The range between high and low levels may be sufficient  There is no such thing as a failed experiment Something is always learned New data prompts us to ask new questions and generates follow-up studies  Remember to keep an open mind Let the data/output guide your conclusions Debunk or validate tribal knowledge Don’t let yourself be “confused by the facts.”
Design Of Experiments Types of Experiments Traditional Approach Six Sigma Approach Very Informal Very Formal • Trial and Error Methods  Introduce a change and see what happens • Running Special Lots or Batches  Produced under controlled conditions • Pilot Runs  Set up to produce a desired effect. • One-Factor-at –a-Time Experiments  Vary one factor and keep all other factors constant • Planned Comparisons of Two to Four Factors  Study separate effects and interactions • Experiment With 5 to 20 Factors  Screening Studies • Comprehensive Experimental Plan With Many Phases  Modeling, multiple factor levels, optimization Very Informal Very Formal • Trial and Error Methods  Introduce a change and see what happens • Running Special Lots or Batches  Produced under controlled conditions • Pilot Runs  Set up to produce a desired effect. • One-Factor-at –a-Time Experiments  Vary one factor and keep all other factors constant • Planned Comparisons of Two to Four Factors  Study separate effects and interactions • Experiment With 5 to 20 Factors  Screening Studies • Comprehensive Experimental Plan With Many Phases  Modeling, multiple factor levels, optimization
Design Of Experiments Barriers to Successful DoE’s  Problem or objective unclear  Results of the experiments unclear  Be present during the DoE  Identify and record unexpected noise or other variables  Measurement Error  Lack of Management Support  Lack of Experimental Discipline  Don’t use a DoE as the first pass to identify key X’s  Manage the constants and the noise  Process map, C&E, Constant or Noise or Experimental  Unstable process prior to running DoE  Process map, C&E, Constant or Noise or Experimental, Manage the C’s and N’s to reduce extraneous variation
Design Of Experiments Objective  Establish the objective for the experiment It should be stated in such a way to provide guidance to those involved in designing the experiment.
Design Of Experiments Planning the Experiment  Team in involvement  Maximize prior knowledge  Pursue measurable objectives  Plan the execution of all phases  Rigorous sample size determination  Allocate sufficient resources for data collection and analysis.
Design Of Experiments The following are some of the objectives of experimentation in an industry:  Improving efficiency or yield  Finding optimum process settings  Locating sources of variables  Correlating process variables with product characteristics  Comparing different processes, machines, materials etc.  Designing new processes and products.
Various Terms Used In Experimentation  Factor: One of the controlled or uncontrolled variables whose influence on the response is being studied. May ne variable or classification data.  Level: The values or the factor being studied usually high(+) and low(-)  Treatment Combination: An experiment run using a set of the specific levels or each input variable  Response Variable: The variable that is being studied. “Y’ factor in the study. Measured output variable.  Interaction: The combined effect of two or more factors that is observed which is in addition to the main effect of each factor individually.
Various Terms Used In Experimentation  Confounding: One or more effects that can not unambiguously be attributed to a single factor or interaction.  Main effect: Change in the average response observed during a change from one level to another for a single factor.  Replication: Replication of the entire experiment. Treatment combinations are not repeated consequently.  Test run: A single combination of factors that yields one or more observation of the response.  Treatment: A single level assigned to a single factor during an experiment.
Trial And Error  Perhaps the most well known and used methodology.  The objective is to provide a quick fix to a specific problem.  The quick fix occurs by randomly and no- randomly making changes to process parameters.  Often changing two or more parameters at the same time.  The result often is a “Band-Aid” fix as the symptoms of the problem are removed, but the cause of the problem goes undetected.  In trial and error experimentation, knowledge is not expanded but hindered.  Implement multiple expensive fixes are not necessary.
One-Factor-At-A-Time (OFAT) The old dogma in experimentation is to hold everything constant and vary only one-factor- at-a-time. ◦ Assumes any changes in the response would be due only to the manipulated factor. But are they? ◦ Is it reasonable to assume that one can hold all variables constant while manipulating one? Experience tells us this is virtually impossible. Imagine there area large number of possible factors affecting the response variable: ◦ How long would OFAT take to identify critical factors and where they should be run for best results? ◦ How much confidence would you have that the knowledge gained would apply in the real world?
OFAT Although OFAT may simplify the analysis of results, the experiment efficiency given up is significant: ◦ Don’t know the effects of changing one factor while other factors are changing (a reality). ◦ Unnecessary experiments may be run. ◦ Time to find casual factors (factors that affect the response) is significant.
Classification Of Factors 1. Experimental Factors are those which we really experiment with by varying them at various levels. 2. Control Factors are those which are kept at a constant (controlled) level throughout experimentation. 3. Error or Noise factors are those which can neither be changed at our will nor can be fixed at one particular level. Effect of these factors causes the error component in the experiment and as such these factors are termed as error or noise factors.
Experimental Design Visualization of The 21 Design (2 Levels – 1 factor) This is often the method used today for process optimization. It is the “only one factor at a time” concept High Factor 1 Low
Experimental Design Visualization of The 22 Design (2 Levels – 2 factor) The most basic of true designs. There are 4 runs. High Factor 1 HighFactor 2Low Low
Experimental Design Visualization of The 23 Design (2 Levels – 3 factor) A little more complicated design but still very practical. There are only 8 runs. High Factor 1 HighFactor 2Low Low High Factor 3 Low
Experimental Design Factor 4Low High Visualization of The 24 Design (2 Levels – 4 factor)
Experimental Design Factor 4Low High Factor 4 High Factor 5 Low Visualization of The 𝟐 𝟓 Design (2 Levels – 5 factors) Here is where it’s time to stop drawing but it represents the complexity associated with a 5 factor design.
Experimental Design Three Factorial Design, without interaction 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 Y - - - 𝒀 𝟏 - - + 𝒀 𝟐 - + - 𝒀 𝟑 - + + 𝒀 𝟒 + - - 𝒀 𝟓 + - + 𝒀 𝟔 + + - 𝒀 𝟕 + + + 𝒀 𝟖
Experimental Design Three Factorial Design, with interaction 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 Y - - - + + + - 𝒀 𝟏 - - + + - - + 𝒀 𝟐 - + - - + - + 𝒀 𝟑 - + + - - + - 𝒀 𝟒 + - - - + + + 𝒀 𝟓 + - + - + - - 𝒀 𝟔 + + - + - - - 𝒀 𝟕 + + + + + + + 𝒀 𝟖
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Six Sigma Green Belt Training

  • 1. SIX SIGMA GREEN BELT TRAINING Shailja Chaudhry National Institute of Technology, Kurukshetra
  • 2. SIX SIGMA OVERVIEW AND EVOLUTION
  • 3. What is Six Sigma A customer focused business improvement process Driven by teamwork, consensus & logical reasoning Structured methodology – DMAIC Focuses on making the process robust & reduce variations Applies to any Process
  • 4. What is Six Sigma  Six sigma is a highly disciplined and quantitative strategic business improvement approach that seeks to increase both customer satisfaction and an organization’s financial health.  Six Sigma helps a company focus on developing and delivering near- perfect products (durable goods or services), to improve customer satisfaction and the bottom line.
  • 5. What Six Sigma is NOT Six Sigma is NOT A Quality Program Cure for World Hunger Only for Technical People Just about Statistics Used when solution is known Used for Firefighting
  • 6. Six-Sigma – A note from Originator of Six Sigma  “Six Sigma is not an improvement program. It is instead a business philosophy that employs a step by step approach to reducing variation, increasing quality, customer satisfaction, and in time, market share”
  • 7. Overview of Six Sigma CULTURAL CHANGE TRANSFORM THE ORGANIZATION GROWTH REDUCE COSTS PAIN, URGENCY, SURVIVAL SIX SIGMA AS A PHILOSOPHY SIX SIGMA AS A PROCESS SIX SIGMA AS A STATISTICAL TOOL
  • 8. What is Six Sigma?  Sigma is a measurement that indicates how a process is performing  Six sigma stands for Six Standard Deviations (Sigma is the Greek letter used to represent standard deviation in statistics) from mean. Six Sigma methodology provides the techniques and tools to improve the capability and reduce the defects in any process.  Six Sigma is structured application of tools and techniques applied on project basis to achieve sustained strategic results.
  • 9. What is Six Sigma A Vision of a Six Sigma Company Organizational Issue • Problem Resolution • Behavior • Decision Making • Process Adjustment • Supplier Relationship • Planning • Design • Employee Training • Chain-of-command • Direction • Manpower Traditional approach • Fixing (symptoms) • Reactive • Experience-based • Tweaking • Cost (piece price) • Short-term • Performance • If Time Permits • Hierarchy • Seat-of-pants • Cost Six Sigma Approach • Preventing (causes) • Data-based • Controlling • Capability • Long-term • Producibility • Mandated • Empowered Teams • Benchmarking and metrics • Asset Traditional approach • Fixing (symptoms) • Reactive • Experience-based • Tweaking • Cost (piece price) • Short-term • Performance • If Time Permits • Hierarchy • Seat-of-pants • Cost Six Sigma Approach • Preventing (causes) • Data-based • Controlling • Capability • Long-term • Producibility • Mandated • Empowered Teams • Benchmarking and metrics • Asset
  • 10. Character of 6s Traditional Quality / Six Sigma Quality Method ISSUE TRADITIONAL APPROACH SIX SIGMA APPROACH Index Data Target Range Method Action • % (Defect Rate) • Discrete Data • Satisfaction for Mfg. Process • Spec Outliner • Experience + Job • Bottom Up • σ • Discrete + Continuous Data • Customer Satisfaction • Variation improvement • Experience + Job + Statistical Ability • Top Down
  • 11. Aligning The Focus Six Sigma Journey Started (Traditional) 1000 Unassigned Projects Six Sigma Project Strategic DirectionTactical Direction Individual Work Group (Lets do it) (Future)
  • 12. What is Six Sigma Definition 2 Sigma 3 Sigma 4 Sigma 5 Sigma 6 Sigma 2σ 3σ 4σ 5σ 6σ Sigma Level Defects/ Million Opportunities% Yield 308,537 66,807 6,210 233 3.4 69.1 93.3 99.4 99.98 99.9997
  • 13. Six Sigma : The Statistical Way LSL USL LSL USL LSL USL Process of Target Excessive Variation Reduce Variation % Center Process Customers feel the variation more than the mean Center Proces s Reduc e Spread Target Target Target
  • 14. Six Sigma – Practical Meaning 99% Good (3.8 Sigma) 99.99966% Good (6 Sigma) • 20,000 lost articles of mail per hour • Unsafe drinking water for almost 15 minutes each day • 5,000 incorrect surgical operations per week • Two short or long landings at most major airports each day • 200,00 wrong drug prescriptions each year • No electricity for almost seven hours each month • Seven articles lost per hour • One unsafe minute every seven months • 1.7 incorrect operations per week • One short or long landing every five years • 68 wrong prescriptions per year • One hour without electricity every 34 years
  • 15. Philosophy of Six Sigma  Know What’s Important to the Customer (CTQ)  Reduce Defects (DPMO)  Center Around Target (Mean)  Reduce Variation (σ)
  • 16. Harvesting the fruit of Six Sigma
  • 18. History of Six Sigma  Quality tools like SPC, Cost of Quality, Control Charts, Process capability etc. are known to industry for long time, much before birth of Six Sigma.  Quality Tools and Quality system implementation was not in conjunction with overall business Goals.  Traditional Quality Tools have limitations to orient the efforts on Quality Improvements to the Organizational direction basically due to approach.  Motorola was the first Company to initiate the Six Sigma breakthrough Strategy.
  • 19. A Little Bit Of History  Six Sigma was developed by Bill Smith, QM at Motorola  It’s implementation began at Motorola n 1987  It allowed Motorola to win the first Baldrige Award in 1988  Motorola recorded more than $16 Billion savings as a result of Six Sigma  Several of the major companies in the world have adopted Six Sigma since then…. Texas Instruments, Asea Brown Boveri, AlliedSignal, General Electric, Bombardier, Nokia Mobile Phones, Lockheed Martin, Sony, Polaroid, Dupont, American Express, Ford Motor,….. The Six Sigma Breakthrough Strategy has become a Competitive Tool
  • 20. Motorola Case Study In early 1980’ Motorola was facing a serious competitive challenge from Japanese Companies. Motorola was losing the market share and customer confidence. Motorola had not done any major changes to their products. The competitors from Japan were offering much better product at much lower price with no field failures.
