ANOVA ANalysis Of VAriance An ANOVA is an analysis of the variation present in an experiment. It is a test of the hypothesis that the variation in an experiment is no greater than that due to normal variation (random noise) of individuals' characteristics and error in their measurement. (Ratio of explained to unexplained variation. Note-1: The output of ANOVA table is the F-test.Note-2: We are comparing MEANS (because we assume variance to be equal)Also see ANOM (analysis of means)Most accurate method for quantifying reproducibility (technician variance) & repeatability (error variance)

Benchmarking Steps of Benchmarking1. Determine current practices and processes to be benchmarked. Understand your and others processes.2. Identify best practices and areas for improvement. Measure your performance & define leaders in best practices. (do not just copy someone’s process)3. Gather data. Analyze best practices. 4. Repeat the cycle.

Box & Whisker or Box Plot

50th percentile is median, 25th percentile is “median” of lower ½BPM Business Process Management: is a method of efficiently aligning an organization with the wants and needs

of clients. It is a holistic management approach that promotes business effectiveness and efficiency while striving for innovation, flexibility and integration with technology. As organizations strive for attainment of their objectives, BPM attempts to continuously improve processes - the process to define, measure and improve your processes – a ‘process optimization' process.

Chart CPM Critical Path Method (See PERT) ACTIVITY oriented Note: can be more vigorous than PERT

Chart PERT Program Evaluation & Review Technique ( see CPM) EVENT oriented)

Chart Scatter – Linear Regression

r Explained variation Total variation (NOTE: as explained variation increases the correlation between x & y increases)

You choose values of x to see how it affects y (x = independent variable, y = dependent variable) Simple linear regression equation is y=ax+b

y=dependent variable, x=independent variable, a=slope, b=intersect (where avg. value of x intersects average value of y)Note: If there is a correlation it DOES NOT ALWAYS mean that one caused the other (I,e. shark attacks and ice cream sales)r calculated means strong correleation if r = + or -1 (45º slope) and no correlation (no slope) if r = 0

COF Coefficient of Variation =Std-dev/Mean (expressed as %)Control Chart Decision Tree

Outliers are values that

are more than 1-1/2 box

lengths away from upper or lower edge of box

Control Chart Defined ATTRIBUTE CHARTS

Attribute charts are useful for both machine- and people-based processes. Data for them is often readily available and they are easily understood. It can thus be easier to start with these, then move on to Variables charts for more detailed analysis.

u- and c-charts give a measure of DEFECTS (can be reworked or does NOT render product unusable). You cannot count the non-occurrences. Failures either happen or they don’t.

u-chart, VARYING sample size (%) the defects within the unit must be independent of one another, for example, 'component failures on a printed circuit board, or defects per square meter of cloth'.

c-chart FIXED sample size is a useful alternative to a u-chart when there are a lot of possible defects on a unit, but there is only a small chance of any one defect occurring. For example, 'flaws in a roll of material, or dents per panel'.

p- and np-charts give a measure of DEFECTIVES (unit failure or cannot be repaired). You can count the occurrences and non-occurrences. Failures may happen more than once.

p-chart, VARYING sample size (%).

np-chart. FIXED sample size

Control Chart Defined VARIABLE CHARTS

Variables charts are useful for machine-based processes, for example in measuring tool wear.

Variables charts are more sensitive to change than Attributes charts, but can be more difficult both in the identification of what to measure and also in the actual measurement.

XmR (aka: I-MR) Individuals & Moving Range. Only use an Individuals (X) chart when few measurements are available (e.g. when they are infrequent or

are particularly costly. Data must be normally distributed. These charts are less sensitive to change than the Averages (X-bar) chart. They can also have an unusual (non-normal) distribution, in which case the Control Chart is not a suitable tool (use a Line Graph instead).

Moving Average Chart: Use to smooth data and estimate future values. Based on X-bar and S chart. Best when mean is stable, poor when process is trending. (Note: 1st subgroup =5 then avg. of 1st group&7, avg. of 7&8, avg. of 8&9….)

The Standard Deviation (s) chart is used to monitor variation. May be used instead of the Range chart. Used for large subgroups (>25).

