Continuous Data Long-Term

1. Label The Normal Curve With The Following: Example: LSL = 8 Area 2 10 18 x x+s 2. Determine Area 1: Find Z1702009 O ##

n n n n

X value X+s value USL & Shade Area To The Right LSL & Shade Area To The Left s

s

USL = 22 Area 1

Z1 =

USL x s

=

( (

)( )

)

=

Look up Z1 in Normal Table

Norm Dist (Z1) = Normal Table Look Up for Z1

=

Area 1 = 1 Look Up

Area 1 = 1 Norm Dist (Z1) = 1 (

)

=

3. Skip this step if there is no LSL 2. Determine Area 2: Find Z2 Z2 = LSL x s =

( (

)( )

)

=

Look up Z2 in Normal Table

Norm Dist (Z2) = Normal Table Look Up for Z2

=

Area 2 = Look Up

Area 2 =

4. Determine Total Area:

Total Area = Area 1 + Area 2 =

(

)+( )

)== =

5. Yield = 1 Total Area

Yield = 1 Total Area = 1 ( = x 100%

6. Process Sigma Comes From Table Look Up Of Yield

SigmaST = Look Up Value in Sigma Table

=

s

Xbar 9.68 S 7.25 USL 2 LSL Sigma = 0.44 Sigma

Enter Values in Yellow Mean (Average) Standard Deviation Delete if no USL (Upper Specification Limit) Delete if no LSL (Lower Specification Limit) Sigma Quality Level

=

-1.05931

Example: The average processing time = 15 days (Xbar = 15) The standard deviation was 2 days (s = 2) A unit processed longer than 18 days was too late to the customer (USL = 18)

=

0.1447292

=

0.8552708

=

N/A

=

N/A

A unit processed faster than 5 days was too early to the customer (LSL = 5)

a2 =

N/A 0.8552708

Sigma Quality Level = 3

)== =

0.1447292 14.4729%

=

0.44

Discrete Method

General Worksheet For Calculating Process Sigma

1 2 3 4 5

Number Of Units Processed Total Number Of Defects Made (Include Defects Made And Later Fixed) Number Of Defect Opportunities Per Unit Solve For Defects Per Million Opportunities (DPMO) Look Up Process Sigma In Abridged Sigma Conversion Table 200 pairs of boots were supplied (N = 200) 35 shoelaces were found broken (D = 35) Each shoe had 1 lace and there were 2 shoes per pair (O = 2)

Page 5

Discrete Method

l Worksheet For Calculating Process SigmaEnter Values Below in Yellow N= 200 100 D= 35 O= 2 DPMO = 87,500 Sigma = 2.86

rocessed efects Made (Include Defects Made And Later Fixed) Opportunities Per Unit Per Million Opportunities (DPMO) igma In Abridged Sigma Conversion Table

Example: 200 pairs of boots were supplied (N = 200) 35 shoelaces were found broken (D = 35) oe had 1 lace and there were 2 shoes per pair (O = 2)

Page 6

Abridged Process Sigma Conversion TableLong-Term Yeild99.99966% 99.9995% 99.9992% 99.9900% 99.8000% 99.9970% 99.9960% 99.9930% 99.9900% 99.9850% 99.9770% 99.9670% 99.9520% 99.9320% 99.9040% 99.8650% 99.8140% 99.7450% 99.6540% 99.5340% 99.3790% 99.181% 98.930% 98.610% 98.220% 97.730% 97.130% 96.410% 95.540% 94.520% 93.320% 91.920% 90.320% 88.50% 86.50% 84.20% 81.60% 78.80% 75.80% 72.60% 69.20% 65.60% 61.80% 58.00% 54.00% 50%

Process Sigma (ST)6.0 5.9 5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5

Defects Per 1,000,0003.4 5 8 10 20 30 40 70 100 150 230 330 480 680 960 1350 1860 2550 3460 4660 6210 8190 10700 13900 17800 22700 28700 35900 44600 54800 66800 80800 96800 115000 135000 158000 184000 212000 242000 274000 308000 344000 382000 420000 460000 500000