  • 21. Motorola Case Study….Continue  When Motorola studies the competitors products, it was revealed that the variation in key product characteristics is very low.  The competitors products were available at lower price.  The competitors products has very low warranty failure rate.  Motorola was not able to match the competitors price mainly due to high cost of Poor quality largely due to high reject rate, high rework / repair rate, high inspection cost, high warranty failure rate etc.  THE TECHNICAL TEAM CONCLUDED THAT THE COPETITORS ARE OFFERING BETTER PRODUCT AT LOWER COST.
  • 22. Motorola Case Study Motorola requested to the competitors from Japan to permit the Team from Motorola to visit them fro Study. Motorola sent the team of managers to Japan to study the “Magic” of Japanese companies. What the team revealed?
  • 23. Motorola Case Study  What Motorola learning was as follows:  Motorola was focusing too much on product Quality i.e. Inspection, rework, repair etc.  The internal defect rate was very high inside Motorola.  The reliability was slow since some of the defects were passing on to the customer as inspection lapses.  A dissatisfied customer was shouting loudly and was taking away min 10 potential customers.  As an effect of this, customers were lost to the competitors.
  • 24. Motorola Case Study What was wrong?  Japanese were concentrating on ◦ Customers ◦ Processes ◦ People  Variation in product and process parameters was known and controlled  All people were well trained and highly motivated  All activities and processes were highly standardized i.e. no person dependence  Defect free lines and robust processes  Very less inspectors  Yet, very low defect rate, internal rejection and customer complaints  VERY HIGH LEVEL OF CUSTOMER SATISFACTION
  • 25. Motorola Case Study WHAT WAS THE SECRET? THE SECRET WAS CONTROL OVER VARIATION Success factor: Proactive Vs. Reactive Quality
  • 26. The Impact Of Added Inspection 3.4 ppm 100,000 ppm 6 ppm If the likelihood of detecting the defect is 70% and we have 10 consecutive inspectors with this level of capability, we would expect about 6 escaping defects out of every 1,000,000 products produced You can save yourself by producing quality not by
  • 27. Motorola Case Study In order to address these issues, Motorola devised the Six Sigma methodology. Dr. Mikel Harry and Mr. Bill Smith were pioneers in Developing and implementing the Six Sigma methodology at Motorola. With implementation of Six Sigma, Motorola could achieve: 4σ level in one and half year time 5σ level in following year
  • 28. Six Sigma Progress 1985 1987 1992 1995 2002 Johnson & Johnson, Ford, Nissan, HoneywellGeneral Electric Allied Signal Motorola Dr Mikel J Harry wrote a Paper relating early failures to quality
  • 29. What can it do? Motorola:  5-Fold growth in Sales  Profits climbing by 20% pa  Cumulative savings of $14 billion over 11 years General Electric  $2 billion savings in just 3 years  The no. 1company in the USA Bechtel Corporation:  $200 million savings with investment of $30 million It is high time, that Indian Companies also start implementing Six Sigma for making breakthrough improvements and to remain
  • 31. Attempting to Define Quality Experts’ definitions of quality fall into two categories:  Level one quality is a simple matter of producing products or delivering services whose measurable characteristics satisfy a fixed set of specifications that are usually numerically defined.  Independent of any of their measurable characteristics, level two quantity products and services are simply those that satisfy customer expectations for their use or consumption. In short, level one quality means get it in the specs, and level two means satisfy the customer.
  • 32. Quality Gap Quality Gap Understanding of Needs Customer Perception of Delivery Customer Expectations Design of Products Capability to Deliver Design Actual Delivery Design Gap Perception Gap Operations Gap Process Gap Understanding the Gap
  • 33. Nine Dimensions of QUALITY According to modern management concepts, quality has nine dimensions: 1) Performance: main characteristics of the product/service 2) Aesthetics: appearance, feel, smell, taste 3) Special features: extra characteristics
  • 34. Nine Dimensions of QUALITY 4) Conformance: how well the product/service conforms to customer’s expectations 5) Safety: risk of injury 6) Reliability: consistency of performance
  • 35. Nine Dimensions of QUALITY 7) Durability: useful life of the product/service 8) Perceived Quality: indirect evaluation of quality (e.g. reputation) 9) Service after Sale: handling of Customer complaints and checking customer satisfaction.
  • 36. Evolution of Quality Historically Contemporary Reactive Quality Quality Checks (QC) - Taking the defectives out of what is produced Proactive Quality “Create process that will produce less or no defects”
  • 37. Old Concept Of Quality Past concepts of quality focused on “conformance to standards”. This definition assumed that as long as the company produced quality products and services, their performance standard was correct regardless of how those standards were met. Moreover, setting of standards and measurement of performance was mainly confined to the production areas and the commercial and other service functions were managed through command control.
  • 38. Value Enrichment The term ‘Value Enrichment’ for the company means that they must strive to produce highest quality products at the lowest possible costs to be competitive in the global markets. For customers, the term ‘Value Enrichment’ means that they have the right to purchase high quality products/services at the lowest cost.
  • 39. Concept Of Value Value to Customers Value = 𝐖𝐡𝐚𝐭 𝐘𝐨𝐮 𝐑𝐞𝐜𝐞𝐢𝐯𝐞 𝐖𝐡𝐚𝐭 𝐘𝐨𝐮 𝐏𝐚𝐲 Price + Inconvenienc e Real + Perceived
  • 40. Definitions VALUE: THOSE ACTIVITIES THAT CONVERT MATERIALS OR IDEAS INTO GOODS OR SERVICES THAT GENERATE CASH
  • 42. Six Sigma and Cost Of Quality Six Sigma has a very significant impact on the cost of quality. As the Sigma level moves up, the cost of quality comes down and vice versa. Traditionally recorded quality cost generally account for only 4 to 5 percent of sales which mainly comprise of scrap, re-work and warranty. There are additional costs of quality which are hidden and do not appear in the account books of the company, as they are intangible and difficult to measure.
  • 43. Visible And Hidden Costs Visible Costs Hidden Costs • Scrap • Rework • Warranty Costs • Conversion efficiency of materials • Inadequate resources utilization • Excessive use of materials • Cost of re-design and re— inspection • Cost of resolving customer problems • Lost customers / Goodwill • High Inventory
  • 44. Cost OF Quality At Various Levels Of Sigma 6 3.4 <10% 5 233 10-15% 4 6210 15-20% 3 66807 20-30% 2 308537 30-40% 1 6,90000 >40% Sigma Defect Rate (PPM) Cost Of Quality Competitive Level World Class Industry Average Non Competitive
  • 45. What is The Cost Of Quality?  Cost of Quality: the cost of ensuring that the job is done right + the cost of not doing the job right. Cost of Conformance + Cost of Non- Conformance(Prevention and Appraisal) (Internal/External Defects)
  • 46. Cost Of Quality Prevention Costs • Quality Planning • Process Evaluation / Improvement • Quality Improvement Meetings • Quality Training External Failure Costs • Complaint Handling • Rework / Correction • Re-Inspection Internal Failure Costs • Rework / Correction • Re-Inspection • Internal Reject • Loss of Business Appraisal Costs • Source Inspection • In / End-Process Inspection • Calibration • Specialist Cost Direct Costs Prevention Costs • Quality Planning • Process Evaluation / Improvement • Quality Improvement Meetings • Quality Training External Failure Costs • Complaint Handling • Rework / Correction • Re-Inspection Internal Failure Costs • Rework / Correction • Re-Inspection • Internal Reject • Loss of Business Appraisal Costs • Source Inspection • In / End-Process Inspection • Calibration • Specialist Cost
  • 48. Fundamental Steps There are 5 fundamental Steps involved in applying the breakthrough strategy for achieving Six Sigma. These steps are :-  Define  Measure  Analyze  Improve  Control
  • 49. Define Phase This phase defines the project. It identifies critical customer requirements and links them to business needs. It also defines a project charter and the business processes to be undertaken for Six Sigma.
  • 50. Define Define D CM A I Define Activities Identify Project, Champion and Project Owner Determine Customer Requirements and CTQs Define Problem, Objective, Goals and Benefits Define Stakeholders/Resource Analysis Map the Process Develop Project Plan Define Quality Tools Project Charter and Plan Effort/Impact Analysis Process Mapping Tree Diagram VOC Kano Model Pareto Analysis
  • 51. Measurement Phase This phase involves selecting product characteristic, mapping respective process, making necessary measurements and recording the results of the process. This is essentially a data collection phase.
  • 52. Measure – Operational Definition Measure M CD A I Measure Activities Determine operational Definitions Establish Performance Standards Develop Data Collection and Sampling Plan Validate the Measurements Measurement System Analysis Determine Process Capability and Baseline Measure Quality Tools Measurement Systems Analysis Check Sheet Process Capability Process FMEA
  • 53. Analysis Phase In this phase an action plan is created to close the “gap” between how things currently work and how the organization would like them to work in order to meet the goals for a particular product or service. This phase also requires organizations to estimate their short term and long term capabilities.
  • 54. Analyze Analyze A CMD I Analyze Activities Benchmark the Process or Product Analysis of the Process Map Brainstorm for likely causes Establish Causal Relationships Using Data Determine Root Cause(s) Using Data Analyze Quality Tools Cause and Effect or Event Diagram Graphical Analysis Statistical Analysis of Data Hypothesis Testing Correlation Regression DOE
  • 55. Improvement Phase  This phase involves improving processes/product performance characteristics for achieving desired results and goals. This phase involves application of scientific tools and techniques for making tangible improvements in profitability and customer satisfaction.
  • 56. Improve Improve I CMD A Improve Activities Develop Solution Alternatives Assess Risks and Benefits of Solution Alternatives Implement error-proofing solutions Validate Solution using a Pilot Implement Solution Determine Solution Effectiveness using Data Improve Quality Tools Brainstorming FMEA Risk Assessment Poka Yoke
  • 57. Control Phase This Phase requires the process conditions to be properly documented and monitored through statistical process control methods. After a “setting in” period, the process capability should be reassessed. Depending upon the results of such a follow-up analysis, it may be sometimes necessary to revisit one or more of the preceding phases.
  • 58. Control – Develop Standards Control CIMD A Control Activities Determine Needed Controls (measurement, design, etc.) Implement and Validate Controls Develop Transfer Plan Realize Benefits of Implementing Solution Institutional Changes Close Project and Communicate Results Control Quality Tools Statistical Process Control Process Map and FMEA Control Plans 5S Control Charts
  • 60. Why Project Selection is Important?  High leverage projects lead to largest Savings  Large returns are expected by management to justify the investment in time and effort  Developing a Six Sigma culture depend upon successful projects having significant business impact
  • 61. How To Focus Projects Process Cost Savings Focus Project Quality focus Product focus (Six Sigma Design) Problem Focus (Least Desirable Use)
  • 62. Project Selection Align with company objectives and business plan (Annual Operating Plan) – Voice of Customer/CT’s Inputs – Quality (CTQ)/Cost (CTC)/ Delivery (CTD) – PPM / COPQ / RTY / Cycle Time Consistent with principles of Six Sigma – Eliminate process defects Concentrate on “Common” issues/opportunities …not “fir-fighting” Large enough to justify the investment
  • 63. Project Desirability • Effort Required:- includes time required of team members and expenditure of money. • Probability of Success:- An assessment that takes into account various risk factors: + Time – uncertainty of the completion date + Effort – uncertainty of the investment required + Implementation – uncertainty of roadblocks
  • 65. Additional Project Considerations  Projects must serve as a learning experience for Green Belts to use the six Sigma tools  Projects scope should not be too large or take too long to implement  Projects scope should be manageable and take at least 255 of the potential Green Belt’s time.  Pareto Chart may be used to Scope the Project  Desirable to have a measurable variable for the primary project output/metrics
  • 66. Additional Project Considerations • Projects must serve as a learning experience for Green Belts to use the six Sigma tools • Projects scope should not be too large or take too long to implement • Projects scope should be manageable and take at least 255 of the potential Green Belt’s time. • Pareto Chart may be used to Scope the Project • Desirable to have a measurable variable for the primary project output/metricsDO NOT try to Solve World Hunger
  • 67. Strategy At Various Levels Almost every Organization can be divided into 3 basic levels:- 1. Business level 2. Operations level 3. Process level. It is extremely important that Six Sigma is understood and integrated at every level.
  • 68. Strategies At Various Levels  Executives at the business level can use Six Sigma for improving market share, increasing profitability and organizations long term viability.  Managers at operations level can use Six Sigma to improve yield and reduce the labor and material cost.  At the process level engineers can use Six Sigma to reduce defects and variation and improve process capability leading to better customer satisfaction.
