Unformatted text preview: Engr 9397
Week 2 Intro Sta3s3cs and Process Mapping If things were done right 99.9 % of the 3me, we’d end up with • 1 hr of unsafe drinking water per month • 2,000 incorrect drug prescrip3ons per year • 32,000 missed heartbeats per person per year Why take measurements? “You manage what you measure” “What gets measured gets done” • Measures are indicators of performance and play a cri3cal role in controlling quality • Eﬀec3ve measurements help to: – jus3fy change to a process – quan3fy performance, results and improvements – Determine priori3es and op3mize resources • Measures need to encompass values of both the customer and the organiza3on Sta3s3cs – some quotes “Sta3s3cs are like bikinis: what they reveal is sugges3ve but what they conceal is vital” “Numbers are like people, torture them enough and they will tell you anything” “There are three kinds of lies: lies, damned lies and sta3s3cs” “Sta3s3cs will prove anything, event the truth” “Sta3s3cs is the art of lying by means of ﬁgures” “Sta3s3cs is never having to say you are certain” “One should use sta3s3cs as a drunken man uses lampposts – for support rather than for illumina3on” Why study sta3s3cs? • It helps us to make informed decisions based on data collected in the face of uncertainty and varia3on What are sta3s3cs? • Sta3s3cs describe a set of tools and techniques used to describe, organize, model and interpret data Why are Sta3s3cs useful in Quality Engineering? • Sta3s3cs, deﬁned as the collec3on, tabula3on, analysis, interpreta3on and presenta3on of numerical data (such as quality measurements), enables one to make decisions about a popula3on using sample data (ex. ac3ons and changes to a product, process or service) • Be]er decision be]er design improved performance lower cost be]er use of resources bigger proﬁts be]er world! Use of Sta3s3cs • Sta3s3cs: the collec3on, analysis, tabula3on, interpreta3on and presenta3on of numerical data • Supports decision making by looking beyond the face value of the data • Sta3s3cal analysis results should illuminate and verify per3nent aspects of the issue or problem being studied Example – Arc3c Pipeline Engineering • The decision to build an arc3c subsea pipeline is a very complex (and expensive) one and involves the use of empirical data and probability to determine the risk jus3ﬁes the cost of the project. How many factors can we think of? Some Basic Terminology • Popula7on: collec3on of all possible elements, object of interest • Sample: popula3on subset
representa3ve sampling allows for predic3ons of en3re popula3on and a degree of conﬁdence to be assigned • Random: unpredictable result, but a known probability of occurrence • Bias: not random, sample does not adequately represent the popula3on • Inference: process of drawing a conclusion using rules Types of Variables • Variable / Con3nuous data: – uncountable quality characteris3cs that can be measured the real number scale (ex. length, speed) – countable quality characteris3cs that are measured using whole numbers and are either present or absent (ex. # parts (a]ribute data ), # of failures) – A mix of discrete and con3nuous data, ofen occurs in engineering applica3ons as a result of a zero lower bound value (quan3ty of water, min/max measures) • Discrete / Categorical data: • Mixed Data Random Variable • Characteris3c whose value may change unpredictably • Associates events, or values of experimental outcomes, with probabili3es • Needed to deﬁne theore3cal distribu3ons, or “probability distribu3ons” • Random variables can be con3nuous, discrete or mixed • With a con3nuous random variable (rv), the probability of any speciﬁc value is zero, whereas a discrete rv has a probability associated to each possible value Types of data • Univariate data: – 1 variable (ex. velocity) • Mul3variate Data: – >1 variable (ex. velocity, temp) • Ungrouped data: – observa3ons have no order • Grouped data: – observa3ons organized based on when values occurred Measurement Error • “You manage what you measure, you measure what you manage” • Validity of measurement is a func3on of sample selec3on, understanding of characteris3cs and measurement technique • Error = Diﬀerence between measured and true value • Measurement error can be classiﬁed into two categories, accuracy and precision Error Terminology • Bias = Average value – True value of measurement • Accuracy (variability) • Precision (repeatability) – Measure of data tendency to center around the actual value – ability to consistently repeat measurements of ~same value – Measurements diﬀer li]le from one another • Usually, excess variability is harder to correct than inaccuracy (why do you think this is so?) • What are some common source of error? • The inﬂuence of errors aﬀect how the sample is distributed Error from Varia3on – Link to Quality Engineering • Varia3on Error can be separated into two kinds: assignable causes + residual error • Varia3on from assignable causes results from sources outside the process, (ex. human error, fa3gue, poor instruc3ons, poor condi3ons) • Residual error is what remains afer assignable