X-bar & Range chart should have 100 individual data points at a minimum to calculate control limits.Control Chart formulas Attributesc-chart

Control Chart formulas Attributesnp-chart

Control Chart formulas Attributesp-chart

Control Chart formulas Attributesu-chart

Note: This is a “running” calculation

Control Chart formulas VARIABLES

X-bar & R chart

for X-bar: [A2 for n=5 is 0.577]

for Range: [D4 for n=5 is 2.114] Note: Zero is LCLR for

n=5

XmR chart

for X-bar: [d2 for n=2 is

1.128]

for Range: [D4 for n=2 is 3.267]

Alternate XmR X-bar

for X-bar: [E2 for n=2 is 2.66]

Control Chart Pre-Control

1. Process must be stable2. Process spread must be less than or equal to specification

spread (Cp ≥ 1.0)3. Ideal is 25 checks until reset is required, if reset is required

before 25 checks increase sample frequency if no reset required in 25 checks relax sample frequency

Rules Setup Ok to run if 5 pieces in a row are inside limits

Rules running: Sample 2 pieces in a row If 1st or 2nd piece is out of Spec limits STOP If 1st piece is inside P/C limits run & don’t measure 2nd piece If 1st piece is not in P/C limits check 2nd piece If 2nd piece is not in P/C limits STOP otherwise run

NOTE: Uses Attribute data (pass/fail)

n A2 d2 D3 D4 E2

2 1.88 1.128 -- 3.267 2.663 1.023 1.693 -- 2.574 1.774 0.729 2.059 -- 2.282 1.465 0.577 2.326 -- 2.114 1.296 0.483 2.534 -- 2.004 1.187 0.419 2.704 0.076 1.924 1.118 0.373 2.847 0.136 1.864 1.059 0.337 2.97 0.184 1.816 1.01

10 0.308 3.078 0.223 1.777 0.98

Control Chart Rules

WECO (Western Electric COmpany rules) General rules for detecting out of control or non-random situations

       Any Point Above +3 Sigma   ---------------------------------------------    +3 LIMIT        2 Out of the Last 3 Points Above +2 Sigma   ---------------------------------------------    +2 LIMIT        4 Out of the Last 5 Points Above +1 Sigma   ---------------------------------------------    +1 LIMIT        8 Consecutive Points on This Side of Control Line  ===================================   CENTER LINE         8 Consecutive Points on This Side of Control Line   ---------------------------------------------    -1 LIMIT        4 Out of the Last 5 Points Below - 1 Sigma  ----------------------------------------------   -2 LIMIT        2 Out of the Last 3 Points Below -2 Sigma   ---------------------------------------------    -3 LIMIT        Any Point Below -3 Sigma 

TREND & SHIFT rules are for X-bar and Range on &R charts and X-bar on XmR charts ONLY

Trend Rules: 7 in a row trending up or down. 14 in a row alternating up and downShift: 7 or more above or below averageOdds of 7 in a row are (1/2)7 = 1/128 but since it could go above or below (50% chance) = 1/64

Control Plan Document describing CTQ characteristics, the critical X’s or Y’s of the part or process. Every completed Six Sigma project should have not only a control chart (if applicable), but a control plan. This ensures that the process doesn't revert to the way it previously operated. This is a living document and is rarely closed.

Cost Of Quality See Quality CostsCpk, Ppk, Cp, Pp, Formulas(see also “Performance Ratio”)

Cpk, Ppk, Cp, Pp, values

Cp & Pp = CapabilityCpk & Ppk = Acceptability Distribution shape = Normality (normal is bell shaped).Cp and Pp – <1.00 incapable

1.00 to 1.33 capable with tight control >1.33 capable

Histogram – bell shaped one peak

can be slightly skewed

Cpk or Ppk – <1.33 not acceptable 1.33 to 1.50 acceptable if process stays centered >1.50 acceptable with wiggle room

Note: Cp is generally larger than CpkNote: For non-normal data use some form of Data Transformation

CTQ Critical To QualityDesign Set Based Design: Incorporated various functions & concepts, Begins with broad based possibilities and

narrows them down to a few alternatives and then to a final solution.Systematic: Step-by-step approach

DATA Types (see also “Scales of Measurement”)

Attribute Data– Discrete / qualitative / categorical / attribute / counted

• Integers (WHOLE numbers: 1, 2, 0, -2) cannot be divided any smaller (i.e. cannot have 2.4 dents in car).

• One or the other (hot cold, yes no…)– Counted

• How many defects• How often do they occur• What kind or category

Variable Data– Continuous / quantitative

• Any real number (any numeric value and can be meaningfully subdivided into finer increments).• Information that can be measured on a continuum or scale.