  • 69. Factors To Control in Improvement Project  Resources Team availability The right tools  Schedule Be realistic Be aggressive Get buy-in  Scope of Work Watch for scope creep Stay focused Anticipate and mitigate risk Control any two areas, the third floats in response
  • 70. Meetings – Make Them Effective  Defined goal for meeting  Notice and agenda  Decision makers prepare and participate  Action Items  Records  Balance Sheet – Focused on process, not topic – What helped us get to our goal – What could have been better – Take appropriate action
  • 71. Skills Needed  People Leadership behaviors Communication  Process Time Management Schedule Coordination Problem Solving Risk analysis and mitigation Tactical Planning  Technical Six Sigma / Lean Tools Business Knowledge
  • 72. Voice of Customer – CTP and CTQ
  • 73. Establishing Customer Focus • Customer – Anyone internal or external to the organization who comes in contact with the product or output of work • Quality – performance to the standard expected by the Customer
  • 74. Variation is the Enemy in Achieving Customer Satisfaction Variation •Uncertainty •Unknown •Disbelief •Risk •Defect Rate
  • 75. What is Variation Variation is any deviation from the expected outcome.
  • 76. Something more on Variation  Any process has variation  There are two kinds of variation Common cause variation Special cause variation  Variation is measured in terms of sigma or standard deviation.
  • 77. Variation and Standard Deviation If a good deal of variation exists in a process activity, that activity will have a very large standard deviation. As a result, the distribution will be very wide and flat. Less Variation More Variation
  • 78. Types of Variation  Special Cause: something different happening at a certain time or place Common Cause: always present to some degree in the process We tamper with the system if we treat all variations as if it were special cause
  • 79. Dealing with Variation  Eliminate special cause variation by recognizing it and dealing with it outside of the process  Reduce common cause variation by improving the process
  • 80. Whom would you Prefer? Operator - 1 Operator - 2
  • 81. Critical To Quality (CTQ) are the key measurable characteristics of a product or process whose performance standards or specification limits must be met in order to satisfy the customer. They align improvement or design efforts with customer requirements.
  • 82. Critical To Quality (CTQ) 1. To put it in layman’s terms, CTQs are what the customer experts of a product... 2. ...the spoken needs of the customer. 3. The customer may often express this in plain English, but it is up to us to convert them to measurable terms using tools such as QFD, DFMEA, etc.
  • 83. Critical To Quality (CTQ) 1. List customer needs. 2. Identify the major drivers for these needs (major means those which will ensure that the need is addressed). 3. Break each driver into greater detail. 4. Stop the breakdown of each level when you have reached sufficient information that enables you to measure whether you meet the customer need or not.
  • 84. Example – CTQ Tree Ease of Operation Ease of Maintenance Ease of Operation and Maintenance Operator Training Time (hrs.) Setup Time (minutes) Operation Accuracy (errors/1000 ops) Mean Time to Restore (MTTR) # Special Tools Required Maintenance Training Time (hrs.) Need CTQsDrivers SpecificGeneral Hard to Measure Easy to Measure
  • 86. Importance of Project Charter A project charter is a written document and works as an agreement between management and the team about what is expected. The charter:  Clarifies what is expected of the team.  Keeps the team focused.  Keeps the team aligned with organizational priorities.  Transfers the project from the champion(s) to the project team.
  • 87. Team Charter  Problem Statement – Currently we carry out reblows to the extent of about 11-15% resulting in lower converter life, lower productivity of converter and increased Ferro-alloy and oxygen consumption.  Scope – All batches and all converters in SMS 1.  Project Goal and Measures – Reblows should be less than 7.5% and 9%.  Expected Business Results – We hope to save Rs. Xxxxx lakhs per year due to this reduction in reblows.
  • 88. Team Charter  Team Members –Supervisor, two operators, technical services, quality control  Support Required –Allow for weekly team meetings –Team budget for quick wins  Schedule –Measure (7wks), Analyze (4wks), Improve (6wks), Check (2wks), Control (1wk), Standardise/Close (1wk)
  • 89. Usual elements of a Project Charter  Project Description – Business Case  Scope – Process/Product  Goals and Measures (Key Indicators)  Expected Business Results  Team Members  Support Required  Expected Customer Benefits  Schedule
  • 91. Measurement Objective The Measure phase aims to set a baseline in terms of process performance through the development of clear and meaningful measurement systems
  • 92. The Measurement Process TOOLS AND TECHNIQUS OF MEASURE Develop Process Measures Collect Process Data Check Data Quality Understand Process Behavior Baseline Process Capability and Potential How do you measure the problem? When and where does the data come from? How does the process currently behave? What is the current performance of the process with respect to the customer Does the data represent what you think it does  Statistics  Operational Definitions  Data Worlds  Process Capability  Cp, Cpk  DPMO  Distributions  First pass yield  Short/long term variation  MSA  Gage R&R  Data Collection Methods  Data Collection Plans  Sampling
  • 93. Statistical and Data World If the data is AttributeCountContinuous Relevant statistical model is … Binomial Distribution Defects per Unit (DPU) Always Poisson – if process is in control Poisson Distribution When does the statistical model apply Common statistics are… Always Binomial – if process is in control Percentage (Proportion) Average (mean), Standard Deviation (sigma) Not always – validity of normality needs to be checked Normal Distribution
  • 95. Statistics  The science of: –Collecting, –Describing –Analyzing –Interpreting data... And Making Decisions
  • 96. What are Statistics?  Descriptive Statistics – Summarize and describe a set of data – Mean, median, range, standard deviation, variance, ....  Analytical Statistical (or Statistics) – Techniques that help us make decisions in the face of uncertainty – Use concepts of descriptive statistics as a base – Hypothesis testing, means comparisons, variance comparisons, proportions comparisons, ...
  • 97. Sample Versus Population  Using a small amount of data (Sample)... to make assumptions (inferences)... on a large amount of data (population).  Population: the total collection of observations or measurements that are if interest.  Sample: A subset of observations and measurements taken form the population.  Why do we use samples? Time Cost Destructive testing (need product left to sell !!) Other?
  • 98. Measures of Central Tendency  What is the Median value of Distribution? – Median  What value represents the distribution? – Mode  What value represents the entire distribution? – Mean (x̄ )  What is the best measures of central tendency?
  • 99. Data Distributions  Mean: Arithmetic average of a set of values – Reflects the influence of all values – Strongly influenced of all values  Median: Reflects the 50% rank – the center number after a set of numbers has been sorted from low to high. – Does not include all values in calculation – Is “robust” to extreme scores  Mode: The value or item occurring most frequently in a series of observations or statistical data.
  • 100. Variable Data Location - MeanMonth # of Units Jan-2006 233 Feb-2006 281 Mar-2006 266 Apr-2006 237 May-2006 260 Jun-2006 250 Jul-2006 237 Aug-2006 275 Sep-2006 218 Oct-2006 279 Nov-2006 227 Dec-2006 246 Jan-2007 258 Feb-2007 272 Mar-2007 229 Apr-2007 240 May-2007 287 Jun-2007 260 Jul-2007 251 Aug-2007 288 Sep-2007 256 Oct-2007 219 Nov-2007 260 Dec-2007 249 n=24 = 𝟔𝟎𝟕𝟖 We have data on the monthly demand history of one of our key product lines. Let’s calculate the statistics for location.  Mean (𝑿)  Add all of the monthly numbers  Divide by the number of months in the sample.  N=24, = 𝟔𝟎𝟖𝟕 𝑿 = 𝟔𝟎𝟖𝟕 𝟐𝟒 = 𝟐𝟓𝟑. 𝟐𝟓 Our average monthly shipment is 253 units 𝝁 = 𝒊=𝟏 𝑵 𝑿𝒊 𝑵 𝒙 = 𝒊=𝟏 𝒏 𝑿𝒊 𝒏 Populatio n Sample
  • 101. Variable Data Location - MedianMonth # of Units Jan-1999 233 Feb-1999 281 Mar-1999 266 Apr-1999 237 May-1999 260 Jun-1999 250 Jul-1999 237 Aug-1999 275 Sep-1999 218 Oct-1999 279 Nov-1999 227 Dec-1999 246 Jan-2000 258 Feb-2000 272 Mar-2000 229 Apr-2000 240 May-2000 287 Jun-2000 260 Jul-2000 251 Aug-2000 288 Sep-2000 256 Oct-2000 219 Nov-2000 260 Dec-2000 249 Month # of Units Sep-1999 218 Oct-2000 219 Nov-1999 227 Mar-2000 229 Jan-1999 233 Jul-1999 237 Apr-1999 237 Apr-2000 240 Dec-1999 246 Dec-2000 249 Jun-1999 250 Jul-2000 251 Sep-2000 256 Jan-2000 258 May-1999 260 Jun-2000 260 Nov-2000 260 Mar-1999 266 Feb-2000 272 Aug-1999 275 Oct-1999 279 Feb-1999 281 May-2000 287 Aug-2000 288 Statistics for location ~ Median (x) • Sort the data from lowest to highest • If there is an even number of observations, the median is the average of the two middle values (𝟐𝟓𝟏 + 𝟐𝟓𝟔) 𝟐 = 𝟐𝟓𝟑. 𝟓
  • 102. Variable Data Location - MedianMonth # of Units Jan-1999 233 Feb-1999 281 Mar-1999 266 Apr-1999 237 May-1999 260 Jun-1999 250 Jul-1999 237 Aug-1999 275 Sep-1999 218 Oct-1999 279 Nov-1999 227 Dec-1999 246 Jan-2000 258 Feb-2000 272 Mar-2000 229 Apr-2000 240 May-2000 287 Jun-2000 260 Jul-2000 251 Aug-2000 288 Sep-2000 256 Oct-2000 219 Nov-2000 260 Dec-2000 249 Month # of Units Sep-1999 218 Oct-2000 219 Nov-1999 227 Mar-2000 229 Jan-1999 233 Jul-1999 237 Apr-1999 237 Apr-2000 240 Dec-1999 246 Dec-2000 249 Jun-1999 250 Jul-2000 251 Sep-2000 256 Jan-2000 258 May-1999 260 Jun-2000 260 Nov-2000 260 Mar-1999 266 Feb-2000 272 Aug-1999 275 Oct-1999 279 Feb-1999 281 May-2000 287 Aug-2000 288 Statistics for location ~  Mode • The most frequently occurring value is the mode 260 is the mode
  • 103. Variable Data Location - Mode  Notes on mean –A measure of central tendency –Limitations:  Reflects the influence of all values  Strongly influenced by extreme values  Median (the centre number after sorting high to low) is robust to extreme values.
  • 104. Variable Data Description – Range, Standard DeviationMonth # of Units Jan-2006 233 Feb-2006 281 Mar-2006 266 Apr-2006 237 May-2006 260 Jun-2006 250 Jul-2006 237 Aug-2006 275 Sep-2006 218 Oct-2006 279 Nov-2006 227 Dec-2006 246 Jan-2007 258 Feb-2007 272 Mar-2007 229 Apr-2007 240 May-2007 287 Jun-2007 260 Jul-2007 251 Aug-2007 288 Sep-2007 256 Oct-2007 219 Nov-2007 260 Dec-2007 249 • Let’s use this same data to calculate the statistics for dispersion  These statistics are Range and Standard Deviation
  • 105. Example – commuting time Commute time (mins) 19.5 22.4 20.7 18.8 18.2 20.0 19.6 19.8 21.0 19.8 20.7 21.9 22.0 22.6 19.4 22.8 18.1 17.5 21.3 19.1 18.4 19.8 21.0 18.5 19.2 19.2 19.4 19.3 24.8 21.2 21.2 18.3 18.2 17.4 19.9 21.0 18.9 16.4 17.6 19.5 19.2 23.9 20.6 21.9 18.7 19.5 20.1 17.1 22.1 19.2 19.6 20.3 20.8 20.7 22.4 19.9 21.1 20.4 16.7 19.1 18.3 22.4 27.1 17.6 18.8 22.5 19.9 21.8 20.4 17.7 21.3 17.8 18.7 15.8 18.9 21.7 20.1 19.6 18.4 21.7 18.7 18.8 20.5 18.6 20.9 22.0 15.8 19.4 20.2 18.7 23.6 21.0 19.9 20.1 18.3 21.9 19.7 21.1 19.9 22.9 • Collect over a hundred occurrences. • Tabulate in chronological order. • Does the data show variation? • Can you make out anything with this arrangement of data? • Let us try and make some sense of this data…
  • 106. Measure of variation – Standard Deviation and Range Category 1 15 2018 22 24 25 19.50 19.75 20.00 20.25 20.50 .. Summary for Commute time Anderson – Darling Normality Test A-Squared P-Value 0.42 0.312 What are the relative merits and demerits of standard deviation over range? Mean St.Dev Variance Skewness Kurtosis N Minimum 1st Quartile Median 3rd Quartile Maximum 95% Confidence Interval for Mean 95% Confidence Interval for Mean 95% Confidence Interval for Mean 19.632 20.006 1.884 3.550 0.54470 1.30256 100 15.754 18.714 19.819 21.186 27.054 20.380 19.448 20.263 1.654 2.189Mean Median One measure of variation (std. dev) Another Measure of variation (Range) Outlier *
  • 107. Variable Data Dispersion – Standard Deviation “s” or “standard deviation”  What does it mean? –Standard deviation is a measure of dispersion (or how our data is spread out). –Range will tell us the difference between the highest and lowest values in a data set, but nothing about how the data are distributed. –We need deviation to statistically describe the distribution of values.
