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Measure Phase Six Sigma Statistics

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 2 Six Sigma Statistics Descriptive Statistics Normal Distribution Assessing Normality Graphing Techniques Basic Statistics Special Cause / Common Cause Wrap Up & Action Items Process Capability Measurement System Analysis Six Sigma Statistics Process Discovery Welcome to Measure

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 3 Purpose of Basic Statistics The purpose of Basic Statistics is to: Provide a numerical summary of the data being analyzed. –Data (n) Factual information organized for analysis. Numerical or other information represented in a form suitable for processing by computer Values from scientific experiments. Provide the basis for making inferences about the future. Provide the foundation for assessing process capability. Provide a common language to be used throughout an organization to describe processes. Relax….it won’t be that bad!

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 4 Statistical Notation – Cheat Sheet An individual value, an observation A particular (1 st ) individual value For each, all, individual values The mean, average of sample data The grand mean, grand average The mean of population data A proportion of sample data A proportion of population data Sample size Population size Summation The Standard Deviation of sample data The Standard Deviation of population data The variance of sample data The variance of population data The range of data The average range of data Multi-purpose notation, i.e. # of subgroups, # of classes The absolute value of some term Greater than, less than Greater than or equal to, less than or equal to

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 5 Parameters vs. Statistics Population Parameters: –Arithmetic descriptions of a population –µ, , P, 2, N Population Sample Sample Statistics: –Arithmetic descriptions of a sample – X-bar, s, p, s 2, n Population: All the items that have the “property of interest” under study. Frame: An identifiable subset of the population. Sample: A significantly smaller subset of the population used to make an inference.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 6 Types of Data Attribute Data (Qualitative) –Is always binary, there are only two possible values (0, 1) Yes, No Go, No go Pass/Fail Variable Data (Quantitative) –Discrete (Count) Data Can be categorized in a classification and is based on counts. –Number of defects –Number of defective units –Number of customer returns –Continuous Data Can be measured on a continuum, it has d ecimal subdivisions that are meaningful –Time, Pressure, Conveyor Speed, Material feed rate –Money –Pressure –Conveyor Speed –Material feed rate

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 7 Discrete Variables Discrete VariablePossible values for the variable The number of defective needles in boxes of 100 diabetic syringes 0,1,2, …, 100 The number of individuals in groups of 30 with a Type A personality 0,1,2, …, 30 The number of surveys returned out of 300 mailed in a customer satisfaction study. 0,1,2, … 300 The number of employees in 100 having finished high school or obtained a GED 0,1,2, … 100 The number of times you need to flip a coin before a head appears for the first time 1,2,3, … (note, there is no upper limit because you might need to flip forever before the first head appears.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 8 Continuous Variables Continuous VariablePossible Values for the Variable The length of prison time served for individuals convicted of first degree murder All the real numbers between a and b, where a is the smallest amount of time served and b is the largest. The household income for households with incomes less than or equal to $30,000 All the real numbers between a and $30,000, where a is the smallest household income in the population The blood glucose reading for those individuals having glucose readings equal to or greater than 200 All real numbers between 200 and b, where b is the largest glucose reading in all such individuals

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 9 Definitions of Scaled Data Understanding the nature of data and how to represent it can affect the types of statistical tests possible. Nominal Scale – data consists of names, labels, or categories. Cannot be arranged in an ordering scheme. No arithmetic operations are performed for nominal data. Ordinal Scale – data is arranged in some order, but differences between data values either cannot be determined or are meaningless. Interval Scale – data can be arranged in some order and for which differences in data values are meaningful. The data can be arranged in an ordering scheme and differences can be interpreted. Ratio Scale – data that can be ranked and for which all arithmetic operations including division can be performed. (division by zero is of course excluded) Ratio level data has an absolute zero and a value of zero indicates a complete absence of the characteristic of interest.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 10 Nominal Scale Qualitative VariablePossible nominal level data values for the variable Blood TypesA, B, AB, O State of ResidenceAlabama, …, Wyoming Country of BirthUnited States, China, other Time to weigh in!

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 11 Ordinal Scale Qualitative VariablePossible Ordinal level data values Automobile SizesSubcompact, compact, intermediate, full size, luxury Product ratingPoor, good, excellent Baseball team classificationClass A, Class AA, Class AAA, Major League

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 12 Interval Scale Interval VariablePossible Scores IQ scores of students in BlackBelt Training 100… (the difference between scores is measurable and has meaning but a difference of 20 points between 100 and 120 does not indicate that one student is 1.2 times more intelligent )

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 13 Ratio Scale Ratio VariablePossible Scores Grams of fat consumed per adult in the United States 0 … (If person A consumes 25 grams of fat and person B consumes 50 grams, we can say that person B consumes twice as much fat as person A. If a person C consumes zero grams of fat per day, we can say there is a complete absence of fat consumed on that day. Note that a ratio is interpretable and an absolute zero exists.)