– Measured • How long did it take• How far does it reach• How much does it weigh

Diagram Activity Network

Diagram Affinity

Wants* Weight **Least Cost 6 8 48*** 4 24 10 60Reliability 10 8 80 8.5 85 4.5 45New features 9 5 45 10 90 3 27No new training 4 9 36 5 20 8 32

161 219 164

* Wants criteria are developed to judge the options** At least one Weight gets a value of 10, the rest are compared to this (as voted on)*** = Weight x value of each category (48 = 6 x 8)Note: Use "MUST" criteria when establishing options (i.e, must improve quality and output)

Refurbish existing machine

Purchase new machine Farm job out

Options that meet "MUST" criteria

Diagram Matrix (also PUGH) * Displays relationships between

Tasks and peopleTasks and tasks

**

*

Analyze the correlations between two ideas

Shows importance of each connecting point relative to every other correlation

Displays the relationship between necessary tasks and people or other tasks, often shows responsibility for tasks

Diagram PDPC Chart

Process Decision Program Chart

* Analyze where problems (or failures) can occur

* Design to prepare for contingency plans

* Helps develop specific actions to be taken to prevent problems from occuring (or if they occur, to minimize their impact)

*Based on PDCA (Plan Do Check Act) model

Diagram Tree (CTQ diagram)

Cost within budgetReturnable if defectiveLong lastingwear repeatably

Shock absorbing solesSoft liningLightweightRoom in toe area

ColorModernLow heelLeather

Attractive style

A shoe that's worth the price

Comfortable fit

Affordability

Diagram Venn

Distribution Binomial

The discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Example-1: There is a test where students will guess at the answers with 1- Each question has 4 multiple choice questions (only 2 states, right or wrong, trials are independent and mutually exclusive (include all possible choices), probability on each trial is the same) 2 - Each test has 25 questions ( Fixed number of trials). Example-2 Output of a

process has x defectives (characteristics).

Distribution Bivariate

Joint distribution of two variables (3D Graph)

Distribution Chi-Square

A continuous probability distribution estimating the variation (standard deviation) of 2 populations. Tests for “goodness of fit” Example: Sample of 10 drawn from normal universe whose variance = 25. The variance of the sample is 35.Question: is there a statistical significance between the two numbers (25 & 35) at a 95% confidence level? (H0: variance of observed = variance of expected). Test for independence & goodness of fit is always right tailed test.

Distribution Exponential

TIME between successive events. In probability theory and statistics, the exponential distributions are a class of continuous probability distributions. An exponential distribution arises naturally when modeling the time between independent events that happen at a constant average rate. i.e. MTBF

Distribution F A continuous probability distribution used to calculate the ratios of variances (sample variances are assumed to be equal) of normally (assumed) distributed populations (is sample-1 from a population with the same standard deviation, within a specified confidence level the same as sample-2). Example: Company A claims it is 99% confident that its machine has less variation than company C’s machine. (Sample size need NOT be the same and we must know standard deviation (or variance) of both machines.Note: The F distribution is the ratio of two chi-square distributions, also it is used in ANOVAAlso known as Snedecor’s distribution.

Distribution Hypergeometric

A discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. Defects (characteristics) are known for the population. Sample size > 10% of N, 2 states, red/black, fixed number of trials. Example: drawing 5 cards from a deck, what is the probability for “x” (1, 2, 3, 4, or 5) red cards being drawn?

Distribution Normal, Z-table

A continuous normally distributed probability distribution with a mean of zero and a standard deviation of one. Bell shaped curve is due to central limit theorem. Large sample (>30 if <30 then use T distribution). Example: Manufacturer tested a large sample of one of their parts and found the length averaged 10.000 inches with a standard deviation of 0.050 inch. Q1 – How many parts are expected to be larger than 10.122 inches?(also called the Gaussian distribution)

Shape is similar to Binomial/Poisson

distribution

Distribution Poisson

A discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate (population average and # of occurrences are known) and independently of the time since the last event. Example: The average number of people on hold during working hours for service calls is 4 during working hours on any give day. Q1 - What is the probability that there is nobody on hold? Q2 - What is the probability that there are 4 or fewer people on hold? Note: Used for number of occurrences (DEFECTS) of some event (for time between events see exponential distribution). Is used with RATES, FREQUENCIES, and FLOW. Mean = variance

Distribution Students t

A continuous probability distribution estimating the mean of a normally distributed population when the sample size is small (<30 if >30 then use Z or Normal distribution). Used to develop a confidence interval for μ. Compare "means" of two data sets (we assume std-dev of the 2 populations is the same) Example: Calls are to be answered within 15 seconds at a call center. The claim is that with 99% confidence the average response time is less than 17 seconds. Random sample of 25 calls reveals that x-bar= 13.5 sec. and s = 0.95sec. Is the claim correct?

Distribution T Test – paired & student

The basic difference is that the paired t requires the 2 sample sets to be dependent where the student's t sample sets must be independent of each other. The sample sizes in the paired t test are equal while in the student's t test they don't have to be equal. The paired t tests the difference between 2 population means where each data point in the first sample is paired with a data point in the second sample (the data sets are dependent on each other such as before and after process change). The student's t test the difference between 2 means from 2 independent populations, in other words, they are unrelated.