  • 108. Variable Data Dispersion – Standard Deviation How we calculate it…  A measure of how far each point deviates from the mean  We square each distance so that all the numbers are positive  The sum of the squares, divided by the sample size, is equal to the variance  The square root of the variance is the standard deviation – Variance can be added; standard deviations cannot 𝜎 = 𝑖=1 𝑛 (𝑥𝑖 − 𝜇)2 𝑁 s = 𝑖=1 𝑛 (𝑥𝑖 − 𝑥)2 𝑛 − 1Population Sample
  • 109. Variable Data Dispersion – Standard Deviation CalculationMonth # of Units Jan-2006 233 Feb-2006 281 Mar-2006 266 Apr-2006 237 May-2006 260 Jun-2006 250 Jul-2006 237 Aug-2006 275 Sep-2006 218 Oct-2006 279 Nov-2006 227 Dec-2006 246 Jan-2007 258 Feb-2007 272 Mar-2007 229 Apr-2007 240 May-2007 287 Jun-2007 260 Jul-2007 251 Aug-2007 288 Sep-2007 256 Oct-2007 219 Nov-2007 260 Dec-2007 249 -20.25 27.75 12.75 -16.25 6.75 -3.25 -16.25 21.75 -35.25 25.75 -26.25 -7.25 4.75 18.75 -24.25 -13.25 33.75 6.75 -2.25 34.75 2.75 -34.25 6.75 -4.25 -20.25 27.75 12.75 -16.25 6.75 -3.25 -16.25 21.75 -35.25 25.75 -26.25 -7.25 4.75 18.75 -24.25 -13.25 33.75 6.75 -2.25 34.75 2.75 -34.25 6.75 -4.25 𝑿𝒊 − 𝑿 (𝑿𝒊−𝑿) 𝟐 𝑺 = 𝒊=𝟏 𝒏 (𝒙𝒊 − 𝒙) 𝟐 𝒏 − 𝟏 𝑿 = 𝟐𝟓𝟑. 𝟐𝟓 Calculate the Mean Count the Samples n = 24 Square each subtraction result Subtract the mean from each value Sum the Squares Calculate the Denominator Complete the Calculation 𝒊=𝟏 𝒏 (𝒙𝒊 − 𝒙) 𝟐= 𝟗. 𝟖𝟑𝟏 𝒏 − 𝟏 = 𝟐𝟒 − 𝟏 = 𝟐𝟑 𝒔 = 𝟗. 𝟖𝟑𝟏 𝟐𝟑 = 𝟐𝟎. 𝟕
  • 110. Variable Data Dispersion – Standard Deviation 𝜎 = 𝑖=1 𝑛 (𝑥𝑖 − 𝜇)2 𝑁  Standard deviation of a population – If your data is from a population versus a sample from a population, use this formula to calculate standard deviation – The difference is the denominator “N” versus “n-1”
  • 111. Fundamental Topic The Normal Curve ◦ Processes have natural variation ◦ Many processes behave “normally” ◦ Characterized by Bell Shaped Curve – Mean near peak – Curve is symmetric ◦ Mean ◦ Standard Deviation Histogram of Diameter, with Normal Curve Diameter Frequency
  • 112. Measures of Variability •The Range is the distance between the extreme values of data set. (Highest – Lowest) •The Variance(S ) is the Average Squared Deviation of each data point from the Mean. •The Standard Deviation (s) is the Square Root of the Variance. •The range is more sensitive to outliners than the variance. •The most common and useful measure of variation is the Standard Deviation.
  • 113. Sample of Statistics versus Population Parameters EstimateStatistics Parameters µ = Population Mean s = Sample Standard Deviation X = Sample Mean σ = Population Standard Deviation
  • 114. Statistical Calculation (Sample) 𝑋 = 𝑖=1 𝑛 𝑋𝑖 𝑛 𝑆2 = 𝑖=1 𝑛 (𝑋𝑖−𝑋)2 𝑛 − 1 𝜎 = 𝑅/𝑑2 s = 𝑖=1 𝑛 (𝑋𝑖 − 𝑋)2 𝑛 − 1 Standard Deviation Standard Deviation VarianceMean n 2 3 4 6 𝒅 𝟐 1.128 1.693 2.059 2.326
  • 115. Statistical Calculation (Population) 𝜇 ≈ 𝑋𝝈 𝟐 = 𝒊=𝟏 𝑵 (𝑿𝒊 − 𝝁) 𝟐 𝑵 𝝈 = 𝒊=𝟏 𝑵 (𝑿𝒊 − 𝝁) 𝟐 𝑵 Standard Deviation VarianceMean
  • 116. Normal Distribution Description of a NORMAL DISTRIBUTION LOCATION: •The Central Tendency •It is usually expressed as the AVERAGE SPREAD: •The dispersion •It is usually expressed as standard deviation (Sigma) LOCATION SPREAD
  • 117. Properties of Normal Distribution •Normal Distribution is Symmetric –Has equal number of points on both sides –Mean Median and Mode Coincide •Normal Distribution is Infinite –The chance of finding a point anywhere on the plus and minus side (around the mean) is not absolutely Zero.
  • 118. Properties Of Normal Distribution Normal Curve & Probability Areas -3𝝈 -2𝝈 -1𝝈 0 1𝝈 2𝝈 3𝝈 68% 95% 99.73%
  • 119. Let’s Summarize… We need data study, predict and improve the processes. Data may be Variable or Attribute. To understand a data distribution, we need to know its Center, Spread and Shape. Normal Distribution is the most common but not the only shape.
  • 120. Standard Deviation - Graphically 0 1 2 3 4 5 Monthly Demand in Units Frequency Month # of Units Jan-1999 233 Feb-1999 281 Mar-1999 266 Apr-1999 237 May-1999 260 Jun-1999 250 Jul-1999 237 Aug-1999 275 Sep-1999 218 Oct-1999 279 Nov-1999 227 Dec-1999 246 Jan-2000 258 Feb-2000 272 Mar-2000 229 Apr-2000 240 May-2000 287 Jun-2000 260 Jul-2000 251 Aug-2000 288 Sep-2000 256 Oct-2000 219 Nov-2000 260 Dec-2000 249 Let’s take our demand data and develop a histogram 1. Set up the scale and limits per subdivision 2. Plot the count of values that fall within each subdivision on the scale
  • 121. Standard Deviation - Graphically 0 1 2 3 4 5 Monthly Demand in Units Frequency If my data is normal… 𝑿 = 𝟐𝟓𝟑. 𝟐𝟓 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕 1𝝈 = 𝟐𝟎. 𝟕
  • 122. Standard Deviation - Graphically 0 1 2 3 4 5 Monthly Demand in Units Frequency If my data is normal… ±3𝝈 = 𝟗𝟗. 𝟕% 𝒐𝒇 𝒅𝒂𝒕𝒂 ±2𝝈 = 𝟗𝟓. 𝟒% 𝒐𝒇 𝒅𝒂𝒕𝒂 ±1𝝈 = 𝟔𝟖. 𝟑% 𝒐𝒇 𝒅𝒂𝒕𝒂
  • 123. Standard Deviation – Simple ApplicationFrequency 800 600 400 200 I have a process with mean of 43 and a standard deviation of 3 1200 1400 1000 37 43 42 41 40 39 38 48 47 46 45 44 49 35 36 51 50 68.3% of the data lies between what points? 95.4% of the data lies between what points? 99.7% of the area lies between what points?
  • 124. Standard Deviation – Simple ApplicationFrequency 800 600 400 200 I have a process with mean of 43 and a standard deviation of 3 1200 1400 1000 37 43 42 41 40 39 38 48 47 46 45 44 49 35 36 51 50 68.3% of the data lies between 40 and 45 𝑋 ± 1𝜎 = 43 ± 3 95.4% of the data lies between 38 and 47 𝑋 ± 2𝜎 = 43 ± 6 99.7% of the area lies between 36 and 49 𝑋 ± 3𝜎 = 43 ± 9 𝑿 = 𝟒𝟑 ±3𝝈 ±2𝝈 ±1𝝈
  • 125. Standard Deviation – Class Exercise  What is the probability that a random sample taken from this process…  Will have a value between 40 and 45?  Will have a value between 36 and 48?  Will have a value between 33 and 51? 68.3% 95.4% 99.7%
  • 127. Probability What is the role of Probability in Statistics?  Any conclusion we reach on a population, based on what we know about a sample, is subject to uncertainty.  This uncertainty is calculated and described using probability theory  Every output (response) from a process adds up to 100% of the
  • 128. Probability Measure  Every event (=set of outcomes) is assigned a probability measure.  The probability of every set is between 0 and 1, inclusive.  The probability of the whole set outcomes is 1.  If A and B are two event with no common outcomes, then the probability of their union is the sum of their probabilities.
  • 129. Probability Measure  Probability of an event A = P (A)  P (A) = 𝑪𝒉𝒂𝒏𝒄𝒆𝒔 𝒐𝒇 𝒇𝒂𝒗𝒐𝒓𝒊𝒏𝒈 𝒆𝒗𝒆𝒏𝒕 𝑻𝒐𝒕𝒂𝒍 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒆𝒗𝒆𝒏𝒕𝒔 Cards  Events: a red card (1/2); a jack (1/13)  Chances of calling correctly on toss of a coin is ½ i.e. 0.5
  • 130. Probability Building an Understanding  We’ll start with a pair of dice  Our customer will only accept combinations that equal 3,4,5,6,7,8,9,10 and 11.  What is the probability of meeting his requirement?