causes have been iden3ﬁed – associated with measurement limits and background variability • Ques3on: If you were tes3ng for the presence of toxins in the local water supply, what are the assignable causes and residual error? Descrip3ve Sta3s3cs (Exploratory Data Analysis) • A thorough sta3s3cal analysis uses graphical, analy3cal and interpre3ve methods • Descrip3ve Sta3s3cs includes: – Graphical (empirical) methods • Ex: frequency diagrams, histograms, bar graphs etc. – Numerical (analy3cal/mathema3cal) methods • Data loca3on is described by measures of central tendency, data spread is described by measures of dispersion Descrip3ve Sta3s3cs Measures of Loca3on • Sample Size: number of data points (n) • Mean (a.k.a. Average) – Most common measurement to locate center of the data distribu3on
histogram balance point – centroid of data – Same units as sample/popula3on data – Sensi3ve to outliers • Trimmed Mean – Mean calculated afer smallest and largest σ% of data removed Median and Mode • Median – Middle value of data if n is odd, the median is the middle value if n is even, the median is the mean of the two middle values – Preferred measure of loca3on, useful when data is skewed • Mode – Largest frequency
used ofen with categorical data (counts units) Descrip3ve Sta3s3cs Measures of Dispersion (Spread) • Variance – Classic measure of data spread, concentra3on of data about the mean – Measured in square units of the observa3ons – 2nd moment about the mean • Standard devia3on (root
mean
squared devia3on) – Measured in original metric – Most common dispersion measure Measures of Dispersion con3nued • Range (r) – Dependent on two observa3ons only
OK for small data sets r = max(xi) – min(xi) • Quar3les – Data divided into 4 equal parts 1st, 2nd and 3rd quar3les (Q1, Q2 and Q3} – Q1 is the lower 25% mark, Q2 = 50%, Q3 = 75% – Gives a be]er indica3on of data characteris3cs (vs. the mean) when there are outliers • Interquar3le Range (IQR) – range of middle 50% of data IQR = Q3
Q1 • Midrange Coeﬃcient of Varia3on • Coeﬃcient of Varia3on cv – dimensionless measure of spread – expresses σ as a percentage of µ – Useful to understand spread in context of mean and when comparing data sets – Not so useful if mean value is close to zero Descrip3ve Sta3s3cs
Example using Minitab Example: Calculate the mean, median, Q1, Q3 and 5% trimmed mean, variance, std devia3on, IQR and Range of the following data: Data: 2, 208, 3, 5, 90, 151, 45, 46, 47, 48, 50 End of Take Home Slides Shape Characteris3cs of Data • Shape characteris3cs are used to summarize the plo]ed form of the data Skewness: – posi3vely skewed data, caused by high outliers, pulls tail/mean to the right mean > the median – nega3vely skewed data, caused by low outliers, pulls tail/mean to the lef mean < the median – When calcula3ng means and std devia3ons, summary sta3s3cs should include measures of skewness Coeﬃcient of skewness – Coeﬃcient of skewness γ: 3rd moment about the mean ÷ std devia3on cubed – If data are symmetrical, skewness = zero – If abs skewness > 1.0, data is considered highly skewed – If abs skewness is <0.5, data is not considered highly skewed Shape Characteris3cs con3nued • Common engineering data characteris3cs include: – Lower bound of zero – Outliers – Skewness – Dependence on uncontrollable variables – Small sample sizes Graphical methods for describing popula3ons • Graphs are a good way to visualize data and quickly get an idea of its centrality and dispersion • Help to gain insights into data structure
easy to iden3fy skewness, # modes, range, pa]erns and trends • Can help to select analysis method, illustrate concepts and communicate results • Helps conﬁrm assump3ons and/or ID assignable causes • Types of graphs: frequency diagrams, cumula3ve frequency diagrams, histograms, stem leaf plots, box plots, normal probability plots and sca]er diagrams Frequency distribu3ons • Shapes of a frequency distribu3on plays an important role in characterizing data, as it reveals insight about the data structure, enabling us to diagnose quality problem and verify assump3ons needed to perform tests • In order to construct, one must sort, bin and tally occurrences into classes • Minitab can save you a lot of work here! Graphical Analysis
Histogram • Shows frequencies within equal intervals, or bins – represents con3nuous data • Useful for looks at large amounts of data, and how data units are distributed within the possible range of values • Class/intervals are represented by bar with an area propor3onal to the class frequency • Histograms with large amounts of data approximate the shape of probability distribu3ons Graphical Analysis– Pareto Graph • Useful in ini3al stage of quality improvement to priori3ze problems • Plots frequency of failure occurrence for a given root cause • Enables team to source key causes of failure (80/20 Pareto’s Law – source of failure unevenly distributed) • The Pareto chart is one of the seven basic tools of quality control – used for ini3al project inves3ga3on and priori3za3on of future eﬀort • Minitab example: shelf assembly failure Graphical Analysis Stem
and