Distribution Weibull

Most common in reliability & statistical applications, used to model time to fail, time to repair, material strength (failure). Scale parameter (determines width of distribution) . If = 1 weibull = exponential, if = 2 weibull = Rayleigh, if = 3-4 weibull approximates Normal.

DMADOV Define Measure Analyze Design Optimize VerifyDMADV Define the project and the goals of the design activity. What is being designed? Why? Use QFD or Analytic

Hierarchical Process to assure that the goals are consistent with customer demands and enterprise strategy. Measure the current state and determine Critical to Stakeholder metrics. Translate customer requirements into project goals.Analyze the root causes and the options available for meeting the goals. Determine the performance of similar best-in-class designs.Design the new product, service or process. Use predictive models, simulation, prototypes, pilot runs, etc. to validate the design concept's effectiveness in meeting goals.Verify the design's effectiveness in the real-world.

(Associated with DFSS Design For Six Sigma)

DMAIC tollgates

Define Measure Analyze Improve Control

DMAIC vs. DMADV

DOE terms Blocking: The arrangement of experimental units into groups (blocks) that are similar to one another.Confounding: Aliasing (factor-A, factor-B, factor-C, AB+C = C is aliased with the interaction AB) (confused)

Covariance: a measure of how much two random variables vary together (are correlated) (Covariate).Collinear: variables that are linear combinations of one another (high correlation among the variables).Orthogonal: Balanced, no confounding, equal no. of data points under each level of each factor (full factorial) (Latin Square: equal no. of rows and columns)Randomization: Conducting experiment in no given order.Replication: Performing the same treatment combination more thanResolution: If resolution = 3 then 1 & 2 interaction, if 4 then 1 & 3 and 2 & 2…. (FIVE FINGER RULE)Input Variables: Independent and predictor variables (NOTE: may be attribute data)Output Variables: Dependent variablesDF: degrees of freedom (n-1) is required for each factor where n is the number of levels for the factor (i.e. 2 factors F1=3 levels & F2 = 4 levels so DF= (n1-1)(n2-2) = (2)(3) = 6

There should be no more than 2-4 factors in a experiment. If more try to screen out some of the list.Items that a statistician may be lacking: 1-unwarranted assumptions of ht process 2-undesirable combinations of the factors 3-violation of known laws of physics 4-too large or small design size 5-inappropriate confounding 6-imprcise measurement 7-unaccaptable prediction error 8-undesirable run order

DPMO / DPMU (Determining baseline sigma) see also TDPU

DPMO - Defects per Million Opportunities - Credit received for meeting SOME of the requirementsDPMU - Defects Per Million Units - Credit given only when ALL requirements are met.

DETERMINE BASELINE SIGMAExample: 20 defects found in 200 units with 3 possible defects per unit

DPMO = Defects/Units x Opportunities = 0.03330.0333 x 106 = 33,333 is between 3.3 & 3.4 sigma*

DPMU = Defects/Units = 20/200 = 0.1 0.1 x 106 = 100,000 is between 2.7 & 2.8 sigma*

*NOTE: Sigma values as shown in Six Sigma Failure Rate tableDPU / DPO Defects Per Unit Defects Per Opportunity

Example: 15 defects found in 250 units with 5 possible defects per unit.Defects/Units = 15/250 = 0.06 DPU -We would expect to find 0.06 defects per unit.DPO = Defects/Units x Opportunities = 15/250 x 5 = 0.012 0.012 in Z table = 2.26, 2.26 + 1.5 shift = 3.76 sigma level

(NOTE: 0.012 x 106 = 12000 DPMO)ERROR Types Type I Error – - (alpha) - Null hypothesis (are =) is rejected when it should not have been (Setting

alpha to .05 vs .01 means more risk of type I error) if is rejected and should have been then 1- Type II Error – - (beta) - Null hypothesis (are =) is not rejected but should have been (Setting alpha

to .05 vs .01 means less risk of type II error) if is not rejected and should not have been then 1-EVOP In EVOP Evolutionary operations experimental designs and improvements are introduced, while an ongoing

full-scale manufacturing process continues to produce satisfactory results. The idea is that process improvement should not interrupt production. Testing & data collection is typically performed by operators.

Failure Project 5 top reasons for project failure:1. Poor team dynamics2. Scope creep3. Poor or no causation (the relation of cause to effect, failure to answer why a problem exists)4. Poor or no champion involvement5. Failure to obtain stakeholder buy-in

Five Competitive Forces

From Michael Porter of Harvard Business School1. The threat of new entrants2. The power of customers3. The power of suppliers4. Substitute products or services5. Industry rivalry

Five S’s Sort, Store, Shine, Standardize, Sustain (safety is considered 6th S)FMEA Failure Mode Effects Analysis

F-failure-DEFECTSM-mode-WHAT GOES WRONGE-effects-CONSEQUENCESA-analysis-RISK ASSESSMENT

FMECA Failure Mode, Effects, and Criticality Analysis Used to chart the probability of failure modes against the severity of their consequences (RPN no).