  • 131. Probability Building an Understanding The customer defines a response of 2 or 12 as a defect Die 1 Roll 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 1 0 5 6 7 8 9 1 0 11 6 7 8 9 1 0 11 1 2 Die2Roll Calculate all possible responses from the combinations of inputs How many total combinations exist? How many times is my response a 2? What is the probability of a response of 2? How many times is my response a 12? What is the probability of a response of 12? What is the probability of a defect? (2 or 12) 𝟏𝑹𝒆𝒔𝒑𝒐𝒏𝒔𝒆 𝒊𝒏 𝟑𝟔 = 𝟏 𝟑𝟔 = 𝟎. 𝟎𝟐𝟕𝟖 = 𝟐. 𝟕𝟖% 𝟔 𝒅𝒊𝒆 𝟏 × 𝟔 𝒅𝒊𝒆 𝟐 = 𝟑𝟔 𝑻𝒐𝒕𝒂𝒍 𝑪𝒐𝒎𝒃𝒊𝒏𝒂𝒕𝒊𝒐𝒏𝒔 𝟏𝑹𝒆𝒔𝒑𝒐𝒏𝒔𝒆 𝒊𝒏 𝟑𝟔 = 𝟏 𝟑𝟔 = 𝟎. 𝟎𝟐𝟕𝟖 = 𝟐. 𝟕𝟖% 𝟎. 𝟎𝟐𝟕𝟖 + 𝟎. 𝟎𝟐𝟕𝟖 = 𝟎. 𝟎𝟓𝟓𝟔 = 𝟓. 𝟓𝟔%
  • 132. Probability Building an Understanding Die 1 Roll 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 1 0 5 6 7 8 9 1 0 11 6 7 8 9 1 0 11 1 2 Die2Roll • Another example  What is the probability of rolling a 7 using a fair pair of dice? Die 1 Die 2 Probability 1 6 0.0278 2 5 0.0278 3 4 0.0278 4 3 0.0278 5 2 0.0278 6 1 0.0278 Total 0.1668 The probability of each roll is included in each block 16.68% Probability
  • 133. Probability Value (Response) Frequency Probability 2 1 0.0278 3 2 0.0556 4 3 0.0833 5 4 0.1111 6 5 0.1389 7 6 0.1667 8 5 0.1389 9 4 0.1111 10 3 0.0833 11 2 0.0556 12 1 0.0278 Total 1.0000 Probability of any given value on Die 1 𝑭𝒐𝒓 𝒓𝒐𝒍𝒍𝒊𝒏𝒈 𝒕𝒉𝒆 𝒅𝒊𝒄𝒆, 𝒘𝒉𝒂𝒕 𝒊𝒔 𝒕𝒉𝒆 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏? 𝟏 𝟔 = 𝟎. 𝟏𝟔𝟔𝟔 = 𝟎. 𝟏𝟔𝟔𝟕 Probability of any given value on Die 2 𝟏 𝟔 = 𝟎. 𝟏𝟔𝟔𝟔 = 𝟎. 𝟏𝟔𝟔𝟕 𝟏 𝟔 × 𝟏 𝟔 = 𝟏 𝟑𝟔 = 𝟎. 𝟎𝟐𝟕𝟕 = 𝟎. 𝟎𝟐𝟕𝟖 Probability of any given combination
  • 134. Probability 0 2 4 6 8 10 12 14 16 18 2 3 4 5 6 7 8 9 10 11 12 This represents the response of our system (In Probability) Response (Dice Total) Probability 0.0278 0.0278 0.1389 0.1111 0.0833 0.0556 0.1667 0.0556 0.0833 0.1111 0.1389
  • 135. Probability  Our customer will only accept combinations that equal 3,4,5,6,7,8,9,10,11  We have a 99.44% probability of meeting the customers specification  The curve of this distribution becomes it’s Probability Density Function 0 5 10 15 20 2 3 4 5 6 7 LSL USL Probability Response (Dice Total) Value (Response) Frequenc y Probabilit y 2 1 0.0278 3 2 0.0556 4 3 0.0833 5 4 0.1111 6 5 0.1389 7 6 0.1667 8 5 0.1389 9 4 0.1111 10 3 0.0833 11 2 0.0556 12 1 0.0278
  • 136. Probability  Our customer will only accept combinations that equal 3,4,5,6,7,8,9,10,11  We have a 99.44% probability of meeting the customers specification  The curve of this distribution becomes it’s Probability Density Function 0 5 10 15 20 LSL USL Probability Response (Dice Total) Value (Response) Frequenc y Probabilit y 2 1 0.0278 3 2 0.0556 4 3 0.0833 5 4 0.1111 6 5 0.1389 7 6 0.1667 8 5 0.1389 9 4 0.1111 10 3 0.0833 11 2 0.0556 12 1 0.0278
  • 137. Probability Theory What is a Probability Density Function? A Mathematical Function  It models the probability density reflected in a histogram With more observations  Class intervals become narrower and more numerous  The histogram of the variable takes on the appearance of a smooth curve The total area under the curve must equal 1. The probability that a random variable will assume a value between any two points is equal in value to the area under the random variable’s probability density function between these two points. h What does this mean to us?
  • 139. Probability Theory Response Intervals Frequency 400 300 200 100 • As the number of data increase, the intervals get smaller When we do this, the curve outlining the data gets smoother
  • 140. Probability Theory What do we know about Probability Distribution?  The area under the curve always equals 1  We can determine the probability that a value of a random variable will fall between 2 points on the curve by calculating the area under the curve between the two points Why would we want to do this? How do we do this?
  • 141. Using Probability Distribution The Standard Normal Distribution  Let’s take a look at the most important PD…the standard normal distribution  We can transform each point on our normal curve into a standard normal curve value using the Z transform 𝒁 𝒑𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒁 𝒔𝒂𝒎𝒑𝒍𝒆
  • 142. Using Probability Distribution • Standard Normal Curve Characteristics The Standard Normal Distribution 𝑿 = 𝟎 𝑿 = 𝟐𝟓𝟑 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟏. 𝟎 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 𝟏𝝈 = 𝟐𝟏 It has a standard deviation of 1.0 𝒁 𝒔𝒂𝒎𝒑𝒍𝒆 = 𝑿 − 𝑿 𝑺 It has a mean of 0.0 The area under the curve equals 1 The curve is symmetrical After the Z Transform The Original Distribution
  • 143. Using Probability Distribution The Standard Normal Distribution The “How” Find the points on the Standard Normal Distribution that correspond to your values Determine the area under the standard normal curve that is between the points you have found If our data is normal, we can use the Standard Normal Distribution This saves us from having to do the calculation for each specific situation!
  • 144. Using Probability Distribution The Standard Normal Distribution A “Why” Example:  The unit sales of Product A follows a normal distribution and has a monthly average of 253 units with a standard deviation of 21 units = 253 S = 21 What is the probability that next months sales will be greater than 300 units? 𝑿
  • 145. Using Probability Distribution The Standard Normal Distribution  What is the probability that next month’s unit sales will be greater than 300? 1. Find the point on the Standard Normal Distribution that corresponds to 300 =25 S = 21 𝒁 𝒔𝒂𝒎𝒑𝒍𝒆 = 𝑋 − 𝑋 𝑆 = 300 − 253 21 = 2.24 𝑿 This is telling us that 300 is 2.24 standard deviations from the mean
  • 146. Using Probability Distribution The Standard Normal Distribution 2) Determine the area under the standard normal curve that is to the right of 2.24 – How? – Use the Table of the Standard Normal Distribution 2.24
  • 148. Using Probability Distribution The Standard Normal Distribution  This table shows the area between 0 (the mean of a standard normal table) and Z  Because the curve is symmetric… The area of each ½ is 0.500 The area to the right of a positive value is 0.500 minus the area between 0 and the Z value  For Z = 2.24 (the equivalent of 300) Locate the row labeled .04 The area is 0.4875  Subtract this area from 0.500 0.500 – 0.4875 = 0.0125 I have a 1.25% probability that my unit sales next month will be greater than 300 units 2.24
  • 149. Normal Distribution If you know your average value ( ) and your standard deviation (s) then for a given specification limit, it is possible to predict rejections (if any), that will occur even if you keep your process in control. Example: = 2.85, s = 0.02 (The dimensions relate to a punched part). Lat us find the percentage rejection if the specified value is 2.85±0.04 i.e. the part is acceptable between 2.81-2.89
  • 150. Normal Distribution Applicable in real life: Acceptable RangeRejections Rejections 2.81 2.85 2.89
  • 151. Normal Distribution Let A, B and C represent the areas under the curve for the following conditions: A – rejections for undersize B – acceptable range C – rejections for oversize Total Area = A+B+C Total Rejections = A+C BA C 2.85 2.892.81
  • 152. Normal Distribution We will introduce a concept called Z which we can use with a one-sided distribution to determine the area under A, B and C and thus the percentage rejections and acceptable components. BA C 2.85 2.892.81
  • 153. Normal Distribution  The area from Normal table corresponding to 2 is 0.02275  Hence Rejection for Over size (Area C) = 2.275%  Similarly one can find the rejection for undersize
  • 155. Binomial Distribution When applicable: When the variable is in terms of attribute data and in binary alternatives such as good or bad, defective or non-defective, success or failure etc. Conditions:  The experiment consists of ‘n’ identical trials  There are only two possible outcomes on each trial. We denote as Success(S) and Failure(F).  The probability of ‘S’ remains the same from trial to trial and is denoted by ‘p’ and the probability of ‘F’ is ‘q’.  p+q = 1  The trials are independent
  • 156. Binomial Distribution For a random experiment of sample size n where there are two categories of events, the probability of success of the condition x in one category (where there is n-x in the other category) is 𝑃(𝑋 = 𝑥) = 𝑛 𝑥 𝑝 𝑥 (𝑞) 𝑛−𝑥 , 𝑥 = 0,1,2, , 𝑛 Where (𝒒 = 𝟏 − 𝒑) is the probability that the vent will not occur. Where 𝑛 𝑥 = 𝑛! 𝑥! 𝑛−𝑥 !
  • 157. Binomial Distribution Consider now that the probability of having the number “2” appear exactly three times in seven rolls of a six die is 𝑷 𝑿 = 𝟑 = 𝒏 𝒙 𝒑 𝒙(𝟏 − 𝒑) 𝒏−𝒙 = 𝟑𝟓 𝟎. 𝟏𝟔𝟕 𝟑 𝟏 − 𝟎. 𝟏𝟔𝟕 𝟕−𝟑 = 𝟎. 𝟎𝟕𝟖𝟒
  • 158. Poisson Distribution When applicable:  No. of accidents in a specified period of time  No. of errors per 100 invoices  No. of telephone calls in a specified period of time  No. of surface defects in a casting  No. of faults of insulation in a specified length of cable  No. of visual defects in a bolt of cloth  No. of spare parts required over a specified period of time  The no. of absenteeism in a specified no. of time  The number of death claims in a hospital per day  The number of breakdowns of a computer per month  The PPM of Toxicant found in water or air emission from a manufacturing plant
  • 159. Poisson Distribution Two Properties of a Poisson Experiment 1) The Probability of an occurrence is he same for any two intervals of equal length. 2) The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.
  • 160. Poisson Distribution Conditions:  The experimental consists of counting the number of times a particular event occurs during a given unit of time or in a given area or volume or weight or distance etc.  The probability that an event occurs in a given unit of time is same for all the units.  The no. of events that occur in one unit of time is independent of the number that occur in other units.  The mean no. of events in each unit will be denoted by .
  • 161. Poisson Distribution The Poisson Random Variable ‘X’ is the number of events that occur in specified period of time. 𝑃 𝑋 = 𝑥 = 𝑒−𝝺 𝞴 𝑥 𝑥! 𝑥 = 0,1,2,3 … A company observed that over several years they had a mean manufacturing line shutdown rate of 0.10 per day. Assuming a Poisson distribution, determine the probability of two shutdowns occurring on the same day. For the Poisson distribution, 𝝺 = 𝟎. 𝟏𝟎 occurrence/day and 𝐱 = 𝟐 results in the probability 𝑃 𝑋 = 2 = 𝑒−𝝺 𝞴 𝑥 𝑥! = 𝑒−0.10.12 2! = 0.004524
  • 162. Poisson Distribution Suppose the number of breakdowns of machines in a day follows Poisson Distribution with an average number of breakdowns is 3. Find the probability that there will be no breakdowns tomorrow. 𝞴 = 3 𝐏(𝐗 = 𝟎) = 𝒆−𝟑 𝟑 𝟎 𝟎! = 𝒆−𝟑 = 𝟎. 𝟎𝟒𝟕𝟗𝟕
  • 163. Poisson Distribution Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? Example: Mercy Hospital
  • 165. Process Accuracy And Precision  We have curves that describe our process  Some questions we may ask… Is my process accurate? Is my process precise?
  • 166. Process Accuracy And Precision  Accuracy describes centering  Is my process mean at my target mean? LSL USL Target
  • 167. Process Accuracy And Precision LSL Target USL •Precision describes spread •How does the spread of my process compare to the customer’s specification limits?
  • 172. Capability In Statistic Terms… LSL USLLSL USL LSL USL LSL USL Mean is not centered in Specification Mean is centered in Specification SmallStandard Deviation LargeStandard Deviation
  • 173. SPC PROCESS The combination of people, equipment, materials, methods, measurement and environment that produce output – a given product or service. Process is transformation of given inputs into outputs
  • 174. SPC VARIATION The inevitable differences among individual outputs of a process. The sources of variation can be grouped into two major classes, Common Causes & Special Causes
  • 175. SPC
  • 176. SPC
  • 177. SPC COMMON CAUSE A source of variation that affects all the individual values of the process output being studied This is the source of the inherent process variation.
  • 178. SPC Common Causes: 1. Plenty in Numbers 2. Results in less Variation 3. Part of the Process 4. Results in constant Variation 5. Predictable 6. Management Controllable 7. Statistics shall apply
  • 179. SPC Examples of Common Causes, MAN MACHINE MATERIAL Differences in Competency (setting, operating & inspection) of Employees working in shifts. Difference in Quality of Product when Production of same Part is being carried out as per plan. UPS provided for Electricity Supply Difference in Mechanical & Chemical Properties in 2 different lots of Material of same grade received from suppliers (Raw Material Manufacturers)
  • 180. SPC SPECIAL CAUSE: A source of variation that affects only some of the output of the process; it is often intermittent and unpredictable. A special cause is some times called assignable cause. It is signaled by one or more points beyond the control limits or a non-random pattern of points within the control limits.