Leaf Diagram • Similar to histogram, but shows more informa3on about the data (no loss) • Stems are chose in a manner similar to a histogram, and the leaves are the values falling within the stems • Useful for visualizing data using intervals and for ﬁnding percen3les • Example: response 3mes in Minitab Graphical Analysis – Sca]er Plots • Plots X against Y – bivariate/paired data • Useful for understanding the eﬀect of one variable on another • In experimental design, a topic to be covered later in the course, we will use tests to determine how the value of a dependent (uncontrolled) variable responds when a independent (control) variable value is changed Graphical Analysis Box
and
Whisker Plot • Compact representa3on of frequency distribu3on • Useful for comparing the centrality and dispersion of mul3ple distribu3ons, iden3fying outliers – Outliers: believed not to be part of the main data, results from measurement (or other) errors – mul3ple deﬁni3ons for outliers • Indicates main data percen3les (X0, X25, X50, X75, X100) Take a Break! • When we come back we’ll do process mapping and start looking at probability distribu3ons Problem Solving using the DMAIC model • Six sigma methodology uses a logical systema3c method to discover the root causes of a problem and control the problem solu3on • Deﬁne: Deﬁne the problem, requirements and objec3ves • Measure: Measure current process, deﬁne/measure key process steps, deﬁne process inputs and outputs • Analyze: Analyze problem root causes, validate cause and eﬀect rela3onships between process variables, determine what changes are to be made to target key process variables • Improve: Implement/test solu3ons, measure the results • Control: Evaluate and monitor improvements, establish standard opera3ng procedures (SOPs) Six sigma performance Metrics • KPIV/KPOV (Key Process input/output Variables): performance measures that deﬁne project and/or strategy success linked to an organiza3on’s bo]om line – (ex. revenue, margins, costs, expenses, warran3es) • Reduce down3me by 9
23% by reducing materials by 20% and improving packaging • Decrease delivery 3mes and reduce defects by reducing inventory space and improving packaging Measure Techniques • Checksheets– keeps track of problem occurrences (easy way to tally/categorize failures) • Issue logs • Pareto Analysis
ranks importance of problem root causes by frequency – helps to quan3fy problem costs • Process Maps: graphical representa3on of all steps involved in a process Measure Techniques: Process Map • Visual representa3on of how current process is performed (process ﬂow) • Process map crea3on: – Deﬁne process boundaries (beginning and end) – Deﬁne process steps (brainstorm or observe) – Order steps into inputsprocess ac3vityoutputs (schedules indicate resources and tasks associated with process ac3vi3es/steps) Process Maps con3nued • During process crea3on, teams ofen ﬁnd s3cky tabs useful – they act as visual aids for the team and form the building blocks of a process diagram, which is reﬁned and transformed into a formal process ﬂow chart, with appropriate symbols and sequences Process map symbol Meaning Opera3on Decision Transport Storage Process Maps Con3nued • Some3mes, job departments, 3tles, names or other categories are wri]en in across the top of the page (some3mes called swimlanes) • This is useful to help employees understand responsibili3es, group interfaces and general ﬂow of the process through the organiza3on • This varia3on on a tradi3onal process ﬂow chart is called a ‘deployment ﬂowchart’ Process map example Process Map Crea3on • Deﬁne the process steps – brainstorm, observe • Sort the steps into the right order (teams may have choices between several alterna3ves) • Take the ordered steps and add appropriate process chart symbols/format • Evaluate the process for completeness, eﬃciencies, problems and risks • Reﬁne the process and improve process performance by iden3fying and implemen3ng con3nuous improvement projects Example – Process Diagrams • Create a process diagram for the crea3on of a new part ordered by an organiza3on from a supplier, to be validated by an Engineer and secured into a larger assembly • Keep in mind the types of informa3on that would be needed at each process step and clearly iden3fy decision
making ac3vi3es Value Stream Process Mapping • Value streams are the ac3ons required to create a product from raw materials un3l point of sale (acquired by the customer) • Value streams focus on mee3ng requirements and helps employees visualize how material and informa3on ﬂow with respect to the process, iden3fy sources of waste and eliminate non
value added ac3vi3es Value Stream Process Mapping • Steps in crea3ng a value stream map – Accurately specify the desired customer value – Walkthrough the process and collect informa3on about the 3mes associated with process steps (cycle 3mes, lead 3me, value crea3on 3me) – Assess the current state to ﬁnd areas of biggest poten3al improvement – Work toward the change ...
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 Winter '11
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