Guru’s of Quality

Deming PDSA 14 step approach to improvement 7 deadly diseases Chain reaction “improve quality, decrease costs, improve productivity, capture the market with better

quality and price, stay in business, and provide jobs”Ishikawa

Quality circle CWQC (Company Wide Quality Control)

Fishbone diagramJuran

Trilogy Proponent of improvement & breakthrough projects

Shewart (Western Electric) Control charts PDCA

Taguchi Robust DOE (use experimentation & testing to establish functional limits) Quality engineering pioneer

Crosby 14 step approach to improvement 4 absolutes (conformance to requirements, prevention, zero defect standards, quality measurement is $)

Histogram & Frequency Distribution

Determining Cell Intervals

No. of data

pointsNo. of cells

Under 50 5 - 750 - 99 6 - 10

100 - 250 7 - 12> 250 12 - 20

Example: 70 data points with 7highest value = 83 83lowest value = 49 -49

= 34(for inclusion) +1

= 35

by 7 so use 7 cells(49-53, 54-58…)

35 is divisable

Histogram will NOT show patterns in data such as Trends, Shifts, 8 Consecutive Points above or below mean, etc.)

House Of Quality (HOQ)

QFD (Quality Function Development)Roof Interaction & Competitor Correlation Key

P = positive & PP = strong positive N = negative & NN = strong negative

IDOV Identify Design Optimize Verify

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Correlation matrixHOW'S

Relationship Matrix

Ranking of HOWS

Interval Estimation (confidence)

Used with inferential statistics to develop a confidence interval The interval within which it is believed (with a given degree of confidence) the population parameter lies

Confidence + Significance = 100 % Example: Confidence = 95%, Significance (alpha ) = 5%Kaizen Continuous improvement philosophy.

By improving the standardized activities and processes, Kaizen aims to eliminate waste Lean 5 steps 1. Specify Value

2. Map the value stream3. Create flow4. Pull5. Pursue perfection

Little’s Law Lead Time = WIP (units) / ACR (units per time period)Management Types

Functional Management (vertically within function): Budget adherence Command and communication throughout the organization

Process Management (horizontally between departments): best for customers

Increases effectiveness and efficiency Increases customer satisfaction Enables achievement of business objectives

Math sequence of execution

Please Excuse My Dear Aunt SallyParenthesesExponentsMultiplyDivideAddSubtract

Measurement Systems Evaluation

Use X-bar & R chart to show measurement system variation. Example below 5 parts measured 3 times (same location on part) and measured by 3 people.The range of measurements within the same piece establishes the control limits and since there should be no variation on the piece being measured the control limits are generated by the range of the measurement variation.

The control limits represent the range (variation) of the measurement system.

Measurement systems variation

BIAS - Difference between the output of the measurement system and the true value, i.e., 1.00000” gage block is measured at 1.0001” to 1.0005” so bias is .00004 (absolute value of: average measured – average of known).

LINEARITY - is the difference in the bias values through the expected operating range of the Gauge.

REPEATABILITY - is the variation in measurements observed over several measurements made under the same conditions.

REPRODUCIBILITY – Average variation in measurement due to factors other than the measurement system or instrument (such as operators, different gages, environmental changes, etc.)

Stability is the consistency of performance over time. Accuracy and sensitivity must be assured before R&R can be performed

MTBF MTBF Mean Time Between Failure (repairable)3 units evaluated over 1000 daysUnit No. Failures

1 2 Failures = 10 = 3.332 4 units 33 4

Total 10 = =

Failure rate = 1/MTBF = 1/300.3 = .00333

300.3 (unit will last an average of 300.3 days before

repair is needed)

Total time periodMean No. of failures

1000 days3.33

(the average unit will need repair 3.33 times in 1000 days

(this means there is a .33% chance that a single unit will fail on a single day)

Exponential DistributionMTTF MTTF Mean Time To Failure (non-repairable)

3 units evaluated over 1000 daysFuse Life (hours)

1 1050 Life = 3400 = 1133.3 hours2 1200 units 33 1150

Total 3400 Hazard rate = 1/MTTF = 1/1133.3 = .00088

Time x hazard rate = probability of failure100 x .00088 = .088 or 8.8% chance of failure after 100 hours

(the fuses have an average life of 1133.3 hours

(Example: it is specified that fuses should have a reliability of 99%

after 100 hours)

Exponential DistributionMUDA The Japanese term for waste.Multi-Vari Analysis

Dramatize variation within a piece (or process, i.e., oven temperature at various locations in oven) i.e, assists in the breakdown of components of variation

Dramatize variation from piece to piece Track time related changes Helps identify areas to look for excessive

variation & also identifies areas not to look for excessive variation.