  • 181. SPC Special Causes: 1. Few in numbers 2. Results in large variation 3. Visitors to the process 4. Variation due to external factors 5. Fluctuating Variation 6. Unpredictable 7. Controllable by Operating personnel 8. Statistics shall not apply Recognize and deal with special causes outside the (Six Sigma) process Implement Corrective and Preventive Action (CAPC)
  • 182. SPC Examples of Special Causes, MAN MACHINE MATERIAL METHOD MEASUREMENT Untrained Employee working on the Machine Production of Product on Conventional Lathe machine where Product Run out requirement is 2 microns. Major & frequent breakdowns of Machine. Frequent Power Failures. Use of different grade of raw material Setting of process Parameters which are not proven. Tool breakage Use of Micrometer having range of 0-25 mm to check O.D. of 25 mm ± 0.1 mm.
  • 183. Types of Control Charts VARIABLE 𝑿, R 𝑿, s 𝑿, mR CUSUM ATTRIBUTE p np c u
  • 184. Control Charts Overview The first step for control charting is to identify the CTQ’s of the process which is required to be brought under control Types of Control Charts Depends on the nature of the variable needed to control:  Variable Control Charts  Attribute Control Charts
  • 186. Variable Control Chart Xbar – Rbar When to use: When studying the behavior of a single measurable characteristic produced in relatively high volumes. How: By plotting sample averages (X-bar) and ranges (R) on separate charts. This allows for independent monitoring of the process average and the variation about that average. Conditions:  Constant sample size.  One characteristic per chart.  Should have no less than 20 samples before calculating control limits.
  • 187. Variable Control Chart Xbar – Rbar 1. Most common type of control chart for analyzing continuous variables. 2. The xbar part of the chart notes the variation between the averages of consecutive sub-groups of data points. 3. The R part of the chart notes the changes of variation within each of the consecutive sub-groups.
  • 188. Variable Control Chart RATIONAL SUBGROUP CONCEPT  Subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized, while the chance for differences due to these assignable causes within a subgroup will be minimized.  Time order is frequently a good basis for forming subgroups because it allows to detect assignable causes that occur over time.  Two general approaches for constructing rational subgroups: ◦ Construction units of production ◦ Random sample of all process output over the sampling interval
  • 189. Control Chart  Reviewing plots & Analysis of trends: Ensure that all points of both X and R charts within control limits. If any point touching to any of the control limits, review process related remark corresponding to particular sub-group. This is assignable cause. Study particular trends if any ◦ Case study: ◦ Consider process of side member sub- assembly where critical dimensional characteristics i.e. concentricity of mounting holes is controlled.
  • 190. Control Chart TRENDS ANALYSIS IN SPC CHARTS ALL POINTS WITHIN CONTROL LIMIT S.NO Trend Type Meaning Precautions for better process control 1. All points within control limits with zigzag pattern Process under control, variation due to random causes. Zigzag pattern changing with each point over judgment Let process continue. Try to make it a natural process 2. 7 more consecutive points on one side of center line Process Centre shifted towards one of the specification limit Do changes to bring process to Centre 3. Cyclic trends Assignable cause happening periodically Study assignable cause and reason. Study to prevent 4. Continuous inclination towards one of the control limits Assignable cause for process drift. If not prevented, product may go out of control Study assignable cause, set process to prevent drifting
  • 191. Control Chart TRENDS ANALYSIS IN SPC CHARTS ALL POINTS WITHIN CONTROL LIMIT S.NO Trend Type Meaning Precautions for better process control 1. All points suddenly going out of control limits Assignable cause present, study specific process event associated with period of specific point Study probable causes for assignable cause taking place try to resolve the same 2. Any point going out of control limits with definite trend Process going out of control due to assignable cause Study the trend type & establish controls to prevent the assignable cause occurring
  • 192. Typical Out-Of-Control Patterns  Point outside control limits  Sudden shift in process average  Cycles  Trends  Hugging the center line  Hugging the control limits  Instability
  • 193. Shift in Process Average
  • 194. Cycles
  • 195. Trend
  • 196. Control Charts PURPOSE OF CONDUCTING SPC STUDIES:  To study and analyze process variation  To find out trends in processes  To identify random & sporadic causes  To manufacture products of consistent quality  To prevent wastage of material
  • 198. Capability vs Stability  Capability has a meaning only when a process is stable.  If a process is out of control, first we need to stabilize the process.  Improvement in the inherent variation can be made only when the process is stable.  Control Charts are used to study stability.  The first job of Six Sigma practitioner is to identify and remove Special Causes of Variation.  Once the process is made predictable, the next job is to identify the causes of inherent variation and remove them.
  • 199. Calculating Capability 𝐶𝑎𝑝𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑎𝑛 𝑏𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑇𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑃𝑟𝑜𝑐𝑒𝑠𝑠 𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 LSL USL𝑿 ± 𝟏𝝈 𝑿 ± 𝟑𝝈 𝑿 ± 𝟐𝝈 𝑪 𝒑 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝝈 𝑪 𝒑 = 𝑻 𝟔𝝈
  • 200. Calculating Capability 𝑴𝒂𝒓𝒈𝒊𝒏𝒂𝒍 𝑪𝒂𝒑𝒂𝒃𝒊𝒍𝒊𝒕𝒚 LSL USL𝑿 ± 𝟏𝝈 𝑿 ± 𝟑𝝈 𝑿 ± 𝟐𝝈 𝑪 𝒑 = 𝟔𝝈 𝟔𝝈 𝑪 𝒑 = 𝑻 𝟔𝝈 𝑪 𝒑 = 𝟏
  • 201. Calculating Capability LSL USL 𝑿 ± 𝟔𝝈 𝑪 𝒑 = 𝟏𝟐𝝈 𝟔𝝈 𝑪 𝒑 = 𝑻 𝟔𝝈 𝑪 𝒑 = 𝟐 • Six Sigma Capability 𝑿 ± 𝟑𝝈 𝟑𝝈 𝟑𝝈
  • 202. Calculating Capability 0 2 4 6 8 10 12 14 16 18 20 𝑪 𝒑𝒌 = 𝑴𝒊𝒏 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 , 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑪 𝒑𝑳 = 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 𝑪 𝒑𝑼 = 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑿 − 𝑳𝑺𝑳 𝑼𝑺𝑳 − 𝑿 • Calculate 𝑪 𝒑 from Upper and Lower side
  • 203. Calculating Performance 0 2 4 6 8 10 12 14 16 18 20 𝑷 𝑷𝒌 = 𝑴𝒊𝒏 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 , 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑷 𝑷𝑳 = 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 𝑷 𝑷𝑼 = 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 𝑿 − 𝑳𝑺𝑳 𝑼𝑺𝑳 − 𝑿 • Calculate 𝑪 𝒑 from Upper and Lower side 𝑷 𝑷 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝝈
  • 204. Calculating Performance 𝑪 𝑷 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝝈 𝑪 𝒑𝒌 = 𝑴𝒊𝒏 𝑿 − 𝑳𝑺𝑳 𝟑𝝈 , 𝑼𝑺𝑳 − 𝑿 𝟑𝝈 • If the formulae are same, what is the difference? • The difference is in Sigma Calculation! • Sigma in Capability covers Short Term Variation. • Sigma in performance covers Long term Variation. • How is the Data Collection Different?
  • 205. Process Capability Ratios ContinuousImprovement LSL USL 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 < 𝟏. 𝟎 LSL USL IncreasedNumberofDefects Process Capability Real Capability 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑 = 𝟐. 𝟎 𝑪 𝒑𝒌 = 𝟐. 𝟎 𝑪 𝒑𝒌 < 𝟐. 𝟎 𝑪 𝒑𝒌 = 𝟏. 𝟎 𝑪 𝒑𝒌 = 𝟎. 𝟎 𝑪 𝒑𝒌 < 𝟎. 𝟎 𝑪 𝒑𝒌 < −𝟏. 𝟎 Understanding 𝑪 𝑷 and 𝑪 𝑷𝑲 𝑪 𝑷 only works for a process that is centered on the target 𝑪 𝑷𝑲 is a better measure for tracking performance
  • 206. Capability Indices 𝑪 𝒑 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝑺 𝑪 𝒑𝒍 = 𝑿 − 𝑳𝑺𝑳 𝟑𝑺 𝑪 𝒑𝒖 = 𝑼𝑺𝑳 − 𝑿 𝟑𝑺 𝑪 𝒑𝒌 = 𝐦𝐢𝐧(𝑪 𝒑𝒖 , 𝑪 𝒑𝒍) 𝑪 𝒑 = 𝟎. 𝟑𝟏𝟕 − 𝟎. 𝟑𝟎𝟕 𝟔 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟏 . 𝟎𝟎𝟕𝟐 = 𝟏. 𝟑𝟖𝟗 𝑪 𝒑𝒖 = 𝟎. 𝟑𝟏𝟕 − 𝟎. 𝟑𝟏𝟓𝟓 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟎𝟏𝟓 . 𝟎𝟎𝟑𝟔 = 𝟎. 𝟒𝟏𝟕 𝑪 𝒑𝒍 = 𝟎. 𝟑𝟏𝟓𝟓 − 𝟎. 𝟑𝟎𝟕 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟎𝟖𝟓 . 𝟎𝟎𝟑𝟔 = 𝟐. 𝟑𝟔𝟏 𝑪 𝒑𝒌 = 𝐦𝐢 𝐧 𝟎. 𝟒𝟏𝟕, 𝟐. 𝟑𝟔𝟏 = 𝟎. 𝟒𝟏𝟕 Exampl e 𝑿 𝒎𝒆𝒂𝒏 = 𝟎. 𝟑𝟏𝟓𝟓 𝒔 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 = 𝟎. 𝟎𝟎𝟏𝟐 𝑳𝑺𝑳 𝒍𝒐𝒘𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟎𝟕 𝑼𝑺𝑳 𝒖𝒑𝒑𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟏𝟕 0.306 0.308 0.310 0.312 0.314 0.316 0.318 0.320 LSL USL
  • 207. Capability Indices 𝑪 𝒑 = 𝑼𝑺𝑳 − 𝑳𝑺𝑳 𝟔𝑺 𝑪 𝒑𝒍 = 𝑿 − 𝑳𝑺𝑳 𝟑𝑺 𝑪 𝒑𝒖 = 𝑼𝑺𝑳 − 𝑿 𝟑𝑺 𝑪 𝒑𝒌 = 𝐦𝐢𝐧(𝑪 𝒑𝒖 , 𝑪 𝒑𝒍) 𝑪 𝒑 = 𝟎. 𝟑𝟐𝟐 − 𝟎. 𝟑𝟎𝟐 𝟔 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟐 . 𝟎𝟎𝟕𝟐 = 𝟐. 𝟕𝟕𝟖 𝑪 𝒑𝒖 = 𝟎. 𝟑𝟐𝟐 − 𝟎. 𝟑𝟏𝟒𝟑 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟎𝟕𝟕 . 𝟎𝟎𝟑𝟔 = 𝟐. 𝟏𝟑𝟗 𝑪 𝒑𝒍 = 𝟎. 𝟑𝟏𝟒𝟑 − 𝟎. 𝟑𝟎𝟐 𝟑 × 𝟎. 𝟎𝟎𝟏𝟐 = 𝟎. 𝟎𝟏𝟐𝟑 . 𝟎𝟎𝟑𝟔 = 𝟑. 𝟒𝟏𝟕 𝑪 𝒑𝒌 = 𝐦𝐢 𝐧 𝟐. 𝟑𝟏𝟗, 𝟑. 𝟒𝟏𝟕 = 𝟐. 𝟑𝟏𝟗 Exampl e 𝑿 𝒎𝒆𝒂𝒏 = 𝟎. 𝟑𝟏𝟒𝟑 𝒔 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 = 𝟎. 𝟎𝟎𝟏𝟐 𝑳𝑺𝑳 𝒍𝒐𝒘𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟎𝟐 𝑼𝑺𝑳 𝒖𝒑𝒑𝒆𝒓 𝒔𝒑𝒆𝒄 𝒍𝒊𝒎𝒊𝒕 = 𝟎. 𝟑𝟐𝟐 0.300 0.305 0.310 0.315 0.320 LSL USL
  • 208. Let’s Summarize  A process cannot be improved till it is Stabilized.  Capability data should be utilized for stable processes  Subgroups should contain consecutive data, not random data.  Performance calculations should be done based on large amount of data representing Long Term Variation.