Defects Per 100,0000.34 0.5 0.8 1 2 3 4 7 10 15 23 33 48 68 96 135 186 255 346 466 621 819 1070 1390 1780 2270 2870 3590 4460 5480 6680 8080 9680 11500 13500 15800 18400 21200 24200 27400 30800 34400 38200 42000 46000 50000

Defects Per 10,0000.034 0.05 0.08 0.1 0.2 0.3 0.4 0.7 1 1.5 2.3 3.3 4.8 6.8 9.6 13.5 18.6 25.5 34.6 46.6 62.1 81.9 107 139 178 227 287 359 446 548 668 808 968 1150 1350 1580 1840 2120 2420 2740 3080 3440 3820 4200 4600 5000

Defects Per 1,0000.0034 0.005 0.008 0.01 0.02 0.03 0.04 0.07 0.1 0.15 0.23 0.33 0.48 0.68 0.96 1.35 1.86 2.55 3.46 4.66 6.21 8.19 10.7 13.9 17.8 22.7 28.7 35.9 44.6 54.8 66.8 80.8 96.8 115 135 158 184 212 242 274 308 344 382 420 460 500

46% 43% 39% 35% 31% 28% 25% 22% 19% 16% 14% 12% 10% 8%

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

540000 570000 610000 650000 690000 720000 750000 780000 810000 840000 860000 880000 900000 920000

54000 57000 61000 65000 69000 72000 75000 78000 81000 84000 86000 88000 90000 92000

5400 5700 6100 6500 6900 7200 7500 7800 8100 8400 8600 8800 9000 9200

540 570 610 650 690 720 750 780 810 840 860 880 900 920

TableDefects Per 1000.00034 0.0005 0.0008 0.001 0.002 0.003 0.004 0.007 0.01 0.015 0.023 0.033 0.048 0.068 0.096 0.135 0.186 0.255 0.346 0.466 0.621 0.819 1.07 1.39 1.78 2.27 2.87 3.59 4.46 5.48 6.68 8.08 9.68 11.5 13.5 15.8 18.4 21.2 24.2 27.4 30.8 34.4 38.2 42 46 50

54 57 61 65 69 72 75 78 81 84 86 88 90 92

Assumptions No analysis would be complete without properly noting the assumptions made. In the above Sigma conversion table, we have assumed that the standard sigma shift of 1.5 is appropriate, the data is Understanding The Formula Defects Per Million Opportunities (DPMO) = ((Total Defects) / (Total Opportunities)) * 1,000,000 Defects (%) = ((Total Defects) / (Total Opportunities)) * 100 Yield (%) = 100 - (Defects Percentage) process Sigma = NORMSINV(1-((Total Defects) / (Total Opportunities))) + 1.5 Alternatively, process Sigma = 0.8406 + SQRT(29.37 - 2.221 * (ln(DPMO))). Examples Here are a couple of examples to help illustrate the calculations. - A long-term 93% yield (e.g. 100 opportunities, 7 defects) equates to a process Sigma long-term value of 1.48 (with no Sigma shift) or a process Sigma short-term value of 2.98 (with a 1.5 Sigma shift). - A long-term about yield (e.g. 1,000 process, we are referring to the process short-term (now). When we talk 99.7% a Lean Six Sigma opportunities, 3 defects) equates to a process Sigma long-term When we talk about DPMO of the process, we are referring to long-term (the future). We refer to 3.4 defects per million opportunities as our goal. This means that we will have a 6 sigma process now in Notice: Sigma with a capital "S" is used above to denote the process Sigma, which is different than the typical statistical reference to sigma with a small "s" which denotes the standard deviation. How to Calculate Process Sigma Consider the power company example from the previous page: A power company measures their performance in uptime of available power to their grid. Here is the 5 step process to calculate your Step 1: Define Your Opportunities An opportunity is the lowest defect noticeable by a customer. Many Six Sigma professionals support the counter point. I always like to think back to the pioneer of Six Sigma, Motorola. They built pagers that did not require testing prior to shipment to the customer. Their process sigma was around six, meaning that only approximately 3.4 pagers out of a million Returning to our power company example, an opportunity was defined as a minute of uptime. That Step 2: Define Your Defects Defining what a defect is to your customer is not easy either. You need to first communicate with your customer through focus groups, surveys, or other voice of the customer tools. To Motorola pager Returning to our power company example, a defect is defined by the customer as one minute of no power. An additional defect would be noticed for every minute that elapsed where the customer didn't Step 3: Measure Your Opportunities and Defects Now that you have clear definitions of what an opportunity and defect are, you can measure them. The power company example is relatively straight forward, but sometimes you may need to set up a formal data collection plan and organize the process of data collection. Be sure to read 'Building a Returning to our power company example, here is the data we collected: Opportunities (last year): Step 4: Calculate Your Yield The process yield is calculated by subtracting the total number of defects from the total number of opportunities, dividing by the total number of opportunities, and finally multiplying the - 500) by 100. Returning to our power company example, the yield would be calculated as:((525,600 result / Alternatively, the yield can be calculated for you by using the iSixSigma Process Sigma Calculator -