Variation = piece (positional), piece-to-piece (cyclical), time (temporal)

1 2 3 4Example: part measured across face at edge

4 2,34

33

22

1 1,41

FIG-1 FIG-2 FIG-3Part is tapered Center is thicker Part is getting larger

Null Hypothesis

Always has some form of equal sign, =, ≤, ≥ If test is for being equal to i.e. H1 = H2 then 2 tailed test If test is for being less (or greater) than or equal to i.e. H1 ≤ H2 then 1 tailed test Null Hypothesis assumes any variation in results is due to chance (common cause) alone. Alternative Hypothesis is what we expect to find (assignable cause from one of 6 M’s). Null always says that there IS NOT or MAY NOT be a difference. Null Hypothesis may be REJECTED but is NEVER ACCEPTED, we only FAIL to REJECT it.

Example above: 95% area is “fail to reject null” area so 5% of the time we are willing take a chance (type I error) that we reject the null when the variation was due to chance alone.

PDCA / PDSA Plan, Do, Check, Act cycle. Shewhart Plan, Do, Study, Act cycle. DemingPLAN: plan ahead for change. Analyze and predict the results.DO: execute the plan, taking small steps in controlled circumstances (i.e. pilot program).CHECK / STUDY: check or study, the results.ACT: take action to standardize or improve the process (i.e. act on resulst of pilot program).

Performance Ratio CR

PEST The acronym for "Political, Economic, Social, and Technological environmental issues that an organization faces (internal)

Power of Statistical test

The power of a statistical test is the probability that the test will reject a false null hypothesis (that it will not make a Type II error). As power increases, the chances of a Type II error decrease. The probability of a Type II error is referred to as the false negative rate (β). Therefore power is equal to 1 − β.

Probability Theory

Probability definition:The probability of an event is a number between 0 (cannot occur) and 1 (will occur)0 < P(A) < 1

INDEPENDENT: Events do not depend on each other (sampling with replacement)Example: Urn contains 5 black & 3 white balls. If 2 balls are drawn (WITH replacement) what is the probability both are black?

Let A be the event that 1st ball is black and B be event that 2nd ball is black. Then P(A) = 5/8 and P(B) = 5/8 P(A and B) = P(A) x P(B) = 5/8 x 5/8 = 25/64

MUTUALLY EXCLUSIVE: events cannot occur together (sampling without replacement such as drawing a ace OR a king from a deck of cards)EXAMPLE:

A is event of drawing a ace and B is event of drawing a king P(A) = 4/52 = 1/13 & P(B) = 4/52 = 1/13 P(A or B) = P(A) + P(B) = 1/13 + 1/13 = 2/13 probability of drawing either a ace or king in a single draw

CONDITIONAL PROBABILITY: the probability of B occurring given event A has occurred. Example: Train arriving on time = A (0.84), leaving on time = B (0.86), arriving AND leaving on time = B+A (0.8) so p of B if A has occurred

0.8/0.84 = 0.9524

Process Mapping

1. Name the process (name is critical i.e. “getting a car” is to vague, use buying a new car, buying a used car, leasing a car)

2. Identify start and stop points (boundaries) (scope and scope creep)3. Identify the output of the process (unqualified noun)4. Identify the customer(s) of the process5. Identify the supplier(s) of the process6. Identify the inputs of the process7. Indicate the 4-7 highest level steps in the process as the exist today.

See SIPOC

Project or Team Charter

A collection of documents that provides purpose and motivation for an improvement team to do its work.SERVES THREE FUNCTIONS:

Establish focus for the team Motivate the teams behavior Motivate emotion

Six elements of a good charter:1. Vibrant business case (what the project does and how it impacts the business objectives, answers

question why is the project worth doing and what are consequences of not doing project)2. Problem statement (be specific and measurable, indicate how long problem has existed, describe

impact of problem, describe gap between current and desired state, and use neutral terms)3. Goals and Objectives (may not have all data in the beginning)4. Project Scope (defined boundaries what is included and what is not included)5. Realistic milestones & deliverables6. Clearly defined roles and responsibilities

Forming, storming, norming and performing (growing a team)Project Management

Planning, Scheduling, ControllingProject is a series of activities and tasks with a specified objective, starting and ending dates, and necessary resources. Resources consumed by the project include time, money, people, and equipment.