  • 210. Discrete Data Capability A discrete defect is an attribute, which can be counted. Such as: Scratches, Spots, Dent Marks, Cracks etc.  In these cases ½ does not make sense.  A defect is non conformance to the standards.  A defective unit can have more than one defect.  A sample of 100, may have 2 defectives but 5 defects.
  • 211. Discrete Data Capability Defect Opportunities:  Defect opportunities are various types of defects, that may occur.  These creates dissatisfaction to the customers.  This is different than defects that occur.  Example : 12 type of defects that can occur on painted part. However, on a part produced, we may observe 0 to up to 12 defects. Thus a part may be defect free or may have1to 12 defects.
  • 212. Discrete Data Capability Example: A sample of 100 nos have been taken. Following are the results of inspection: No of Defectives – 3 No of defects – 10 No of Opportunities - 12
  • 213. Discrete Data Capability Example: The capability can be calculated as follows: No of units = U =100 Defects = D =10 No of Opportunities = O = 12 Total defect opportunities = UxO = 100x12 =1200 DPO = Defects per opportunity = 10/1200 =1/2 = 0.0083
  • 214. Discrete Data Capability Example: Defect per million opportunities (DPMO) =DPO x 1,000,000 =0.0083 x 1,000,000 =8300 DPMO From the tables, the corresponding sigma level is 3.9.
  • 215. Discrete Data Capability The same formula also can be expressed as DPMO = 𝑵𝑶. 𝑶𝑭 𝑫𝑬𝑭𝑬𝑪𝑻𝑺×𝟏𝟎 𝟔 𝑵𝑶 𝑶𝒇 𝑼𝑵𝑰𝑻𝑺×𝑶𝑷𝑷./𝑼𝑵𝑰𝑻
  • 216. Discrete Data Capability – Example of DPMO Suppose we observe 200 letters delivered incorrectly to the wrong addresses in a small city during a single day when a total of 200,000 letters were delivered. What is the DPMO in this situation? DPMO = 𝟐𝟎𝟎 × 𝟏𝟎 𝟔 𝟐𝟎𝟎,𝟎𝟎𝟎 ×𝟏 = 𝟏, 𝟎𝟎𝟎 So, for every only million letters delivered this city’s postal managers can expect to have 1,000 letters incorrectly sent to the wrong address. What is the Six Sigma Level for this Process?
  • 217. DPMO Example  IRS tax form advice  Survey of responses indicates predicted error rate  If 40% then: DPO = 0.40 DPMO = 0.40 defects/opportunity * 1,000,000 opportunities/million opportunities 400,000 DPMO = 1.75 Sigma
  • 218. DPMO Example Example of Rolled throughput yield  If there are five processes with following yields:  Rolled throughput yield for this process is = 0.9 × 0.99 × 0.95 × 0.96 × 1 = 0.7279 = 0.73 = 73% Process No. Yield in % 1 90 2 99 3 95 4 96 5 100
  • 219. DPMO - Exercise  You have 100 documents You take a sample of 10 documents There are 10 opportunities for defect on each document. 5 defects were found. What is DPMO Attendance Policy June 23, 2000 Crane Operational Excellence Program All Operational Excellence Leaders should be aware.
  • 220. Complexity and Capability Payroll and Labor Tracking Process Does complexity have an important impact on process capability and quality? There are many opportunities for defects… Step 1 97.4% Read and record daily start and stop time 𝒀 𝑹𝑻 Output 79.1% Step 6 99.9% Create payroll checks Step 5 95.5% Transfer hour totals to payroll generatio n system Step 4 91.8% Total weekly work hours and job accounts . Submit time card Step 3 98.0% Total daily work hours Step 2 94.6% Read and record daily start and stop time Rolled Throughput Yield Example =
  • 221. Complexity and Capability Payroll and Labor Tracking Process Our goal, reduce the total number of opportunities and increase the capability of remaining opportunities Step 1 97.4% Output 79.1% Step 6 99.9% Step 5 95.5% Step 4 91.8% Step 3 98.0% Step 2 94.6% Rolled Throughput Yield Example = 𝒀 𝑹𝑻 Output 98.9% Step 3 99.9% Print payroll checks from computer generated database Step 2 99.4% Scan employee badge and job card for labor start and stop time Step 1 99.6% Scan employee badge for start and stop time =
  • 222. Complexity and Capability Notice any Difference? Step 1 93.32% Output 81.26% Step 3 93.32% Step 2 93.32% Step 2 99.999997% Rolled Throughput Yield Example = = Output 79.1% Step 2 99.999997% Step 2 99.999997%x x xx A Three Sigma Process A Six Sigma Process
  • 223. Sigma Levels SIGMA Defect per Million Opportunities (DPMO) 1 690,000 2 308,537 3 66,807 4 6,210 5 233 6 3.4
  • 225. Hypothesis Testing Concept  Hypothesis testing is one of the most scientific ways of decision making.  It works very much like a court case.  We have a suspect, we have to take decision whether He / She is innocent or guilty.  Suppose there is person charged with murder, and both sides (defense and prosecution) do not have any evidence, what would be decision?  Innocent unless proven guilty?  Guilty unless proven Innocent? Null Hypothesis
  • 226. Null Hypothesis  Null hypothesis is represented by Ho  It is statement of Innocence.  It is something that has to be assumed if you cannot prove otherwise.  It is statement of No Change or No Difference.
  • 227. Null Hypothesis – A Court Case  Just Like a court case, we first assume the accused (X) is innocent and then try to prove it otherwise based on evidence (Data).  If evidence (Data) does not show sufficient difference, we cannot reject the innocence(Ho)  But if Evidence (Data) is strong enough, we reject the Innocence (Ho) and pronounce the suspect Guilty (Ha).  The statement that will be considered valid if null hypothesis is rejected is called Alternate Hypothesis (Ha)
  • 228. Null hypothesis – A Concept  Hypothesis testing is a philosophy that real life situations.  You cannot prove two things equal.  You cannot prove two things different by proving only one difference  If you cannot prove 2 things different, you have to assume that they are equal.  But if you cannot prove them Different, are they really Equal?  What is the RISK involved?
  • 229. Hypothesis Testing Concept In Truth, the Defendant is: Correct Decision Innocent individual goes Free Incorrect Decision Guilty Individual Goes Free Incorrect Decision Innocent Individual Is Disciplined Correct Decision Guilty Individual Is Disciplined 𝑯 𝑨: Guilty𝑯 𝒐: Innocent Verdict Innocent Guilty
  • 230. Hypothesis Testing Concept Correct Decision Incorrect Decision Type II Error Probability = 𝛽 Incorrect Decision Type I Error Probability = 𝛼 Correct Decision 𝑯 𝑨is True𝑯 𝒐is True 𝑯 𝒐is True 𝑯 𝑨is True Decision True, But Unknown State of the World
  • 231. Hypothesis Testing Concept Hypothesis testing Justice System  State the Opposing Conjectures, Ho and HA.  Determine the amount of evidence required, n, and the risk of committing a “type error”,  What sort of evaluation of the evidence is required and what is the justification for this? (type of test)  What are the conditions which proclaim guilt and those which proclaim innocence/ (Decision Rule)  Gather & Evaluate the evidence.  What is the verdict? (Ho or HA?)  Determine “Zone of Belief” : Confidence Interval.  What is appropriate justice? – Conclusions
  • 232. Hypothesis Testing 1. Null Hypothesis (Ho) – statement of no change or difference. The statement is assumed true until sufficient evidence is presented to reject it. 2. Alternate Hypothesis (Ha) – statement of change or difference. This statement is considered true if Ho is rejected. 3. True I Error – the error in rejecting Ho when it is in true fact, there is no difference. 4. Alpha Risk – the maximum risk or probability of making a Type I Error. This Probability is always greater then zero, and is usually established at 5%. The researcher makes the decision to the greatest level of risk that is acceptable for a rejection of Ho. Also known as significant level. 5. Type II Error – The error in failing to reject Ho when it in fact false, or saying there is no difference when there really is a differerence.
  • 233. Hypothesis Testing Concept 6) Beta Risk – The risk probability or making a Type II Error, or overlooking an effective treatment or solution to the problem. 7) Significant Difference – The term where a difference is too large to be reasonably attributed to chance.
  • 234. 𝛼 𝑎𝑛𝑑 𝛽 Risks  𝛼 Risk is also called producer’s risk.  𝛽 Risk is also called consumer’s risk.  Can we commit both type I and type II error at the same time?  As it necessary that we will have both 𝛼 and 𝛽risks?  Are𝛼 and 𝛽 risks equal?  Is 𝛼 and 𝛽 = 1?  Is there any relationship between 𝛼 and 𝛽?  Which risk is more important?
  • 235. 𝛼 𝑎𝑛𝑑 𝛽 Risks  An 𝛼 Risk of 5% is generally accepted.  An 𝛽 Risk of 10% is generally Accepted.  Since Ha cannot be proved, our attempt is to try and reject it.  What risk do we get in trying to reject the Ho.  Minitab represents 𝛼 risk by p-panel!
  • 236. Steps in Hypothesis Testing  Define Ho  Define Ha  Select Appropriate Test.  Decide Significance Level (𝛼 and 𝛽)  Decide Sample Size  Collect Data  Conduct Test  Interpret!
  • 237. Define Ho/Ha For following Cases To find if a distribution is normal or not.  Ho =>  Ha =? To find if the defects from three machines are same or different  Ho =>  Ha => To find if 2 groups of students from different streams have differing IQ  Ho =>  Ha =>
  • 239. Statistical Error Definitions  Null Hypothesis Ho:  “Status quo”  “Nothing is different”  Equality  We fail to reject Ho based on statistical evidence  Alternate Hypothesis Ha:  “Something is different”  Statement about the population that requires strong evidence to prove  If we reject Ho, we in practice accept Ha.  Alpha Risk (𝛼)  Also called type I Error  Hypothesis the null hypothesis when it is fact true.  Beta Risk (𝛽)  Also called a Type II Error  Accepting the null Hypothesis when it is in fact false.
  • 240. Statistical Error Typical 𝛼 𝑎𝑛𝑑 𝛽Risks  Typically, the 𝛼 level is set at 0.05 and the 𝛽 level is set at 0.10  They can be set at any level depending on what you want to know The risk is also called the “p-value” 1-𝛼 = confidence that an observed outcome in the sample is “real” We typically look for a p-value of 0.05 because:  1-0.05 = 0.95 (or 95% confidence)
  • 241. The Central Limit Theorem Normally  Why are distributions normal? When all factors are random Some measurements are actually averages over time of “micro-measurements” In other words, what we see as a measurement is actually an average The Central Limit Theorem explains why a distribution of averages tends to be normal
  • 242. Confidence Sample statistics estimate the mean or standard deviation of a population The “True” population mean and standard are unknown Confidence limits, levels, and intervals are used to determine the population statistics For means… We use t distribution to calculate limits, levels, and intervals For Standard Deviations… We use the 𝑐2 distribution to calculate limits, levels, and intervals
  • 243. Definition Confidence Level:  The level of risk we are willing to take  How sure we want to be that the population mean or standard deviation falls between the confidence level is typical  95% confidence level is typical.  95% chance that the population mean or standard deviation falls between the limits.  5% chance (alpha risk) that the population mean or standard deviation isn’t contained within the calculated limits.  Risk (𝛼/2) Risk (𝛼/2) Risk (𝛼)
  • 244. Definition Confidence Limit Upper and Lower limits that bracket the “true” mean or standard deviation of a population  Calculation from the sample data and the appropriate test statistic.  Test statistic is dependent on the risk we accept that our results will be wrong.