Step 5: Look Up Process Sigma The final step (if not using the iSixSigma Process Sigma Calculator) is to look up your sigma on a

Unit A unit is any item that is produced or processed which is liable for measurement or evaluation Opportunity Any area within a product, process, service, or other system where a defect could be produced or where you fail to achieve the ideal product in the eyes of the customer. In a product, the areas where defects could be produced are the parts or connection of parts Opportunities are the things which must go right to satisfy the customer. It is not the number Defect Any type of undesired result is a defect. A failure to meet one of the acceptance criteria of your customers. A defective unit may have A defect is a failure to conform to requirements', whether or not those requirements have The non-conformance to intended usage requirement. Defective The word defective describes an entire unit that fails to meet acceptance criteria, regardless of the number of defects within the unit. A unit may be defective because of one or more Defects Per Unit - DPU DPU or Defects Per Unit is the average number of defects observed when sampling a DPU = Total # of Defects / Total population Consider 100 electronic assemblies going through a functional test. If 10 of these fail the first time around, we have a first pass yield of 90%. Let's say the 10 fails get reworked and retested and 5 pass the second time around; the 5 remaining fails pass on the third attempt. DPU takes a fundamentally different approach to the traditional measurement of yield. It is simply a ratio of the number of defects over the number of units tested (don't worry about In the above example, the DPU is 15/100 or 0.15. There are 100 units which were found to One interesting feature of DPU is that if you have sequential test nodes, i.e. if the above 100 units had to go through 'Final Test' and threw up a DPU figure of 0.1 there, you simply add the DPU figures from both nodes to get the overall DPU of 0.25 (this is telling you that there If out of the 100 loans applications there are 30 defects, the FTT yield is .70 or 70 percent. Further investigation finds that 10 of the 70 had to be reworked to achieve that yield so our To consider the defects per unit in this process we divide the number of defects by the result No.of defects/(no. of units)*(no. of opportunities for a defect)= 30/100*3 = 30/300 = .1 or we would say that there is a 10 percent chance for a defect to occur in this process. Defects Per Million Opportunities - DPMO Defects per million opportunities (DPMO) is the average number of defects per unit observed during an average production run divided by the number of opportunities to make a defect on Defects Per Million Opportunities. Synonymous with PPM. To convert DPU to DPMO, the calculation step is actually DPU/(opportunities/unit) * Yield Yield is the percentage of a process that is free of defects. OR Yield is defined as a percentage of met commitments (total of defect free events) over the total number of