PUGH Analysis See “Diagram Matrix” - based on the use of matrix based process to refine conceptual designQFD Quality Function Development (see House of Quality)Quality Costs Prevention Costs: activities designed to prevent poor quality (repair, capability studies, design review,

field testing, training, fixturing, surveys, evaluating vendors) Appraisal Costs: costs associated with measuring, evaluating, auditing (calibration, inspection,

laboratory testing, test equipment maintenance) Failure Costs: costs resulting from poor quality

1. Internal: Prior to delivery or shipment of product or service2. External: After delivery or shipment of product or service.

Hidden Failure Costs: - Engineering Time – Management Time – Downtime – Increased Inventory – Decreased Capacity – Lost Opportunity

QxA=E Q = quality of solutions (scale of 1-10) A = Acceptance of solutions E = Excellence of the results

E must = 60, in order for this to happen A cannot be ignored in favor of Q (your solutions must be accepted)

Regression Analysis

See Chart scatter

ROI & ROAReturn On Investment ROI Return On Assets

Roles & Titles Six Sigma

Sponsor Senior executive who sponsors the overall Six Sigma Initiative.

LeaderSenior-level executive who is responsible for implementing Six Sigma within the business.

Master Black Belt

Highly experienced and successful Black Belt who has managed several projects and is an expert in Six Sigma methods/tools. Responsible for coaching/mentoring/training Black Belts and for helping the Six Sigma leader and Champions keep the initiative on track.

Black Belt

Full-time professional who acts as a team leader on Six Sigma projects. Typically has four to five weeks of classroom training in methods, statistical tools, and (sometimes) team skills.

Green Belt

Part-time professional who participates on a Black Belt project team or leads smaller projects. Typically has two weeks of classroom training in methods and basic statistical tools.

ChampionMiddle- or senior-level executive who sponsors a specific Six Sigma project, ensuring that resources are available and cross-functional issues are resolved.

Process OwnerProfessional responsible for the business process that is the target of a Six Sigma project. (also Champions & Sponsors)

Team MemberProfessional who has general awareness of Six Sigma (through no formal training) and who brings relevant experience or expertise to a particular project.

RPN R isk P riority N umber Example – flat tire on car:-Function: hold air -Occurrence: 2 (P) = probability of occurrence (1=small chance)* -Failure mode: does not hold air -Prevention: maintenance-Effects of failure: stranded -Detection control: warning light-Severity: 8 (S) (10=severe)* -Detection: 1 (D) = effectiveness of detection control (1=effective)*-Cause: bent rim RPN = PxSxD (*scale of 1-10) this example = 2x8x1 = RPN = 16

Scales of Measurement

Nominal scale Identifies category to which a unit belongs (i.e. Sex: 0-male, 1=female) Assigned numbers have no value

Ordinal scale Units are ranked (i.e. a scale of 1 to 10) Only the order of numbers is meaningful

Interval and ratio scales Actual Measurement where numbers have a value (i.e. height in inches) Scales dependent on method of measurement not property being measured (i.e. temperature measured

relative to absolute zero)Scree Plot In this example:

C, G, A, & F are significant D, E, & B are noise.

Seven Management Tools & basic tools for TQM

The seven basic Management Tools: The seven basic tools for TQM are:

1 Interrelationship digram 1Cause and effect diagrams2 Affinity diagram 2 Flow charts/process flow diagrams3 Tree diagram 3 Pareto charts4 Prioritization matrices 4 Run charts5 Matrix diagram 5 Histograms6 PDPC process Decision Program Chart 6 Scatter diagrams7 Activity network diagram 7 Control charts

SI The International System of Units. Meter, Kilogram, Second, Ampere, Kelvin, Mole, CandelaSigma Value (ppm defects)

Sigma σ

Percent Acceptable

ppm Defects With 1.5 σ

process ShiftSigma

σPercent

Acceptable

ppm Defects Without 1.5

σ Shift1 30.2% 697,672 1 68.3% 317,3112 69.1% 308,770 2 95.5% 45,5003 93.3% 66,811 3 99.7% 2,7004 99.4% 6,210 4 100.0% 63.375 99.98% 233 5 100.00% 0.5746 99.9997% 3.4 6 100.0000% 0.002

SIPOC SIPOC stands for Suppliers, Inputs, Process, Output, and Customers. You obtain inputs from suppliers, add value through your process, and provide an output that meets or exceeds your customer's requirements. Process Mapping Steps:

1. Name the PROCESS2. Identify the START and STOP points3. Identify OUTPUT of process4. Identify the CUSTOMER(s) of the process5. Identify the SUPPLIER(s) of the process6. Identify the INPUTS of the process

Indicate the 5 to 7 highest level steps in the process AS THEY EXIST TODAYSix M’s Machines, Materials, Methods, Measurement, Mother Nature (environment), Man (people) – Fish bone diagramSix Sigma Improve effectiveness and efficiencySMART Specific, Measurable, Action oriented, Realistic, Time bound