  • 245. Definition Confidence Interval The interval defined by the upper and lower confidence limits. A range of values based on  Sample mean or sample standard deviation  Sample size  Confidence level  Appropriate test statistic Contains  Population mean or  Population standard deviation
  • 247. Confidence Limit Formulas Means Lower Confidence Limit Upper Confidence Limit 𝑋 - 𝑡( 𝛼 2 ,𝑛−1) 𝑠 √𝑛 𝑋 + 𝑡( 𝛼 2 ,𝑛−1) 𝑠 √𝑛 𝑋 = 𝑚𝑒𝑎𝑛 𝑠 = 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑛 = 𝑆𝑎𝑚𝑝𝑙𝑒 𝑆𝑖𝑧𝑒 𝑡 = 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑠𝑡𝑢𝑑𝑒𝑛𝑡′ 𝑠 𝑡 − 𝑡𝑎𝑏𝑙𝑒 𝑓𝑜𝑟 𝛼 2 𝑎𝑛𝑑 𝑛 − 1 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚 (𝑓𝑜𝑟 𝑎 95% 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙, 𝛼 2 = 0.025) 𝛼 = 𝐴𝑙𝑝ℎ𝑎 𝑅𝑖𝑠𝑘
  • 248. Confidence Limit Formulas Means Confidence Interval Lower Confidence Limit ≤ Mean ≤ Upper Confidence Limit ≤ 𝝁 ≤𝑋 - 𝑡( 𝛼 2 ,𝑛−1) ( 𝑠 𝑛 ) 𝑋 + 𝑡( 𝛼 2 ,𝑛−1) ( 𝑠 √𝑛 )
  • 249. Confidence Limit - Example The tensioning device (rubber band) used on the Silobuster has come under scrutiny  Two sets of tensioners are measured and descriptive statistics are run. What is the 95% confidence interval for the variation? ≤ 𝝁 ≤ Set 1 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 25 Set 2 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 100 We are 95% confident that the interval 0.2479 to 0.2521 brackets the true process standard deviation (0.0042 width) We are 95% confident that the interval 0.2490 to 0.2510 brackets the true process standard deviation (0.0020 width) 𝑿 - 𝒕( 𝜶 𝟐 ,𝒏−𝟏)( 𝒔 𝒏 ) 𝑿 + 𝒕( 𝜶 𝟐 ,𝒏−𝟏) ( 𝒔 √𝒏 ) . 𝟐𝟓𝟎 ± 𝟐. 𝟎𝟔𝟑𝟗 . 𝟎𝟎𝟓 𝟐𝟓 = . 𝟐𝟒𝟕𝟗 𝒕𝒐 . 𝟐𝟓𝟐𝟏 . 𝟐𝟓𝟎 ± 𝟏. 𝟗𝟖𝟒𝟐 . 𝟎𝟎𝟓 𝟏𝟎𝟎 = . 𝟐𝟒𝟗𝟎 𝒕𝒐 . 𝟐𝟓𝟏𝟎
  • 250. Basic Concepts Confidence Limits For Standard Deviation
  • 251. Confidence Limit Formulas Variation Confidence Interval - Lower Confidence Limit ≤ Standard Deviation ≤ Upper Confidence Limit 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒍𝒐𝒘𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 ≤ 𝝈 ≤ 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒖𝒑𝒑𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 Population Standard Deviation
  • 252. Confidence Limit Formulas Variation The tensioning device (rubber band) used on the Silobuster has come under scrutiny  Two sets of tensioners are measured and descriptive statistics are run. What is the 95% confidence interval for the variation? 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒍𝒐𝒘𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 ≤ 𝝈 ≤ 𝒔 𝒏 − 𝟏 𝒙 𝟐 𝒖𝒑𝒑𝒆𝒓 𝒗𝒂𝒍𝒖𝒆 𝟎. 𝟎𝟎𝟓 𝟐𝟒 𝟑𝟗.𝟑𝟔 = 0.0039 and𝟎. 𝟎𝟎𝟓 𝟐𝟒 𝟏𝟐.𝟒𝟎 = .0070 𝟎. 𝟎𝟎𝟓 𝟗𝟗 𝟏𝟐𝟖𝟒𝟐 = 0.0044 and 𝟎. 𝟎𝟎𝟓 𝟗𝟗 𝟕𝟑.𝟑𝟔 = .0058 Set 1 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 25 Set 2 Mean: 0.250” Standard Deviation: 0.005” Sample Size: 100 We are 95% confident that the interval 0.0039 to 0.0070 brackets the true process standard deviation (0.0031 width) We are 95% confident that the interval 0.0044 to 0.0058 brackets the true process standard deviation (0.0014 width)
  • 253. TEST OF HYPOTHESIS - roadmap You want to compare the averages/ medians of samples of data to decide if they are statistically different Are samples normally distributed Compare median values instead if average How many samples do you wan to compare Kruskall Wallis Test For samples that do not have any outliners One-way ANOVA For comparing averages of three or more samples against one another 1 Sample t-test Comparing av. of one sample against target Paired t-test For comparing averages of two samples that contain data that is linked in pairs Two Sample t-test For comparing averages of two samples against each other Mood’s Median Test For samples that have some outliners Transform Data Yes 2 1 or 3 or more No No or
  • 255. EXERCISE  Represent the following data in graphical form: Temperature 100 100 120 120 Response 275 285 270 325 Pressure 250 300 250 300
  • 256. EXERCISE - continued a) Determine what parameter settings yield the largest response. b) Determine what parameter settings of pressure would be bets if it were important to reduce the variability of the responses that results from frequent temperature variations between two extremes.
  • 257. EXERCISE - continued 0 100 200 300 400 500 600 700 100 120 Pressure = 300 Pressure = 250 Response Temperature
  • 258. Design Of Experiments Design of Experiments (DOE) is a valuable tool to optimize product and process designs, to accelerate the development cycle, to reduce development costs, to improve the transition of products from research and development to manufacturing and to effectively trouble shoot manufacturing problems. Today, Design of Experiments is viewed as a quality technology to achieve product excellence at lowest possible overall cost.
  • 259. Design of Experiments General Comments  Keep your experiments simple  Don’t try to answer all the questions in one study  Use 2 level designs to start  Try potential business results to the project  The best time to design an experiment is after the previous one is finished  Always verify results in a follow-up study ( verification)  Be ready for changes  A final report is a must to share the knowledge  Avoid DoE infatuation…do your homework first!  Measure & Analyze to reduce potential variables  Use Graphical Analysis  Use the basic tools of Operational Execllence
  • 260. Design of Experiments Be Proactive  DOE is a proactive tool  If DOE output is inconclusive: You may be working with the wrong variables Your measurement system may not be capable The range between high and low levels may be sufficient  There is no such thing as a failed experiment Something is always learned New data prompts us to ask new questions and generates follow-up studies  Remember to keep an open mind Let the data/output guide your conclusions Debunk or validate tribal knowledge Don’t let yourself be “confused by the facts.”
  • 261. Design Of Experiments Types of Experiments Traditional Approach Six Sigma Approach Very Informal Very Formal • Trial and Error Methods  Introduce a change and see what happens • Running Special Lots or Batches  Produced under controlled conditions • Pilot Runs  Set up to produce a desired effect. • One-Factor-at –a-Time Experiments  Vary one factor and keep all other factors constant • Planned Comparisons of Two to Four Factors  Study separate effects and interactions • Experiment With 5 to 20 Factors  Screening Studies • Comprehensive Experimental Plan With Many Phases  Modeling, multiple factor levels, optimization Very Informal Very Formal • Trial and Error Methods  Introduce a change and see what happens • Running Special Lots or Batches  Produced under controlled conditions • Pilot Runs  Set up to produce a desired effect. • One-Factor-at –a-Time Experiments  Vary one factor and keep all other factors constant • Planned Comparisons of Two to Four Factors  Study separate effects and interactions • Experiment With 5 to 20 Factors  Screening Studies • Comprehensive Experimental Plan With Many Phases  Modeling, multiple factor levels, optimization
  • 262. Design Of Experiments Barriers to Successful DoE’s  Problem or objective unclear  Results of the experiments unclear  Be present during the DoE  Identify and record unexpected noise or other variables  Measurement Error  Lack of Management Support  Lack of Experimental Discipline  Don’t use a DoE as the first pass to identify key X’s  Manage the constants and the noise  Process map, C&E, Constant or Noise or Experimental  Unstable process prior to running DoE  Process map, C&E, Constant or Noise or Experimental, Manage the C’s and N’s to reduce extraneous variation
  • 263. Design Of Experiments Objective  Establish the objective for the experiment It should be stated in such a way to provide guidance to those involved in designing the experiment.
  • 264. Design Of Experiments Planning the Experiment  Team in involvement  Maximize prior knowledge  Pursue measurable objectives  Plan the execution of all phases  Rigorous sample size determination  Allocate sufficient resources for data collection and analysis.
  • 265. Design Of Experiments The following are some of the objectives of experimentation in an industry:  Improving efficiency or yield  Finding optimum process settings  Locating sources of variables  Correlating process variables with product characteristics  Comparing different processes, machines, materials etc.  Designing new processes and products.
  • 266. Various Terms Used In Experimentation  Factor: One of the controlled or uncontrolled variables whose influence on the response is being studied. May ne variable or classification data.  Level: The values or the factor being studied usually high(+) and low(-)  Treatment Combination: An experiment run using a set of the specific levels or each input variable  Response Variable: The variable that is being studied. “Y’ factor in the study. Measured output variable.  Interaction: The combined effect of two or more factors that is observed which is in addition to the main effect of each factor individually.
  • 267. Various Terms Used In Experimentation  Confounding: One or more effects that can not unambiguously be attributed to a single factor or interaction.  Main effect: Change in the average response observed during a change from one level to another for a single factor.  Replication: Replication of the entire experiment. Treatment combinations are not repeated consequently.  Test run: A single combination of factors that yields one or more observation of the response.  Treatment: A single level assigned to a single factor during an experiment.
  • 268. Trial And Error  Perhaps the most well known and used methodology.  The objective is to provide a quick fix to a specific problem.  The quick fix occurs by randomly and no- randomly making changes to process parameters.  Often changing two or more parameters at the same time.  The result often is a “Band-Aid” fix as the symptoms of the problem are removed, but the cause of the problem goes undetected.  In trial and error experimentation, knowledge is not expanded but hindered.  Implement multiple expensive fixes are not necessary.
  • 269. One-Factor-At-A-Time (OFAT) The old dogma in experimentation is to hold everything constant and vary only one-factor- at-a-time. ◦ Assumes any changes in the response would be due only to the manipulated factor. But are they? ◦ Is it reasonable to assume that one can hold all variables constant while manipulating one? Experience tells us this is virtually impossible. Imagine there area large number of possible factors affecting the response variable: ◦ How long would OFAT take to identify critical factors and where they should be run for best results? ◦ How much confidence would you have that the knowledge gained would apply in the real world?
  • 270. OFAT Although OFAT may simplify the analysis of results, the experiment efficiency given up is significant: ◦ Don’t know the effects of changing one factor while other factors are changing (a reality). ◦ Unnecessary experiments may be run. ◦ Time to find casual factors (factors that affect the response) is significant.
  • 271. Classification Of Factors 1. Experimental Factors are those which we really experiment with by varying them at various levels. 2. Control Factors are those which are kept at a constant (controlled) level throughout experimentation. 3. Error or Noise factors are those which can neither be changed at our will nor can be fixed at one particular level. Effect of these factors causes the error component in the experiment and as such these factors are termed as error or noise factors.
  • 272. Experimental Design Visualization of The 21 Design (2 Levels – 1 factor) This is often the method used today for process optimization. It is the “only one factor at a time” concept High Factor 1 Low
  • 273. Experimental Design Visualization of The 22 Design (2 Levels – 2 factor) The most basic of true designs. There are 4 runs. High Factor 1 HighFactor 2Low Low
  • 274. Experimental Design Visualization of The 23 Design (2 Levels – 3 factor) A little more complicated design but still very practical. There are only 8 runs. High Factor 1 HighFactor 2Low Low High Factor 3 Low
  • 275. Experimental Design Factor 4Low High Visualization of The 24 Design (2 Levels – 4 factor)
  • 276. Experimental Design Factor 4Low High Factor 4 High Factor 5 Low Visualization of The 𝟐 𝟓 Design (2 Levels – 5 factors) Here is where it’s time to stop drawing but it represents the complexity associated with a 5 factor design.
  • 277. Experimental Design Three Factorial Design, without interaction 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 Y - - - 𝒀 𝟏 - - + 𝒀 𝟐 - + - 𝒀 𝟑 - + + 𝒀 𝟒 + - - 𝒀 𝟓 + - + 𝒀 𝟔 + + - 𝒀 𝟕 + + + 𝒀 𝟖
  • 278. Experimental Design Three Factorial Design, with interaction 𝑿 𝟏 𝑿 𝟐 𝑿 𝟑 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 𝑿 𝟏 Y - - - + + + - 𝒀 𝟏 - - + + - - + 𝒀 𝟐 - + - - + - + 𝒀 𝟑 - + + - - + - 𝒀 𝟒 + - - - + + + 𝒀 𝟓 + - + - + - - 𝒀 𝟔 + + - + - - - 𝒀 𝟕 + + + + + + + 𝒀 𝟖