【正文】 PROVEMENT ROADMAP Uses of Probability Distributions Breakthrough Strategy Characterization Phase 1: Measurement Phase 2: Analysis Optimization Phase 3: Improvement Phase 4: Control ?Establish baseline data characteristics. Project Uses ?Identify and isolate sources of variation. ?Use the concept of shift drift to establish project expectations. ?Demonstrate before and after results are not random chance. 169。The National Graduate School of Quality Management v7 ? 40 COIN TOSS EXAMPLE ? Take a coin from your pocket and toss it 200 times. ? Keep track of the number of times the coin falls as “heads”. ? When plete, the instructor will ask you for your “head” count. 169。The National Graduate School of Quality Management v7 ? 44 COIN TOSS PROBABILITY DISTRIBUTION 6 5 4 3 2 1 0 1 2 3 4 5 6 NUMBER OF HEADS PROCESS CENTERED ON EXPECTED VALUE s SIGMA (s ) IS A MEASURE OF “SCATTER” FROM THE EXPECTED VALUE THAT CAN BE USED TO CALCULATE A PROBABILITY OF OCCURRENCE SIGMA VALUE (Z) CUM % OF POPULATION 58 65 72 79 86 93 100 107 114 121 128 135 142 .003 .135 1 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0 7 0 6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 0 F r e q u e n c y If we know where we are in the population we can equate that to a probability value. This is the purpose of the sigma value (normal data). % of population = probability of occurrence 169。The National Graduate School of Quality Management v7 ? 48 Descriptive Statistics Descriptive Statistics is the branch of statistics which most people are familiar. It characterizes and summarizes the most prominent features of a given set of data (means, medians, standard deviations, percentiles, graphs, tables and charts. Descriptive Statistics describe the elements of a population as a whole or to describe data that represent just a sample of elements from the entire population Inferential Statistics 169。The National Graduate School of Quality Management v7 ? 52 SO WHAT MAKES A DISTRIBUTION UNIQUE? CENTRAL TENDENCY Where a population is located. DISPERSION How wide a population is spread. DISTRIBUTION FUNCTION The mathematical formula that best describes the data (we will cover this in detail in the next module). 169。The National Graduate School of Quality Management v7 ? 55 WHAT IS THE MEDIAN? ORDERED DATA SET 5 3 1 1 0 0 0 0 0 1 3 6 5 4 3 2 1 0 1 2 3 4 5 6 4 If we rank order (descending or ascending) the data set for this distribution we could represent central tendency by the order of the data points. If we find the value half way (50%) through the data points, we have another way of representing central tendency. This is called the median value. Median Value Median 50% of data points 169。The National Graduate School of Quality Management v7 ? 59 SO WHAT’S THE BOTTOM LINE? MEAN Use on all occasions unless a circumstance prohibits its use. MEDIAN AND MODE Only use if you cannot use mean. 169。The National Graduate School of Quality Management v7 ? 62 WHAT IS THE VARIANCE/STANDARD DEVIATION? The variance (s2) is a very robust metric which requires a fair amount of work to determine. The standard deviation(s) is the square root of the variance and is the most monly used measure of dispersion for larger sample sizes. ( ) s X X n i 2 2 1 61 67 12 1 5 6 = = = ? . . DATA SET 5 3 1 1 0 0 0 0 0 1 3 6 5 4 3 2 1 0 1 2 3 4 5 6 4 X X n i = = = ? 2 12 X X i 5()= 3()= 1()= 1()= 0()=.17 0()=.17 0()=.17 0()=.17 0()=.17 1()= 3()= 4()= ( )X Xi 2()2= ()2= ()2=.69 ()2=.69 (.17)2=.03 (.17)2=.03 (.17)2=.03 (.17)2=.03 (.17)2=.03 ()2= ()2= ()2= 169。 the range, the standard deviation and the variance. 169。The National Graduate School of Quality Management v7 ? 57 MEASURES OF CENTRAL TENDENCY, SUMMARY MEAN ( ) (Otherwise known as the average) X X n i = = = ? 2 12 17 . X ORDERED DATA SET 5 3 1 1 0 0 0 0 0 1 3 6 5 4 3 2 1 0 1 2 3 4 5 6 4 ORDERED DATA SET 5 3 1 1 0 0 0 0 0 1 3 6 5 4 3 2 1 0 1 2 3 4 5 6 4 ORDERED DATA SET 5 3 1 1 0 0 0 0 0 1 3 6 5 4 3 2 1 0 1 2 3 4 5 6 4 MEDIAN (50 percentile data point) Here the median value falls between two zero values and therefore is zero. If the values were say 2 and 3 instead, the median would be . MODE (Most mon value in the data set) The mode in this case is 0 with 5 occurrences within this data. Median n=12 n/2=6 n/2=6 } Mode = 0 Mode = 0 169。 the mean, the median and the mode. 169。The National Graduate School of Quality Management v7 ? 50 WHAT DOES IT MEAN? 6 5 4 3 2 1 0 1 2 3 4 5 6 NUMBER OF HEADS s SIGMA VALUE (Z) CUM % OF POPULATION 58 65 72 79 86 93 100 107 114 121 128 135 142 .003 .135 1 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0 7 0 6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 0 F r e q u e n c y And the first 50 trials showed “Head Counts” greater than 130? WHAT IF WE MADE A CHANGE TO THE PROCESS? Chances are very good that the process distribution has changed. In fact, there is a probability greater than % that it has changed. 169。The National Graduate School of Quality Management v7 ? 46 Probability and Statistics ? “ the odds of Colorado University winning the national title are 3 to 1” ? “Drew Bledsoe’s pass pletion percentage for the last 6 games is .58% versus .78% for the first 5 games” ? “The Senator will win the election with 54% of the popular vote with a margin of +/ 3%” ? Probability and Statistics influence o