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 14 Converting Attribute Data to Continuous Data Continuous Data is always more desirable In many cases Attribute Data can be converted to Continuous Which is more useful? –15 scratches or Total scratch length of 9.25” –22 foreign materials or 2.5 fm/square inch –200 defects or 25 defects/hour

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 15 Descriptive Statistics Measures of Location (central tendency) –Mean –Median –Mode Measures of Variation (dispersion) –Range –Interquartile Range –Standard deviation –Variance

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 16 Descriptive Statistics Open the MINITAB™ Project “Measure Data Sets.mpj” and select the worksheet “basicstatistics.mtw”

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 17 Measures of Location Mean is: Commonly referred to as the average. The arithmetic balance point of a distribution of data. Population Sample Descriptive Statistics: Data Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Data Variable Maximum Data Stat>Basic Statistics>Display Descriptive Statistics…>Graphs… >Histogram of data, with normal curve

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 18 Measures of Location Median is: The mid-point, or 50 th percentile, of a distribution of data. Arrange the data from low to high, or high to low. –It is the single middle value in the ordered list if there is an odd number of observations –It is the average of the two middle values in the ordered list if there are an even number of observations Descriptive Statistics: Data Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Data Variable Maximum Data

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 19 Measures of Location Trimmed Mean is a: Compromise between the Mean and Median. The Trimmed Mean is calculated by eliminating a specified percentage of the smallest and largest observations from the data set and then calculating the average of the remaining observations Useful for data with potential extreme values. Stat>Basic Statistics>Display Descriptive Statistics…>Statistics…> Trimmed Mean Descriptive Statistics: Data Variable N N* Mean SE Mean TrMean StDev Minimum Q1 Median Data Variable Q3 Maximum Data

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 20 Measures of Location Mode is: The most frequently occurring value in a distribution of data. Mode = 5

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 21 Measures of Variation Range is the: Difference between the largest observation and the smallest observation in the data set. A small range would indicate a small amount of variability and a large range a large amount of variability. Interquartile Range is the: Difference between the 75 th percentile and the 25 th percentile. Descriptive Statistics: Data Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Data Variable Maximum Data Use Range or Interquartile Range when the data distribution is Skewed.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 22 Measures of Variation Standard Deviation is: Equivalent of the average deviation of values from the Mean for a distribution of data. A “unit of measure” for distances from the Mean. Use when data are symmetrical. Population Sample Descriptive Statistics: Data Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Data Variable Maximum Data Cannot calculate population Standard Deviation because this is sample data.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 23 Measures of Variation Variance is the: Average squared deviation of each individual data point from the Mean. Sample Population

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 24 Normal Distribution The Normal Distribution is the most recognized distribution in statistics. What are the characteristics of a Normal Distribution? –Only random error is present –Process free of assignable cause –Process free of drifts and shifts So what is present when the data is Non-normal?

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 25 The Normal Curve The normal curve is a smooth, symmetrical, bell-shaped curve, generated by the density function. It is the most useful continuous probability model as many naturally occurring measurements such as heights, weights, etc. are approximately Normally Distributed.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 26 Normal Distribution Each combination of Mean and Standard Deviation generates a unique normal curve: “Standard” Normal Distribution –Has a μ = 0, and σ = 1 –Data from any Normal Distribution can be made to fit the standard Normal by converting raw scores to standard scores. –Z-scores measure how many Standard Deviations from the mean a particular data-value lies.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 27 Normal Distribution The area under the curve between any 2 points represents the proportion of the distribution between those points. Convert any raw score to a Z-score using the formula: Refer to a set of Standard Normal Tables to find the proportion between μ and x. x The area between the Mean and any other point depends upon the Standard Deviation.

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© OpenSourceSixSigma, LLCOSSS LSS Green Belt v9.1 - Measure Phase 28 The Empirical Rule The Empirical Rule… % of the data will fall within +/- 1 standard deviation % of the data will fall within +/- 2 standard deviations % of the data will fall within +/- 3 standard deviations % of the data will fall within +/- 4 standard deviations % of the data will fall within +/- 5 standard deviations % of the data will fall within +/- 6 standard deviations

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