Sum of Squares Scree Plot

0

20

40

60

80

100

120

C G A F D E B

Factors

SS

Note: SS= ( value)²n (factors)

E 6.3 5B 4.0 2

F 23.0 66D 13.0 21

G 27.3 93A 25.0 78

Factor Delta SSC 29.1 106

Stage Gate Process

Standard Deviation σ & S

= square root of variance

STATISTICS Types of

QUANTITATIVE: asks “how many” classified into: DISCRETE data - 3 dents per unit for example (values cannot be subdivided meaningfully) CONTINUOUS data – 2.007 inches in diameter for example (time, distance, cost, etc) Based on sampling and probability methods

QUALITATIVE; Deals with attributes and making inferences and conclusions. asks “why” i.e. cannot be measured only described. Not based on probability (on a scale of 1 to 10 how do you…)

SWOT Strengths Weakness Opportunities Threats (Michael Porter)Symbols μ = Population MEAN

σ = Population STANDARD DEVIATIONσ² = Population VARIANCEN = Total number of individual observations in the populationn = Individual observations & sample sizeXi = Individual observations in the population or sample

= Sample MEANr = Correlation coefficient (scatter diagram / linear regression)S = Sample STANDARD DEVIATIONS² = Sample VARIANCE = Alpha (type I error - rejecting a null hypothesis when it is actually true) = Beta (type II error - failing to reject a null hypothesis when the alternative hypothesis is true)

=Chi SquareTakt time Takt time can be defined as the maximum time allowed to produce a product in order to meet demand.

Example: customer buys 16 units a day so your takt time is 1/2 hour (1 shift), time needed to satisfy customer demand. The customer should PULL products through the process, not you producing products in hopes that they will sell (pushing).

TDPU Total Defects Per Unit – used for multi-step process TDPU = Inverse of RTY (Rolled Throughput Yield)Example: Yield rates are as follows Y1=99.8% Y2=97.4% Y3=96.4% so RTY=(.998)(.974)(.964)=0.93706 - TDPU = Inverse (RTY) = In(.93706) = 0.065

Team Roles Most important: Member, Leader, Facilitator, Recorder,& Timekeeper, Process owner,TOC Theory Of Constraints

Increase THROUGHPUT Reduce INVENTORY Reduce OPERATING EXPENSES Balance FLOW through the plant (process) Note: NOT balance capacity

The key steps in implementing an effective process of ongoing improvement according to TOC are:0. (Step Zero) Make the GOAL explicit. Frequently, this is something like, "Make money now and in the future." 1. Identify the CONSTRAINT (the thing that prevents the organization from obtaining more of the goal) 2. Exploit the CONSTRAINTt (make sure the constraint is doing things that the constraint uniquely does, and not doing things that it should not do) 3. Subordinate all other processes to above decision or CONSTRAINT (Let the resources who have excess capacity spend some of that capacity to help the Constraint) 4. Elevate the constraint (if required, permanently increase capacity of the constraint; "buy more") 5. If, as a result of these steps, the constraint has moved, return to Step 1. Don't let inertia become the constraint. Drum - The constraint(s), linked to market demand, is the drumbeat for the entire plant. Buffer - Time/inventory that ensures that the constraint(s) is protected from disturbances occurring in the system. Rope - Material release is "tied" to the rate of the constraint(s).

TPM Total Productive Maintenance – complete elimination of failures, defect, waist & loss due to equipment related operations.

Variance σ² & S²

=standard deviation² (squared)

x x

=Loading Time

Loading Time - DowntimeLoading Time

Rate ofQuality Products

Availability = Operation Time

Overall Equipment = Availability PerformanceEfficencyEffectiveness

Variation Two types Common Cause – no undue influence Special Cause – one or more of the 5M’s & 1P have an undue influence

Waste, in manufacturing Seven (7) (or 8)

1. Overproduction: One of the important concepts of lean manufacturing is the pull scheduling. This means the production is pulled by the process above them.

2. Waiting: Waiting for processing.3. Higher Transportation: Transportation adds up to the lead-time, uses space and money, create delays.4. Inappropriate Tooling (or process): The best tools (or processes) for the particular purpose are

required, money is not saved buying cheap tooling or using a inefficient process.5. Higher WIP: Work in progress is a direct result of over production and waiting. It hides the quality

defects and other problems in the system.6. Movement & Ergonomic Imperfections: High amount of motion in work is inefficient and affects the

health of the worker. This is a long term cost for the company.7. Defects: No need to explain it more.8. Underutilization Of Human Talents: By using human resources effectively every other waste (#1-7)

can be eliminated from the system. ( This is added to the 7 wastes)