1. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-271 Basic Statistics

2. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-272 Cornerstones of a successful use of 6  Change Management Results Methodology World Class Business Performance

3. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-273 Different types of data Continuous Data nOften obtained by use of a measuring system e.g. dB, Watts, volts etc. nThe usefulness of the data depends on the quality of the measurement system Discrete Data n Includes percentages, attribute & counts Percentages: The proportion of items with a given characteristic; need to be able to count both occurrences and and non-occurrences, e.g. Yield Attribute data Gives only conforming or non conforming information, such as Pass/Fail, Red/Green, 1 / 0, etc. Counts Number of events per hour, per shift or other delimitations n Occurrences must be independent

4. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-274 Yield Defects Lower Tolerance Limit Upper Tolerance Limit Yield Yield = Pass / Trials p(d) = (1- Yield)/100

5. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-275 Discrete data - First Time Right (First Time Yield) Discrete data - First Time Right (First Time Yield) Measures the units that avoid the hidden costs. Step A Step B SCRAP Rework Ship It! Rework Fix It? Good? Fix It? Good? COPQ Yes No Yes No

6. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-276 Discrete data - Rolled Thru Yield Most processes are complex interrelationships of many sub-processes. The overall performance is usually of interest to us. First Process Second Process Second Process Third Process Terminator FTY First Process 99% FTY Second Process 89% FTY Third Process 95% First pass yield or rolled through yield for these three processes is 0.99 x 0.89 x 0.95 =.837, almost 84% Rolled yield is a realistic assessment of the cumulative effect of sub-processes Rework

7. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-277 YIELD Yield \No of op. 310100100010000 0,80,5120000,1073740,0000000,0000000,000000 0,950,8573750,2146390,0059210,0000000,000000 0,99990,9997000,9990000,9900490,9048330,367861 0,9999970,9999910,9999700,9997000,9970040,970445 (process yield) no of operations

8. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-278 DPMO - Defects Per Million Opportunities DPMO  Measurable  The number of opportunities for a defect to occur, is related to the complexity involved. Opportunit y DPO - Defects Per Opportunity DPO Is it fair to compare processes and products that have different levels of complexity?

9. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-279 Yeild to DPMO? Y=e (-dpu) dpu=-lnY dpu = defects per unit = DPMO*(opportunities/unit)/1 000 000

10. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2710 6666 5555 4444 3333 100 opp.1000 opp.10000 opp. Product yield vs dpmo The automation wall The Design & supply wall 100000 opp.

11. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2711 The Normal Curve Context The normal distribution provides the basis for many statistical tools and techniques. Definition A probability distribution where the most frequently occurring value is in the middle and other probabilities tail off symmetrically in both directions. This shape is sometimes called a bell-shaped curve. Characteristics Curve theoretically does not reach zero; thus the sum of all finite areas total less than 100% Curve is symmetric on either side of the most frequently occurring value The peak of the curve represents the center, or average, of the process For all practical purposes, the area under the curve represents virtually 100% of the product the process is capable of producing

12. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2712 Variation Output power Measurement no Average, Mean-value (x, m or µ, M) Standard deviation (std, s,  ) Common cause variation Special cause variation

13. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2713 n1:st 12 24 34 44 55 65 73 82 94...... 973 984 993 1001 2:nd 5 1 5 6 3 5 3 2 5... 2 2 1 1 Sum 1:2 7 5 9 10 8 6 3 9... 5 7 4 3 3:rdSum 1:3 18 510 312 413 6 514 28 47 312...... 17 29 26 47 4:thSum 1:4 412 516 114 619 417 2 211 310 214...... 512 413 17 28 5:thSum 1:5 416 218 115 221 220 521 617 414 317...... 5 215 613 5 6:thSum 1:6 521 3 6 222 2 224 320 6 421...... 118 520 518 519

14. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2714 Calculations Arithmetic Mean Average Median Middle value, so that half of the data are above and half of the data are below the median. For a SampleFor the whole Population = Average of entire population or “true” mean = (x bar) Average of sample or “best estimate” for mean N = Number of observations in entire population n = Number of observations in the sample = Value of measurement x at position i

15. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2715 Every Normal Curve can be defined by two numbers: Mean: a measure of the center Standard deviation: a measure of spread  

16. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2716 0 2 4 6 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8   Observationvalue X 1 0,4 X 2 0,3 X 3 0,4 X 4 0,6 X 5 0,5 X 6 0,4 X 7 0,2 X 8 0,3 X 9 0,5 X 10 0,4  x-m) 2   n-1 sample = n-1 population = n The range method: N<10: Range/3 N>10 Range/4

17. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2717 Parameters to describe the spread (variability) Range Difference between highest and lowest value of the distribution Influenced by Outliers Variance (  2 ) Average squared difference of data point from the average Standard Deviation Square root of the variance Commonly used parameter for variability

18. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2718 How to calculate Range Variance (  2 ) Standard Deviation  2 = Variance of entire population, or “true” variance S 2 = Variance of sample, or “best estimate” for  2  = Standard deviation of entire population, or “true” standard deviation. S= Standard deviation of sample, or “best estimate” for  SamplePopulation

19. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2719 Exercise Calculate Range, Variance and Standard deviation. Draw a normal probability plot of the result. R = Formulas Data

20. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2720 Each diagram has an average of 10, range of 18 and a variation of approx. 5,8. Imagine only looking at the result and not on the graphs. Average, Range & Spread

21. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2721 The normal distribution   99.9999998% 99.999943% 99.9937% 99.73% 95.45% 68.27% -6  -5  -4  -3  -2  -1  0 1  2  3  4  5  6 

22. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2722 Z Area under the normal curve is equal to the probability (p, also named dpo) of getting an observation beyond Z (see the Z-table) The Z-table

23. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2723 Normalizing standard deviations The expected probability of having a specific value Observed value - Mean Value = Z-value Standard deviation ( the Z-table gives the probability occurrence) | x - M | std = Z

24. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2724 Z-VALUES AND PROBABILITIES -1  +1  68,3% -2  +2  95,4% -3  +3  99,7% -6  +6  99,999997%

25. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2725 Z – Table Area

26. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2726 Z – Table Area

27. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2727 * 6 C P Tolerance width divided by 6 times the standard deviation. A C P value greater than 2 is good (thumb rule) T Ö - T U C P = ----------- 6   Tolerance width Capability

28. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2728 * 3 Cpk Difference between nearest tolerance limit and average, divided by 3 times the standard deviation. A Cpk value greater than 1,5 is good (thumb rule) Min(T Ö  alt.   T U ) C PK = ---------------------- 3   TÖTÖ TUTU Capability

29. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2729 Continuous data and possible Pitfalls nCan be divided in to two types of variation nCommon cause(e.g. within batch variation) nSpecial cause-The shift between and (e.g. batch variation) -Outliers or non-rare occasions will appear and may ruin the analyze

30. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2730 Long-Term Capability TargetUSLLSL Time 1 Time 2 Time 3 Time 4 Short-Term Capabilities (within group variation) (between group variation) (all variation) „Shift Happens“

31. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2731 Z long term and Z short term The sample and the population sigma are often almost the same, but the average will probably differ. Therefore is z ST (z B ) and shift & drift preferably used to estimate the “true” fault rate. Shift & Drift = Z short term - Z long term What will the long term fault rate be in exercise 5 with a S&D of 1.5  ?

32. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2732 ZBZB Lower Tolerance Limit Upper Tolerance Limit Z B – From table with P tot Rev C Peter Häyhänen 9805 P tot =P upper +P lower

33. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2733 Short-term LSL USL -6  -5  -4  -3  -2  -1  0 1  2  3  4  5  6  Short-term 99.9999998% or 0.002 ppm  1.5  99.99966% or 3.4 ppm Is Six Sigma corresponding to a defect level of 3,4ppm? Yes, with a S&D of 1,5!!

34. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2734 Shift & Drift Z short term in a typical process 4,02 (based on approx. 30 values).

35. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2735 Shift & Drift Z long term in a typical process 3,03 (measurments from one and a half year of production, “all values”)

36. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2736 P overall = 1200ppm  Z = 3,03  P sample = 29ppm  Z = 4,02  Shift & Drift = Z short term - Z long term Shift & Drift = 4,02  - 3,03  Shift & Drift = 0,99  Shift & Drift

37. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2737 Minitab Capability Output

38. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2738 Nomenclature dpmo- defects per million opportunities Yield - % of the number of approved units divided by the total number of units p(d) - probability for defects (1-Yield) Fty - First time yield, the yield when the units are tested for the first time TpY - Throughput yield, the yield in every unique process step Yrt - Yield rolled through, multiplied throughput yield DPU - Defects per units DPO - Defects per opportunity Opp - Opportunity, measurable opportunity for defect

39. DFSS Basic Staistics EAB/JN Stefan Andresen 2004-09-2739 Nomenclature Zst- Single side short term capability, calculated with the help of the target Zb - An estimate of the overall short term capability, used to calculate Zlt Zlt - A rating of the long term capability, normally based on S&D & Zb pl - Probability for defect beneath lower specification limit pu - Probability for defect above upper specification limit p - Summarized probability for defect, pl + pu S&D - An approximation of the drift in average, fundamentally 1,5  LSL - Lower specification limit USL - Upper specification limit