First Time Yield - FTY Rolled Throughput Yield - RTY Z Score A measure of the distance in standard deviations of a sample from the mean. Calculated as (X

Table of the Standard Normal (z) Distributionz0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4

0.000.0000 0.0398 0.0793 0.1179 0.1554 0.1915 0.2257 0.2580 0.2881 0.3159 0.3413 0.3643 0.3849 0.4032 0.4192 0.4332 0.4452 0.4554 0.4641 0.4713 0.4772 0.4821 0.4861 0.4893 0.4918 0.4938 0.4953 0.4965 0.4974 0.4981 0.4987 0.4990 0.4993 0.4995 0.4997

0.010.0040 0.0438 0.0832 0.1217 0.1591 0.1950 0.2291 0.2611 0.2910 0.3186 0.3438 0.3665 0.3869 0.4049 0.4207 0.4345 0.4463 0.4564 0.4649 0.4719 0.4778 0.4826 0.4864 0.4896 0.4920 0.4940 0.4955 0.4966 0.4975 0.4982 0.4987 0.4991 0.4993 0.4995 0.4997

0.020.0080 0.0478 0.0871 0.1255 0.1628 0.1985 0.2324 0.2642 0.2939 0.3212 0.3461 0.3686 0.3888 0.4066 0.4222 0.4357 0.4474 0.4573 0.4656 0.4726 0.4783 0.4830 0.4868 0.4898 0.4922 0.4941 0.4956 0.4967 0.4976 0.4982 0.4987 0.4991 0.4994 0.4995 0.4997

0.030.0120 0.0517 0.0910 0.1293 0.1664 0.2019 0.2357 0.2673 0.2969 0.3238 0.3485 0.3708 0.3907 0.4082 0.4236 0.4370 0.4484 0.4582 0.4664 0.4732 0.4788 0.4834 0.4871 0.4901 0.4925 0.4943 0.4957 0.4968 0.4977 0.4983 0.4988 0.4991 0.4994 0.4996 0.4997

0.040.0160 0.0557 0.0948 0.1331 0.1700 0.2054 0.2389 0.2704 0.2995 0.3264 0.3508 0.3729 0.3925 0.4099 0.4251 0.4382 0.4495 0.4591 0.4671 0.4738 0.4793 0.4838 0.4875 0.4904 0.4927 0.4945 0.4959 0.4969 0.4977 0.4984 0.4988 0.4992 0.4994 0.4996 0.4997

0.050.0190 0.0596 0.0987 0.1368 0.1736 0.2088 0.2422 0.2734 0.3023 0.3289 0.3513 0.3749 0.3944 0.4115 0.4265 0.4394 0.4505 0.4599 0.4678 0.4744 0.4798 0.4842 0.4878 0.4906 0.4929 0.4946 0.4960 0.4970 0.4978 0.4984 0.4989 0.4992 0.4994 0.4996 0.4997

0.060.0239 0.0636 0.1026 0.1406 0.1772 0.2123 0.2454 0.2764 0.3051 0.3315 0.3554 0.3770 0.3962 0.4131 0.4279 0.4406 0.4515 0.4608 0.4686 0.4750 0.4803 0.4846 0.4881 0.4909 0.4931 0.4948 0.4961 0.4971 0.4979 0.4985 0.4989 0.4992 0.4994 0.4996 0.4997

0.070.0279 0.0675 0.1064 0.1443 0.1808 0.2157 0.2486 0.2794 0.3078 0.3340 0.3577 0.3790 0.3980 0.4147 0.4292 0.4418 0.4525 0.4616 0.4693 0.4756 0.4808 0.4850 0.4884 0.4911 0.4932 0.4949 0.4962 0.4972 0.4979 0.4985 0.4989 0.4992 0.4995 0.4996 0.4997

0.080.0319 0.0714 0.1103 0.1480 0.1844 0.2190 0.2517 0.2823 0.3106 0.3365 0.3529 0.3810 0.3997 0.4162 0.4306 0.4429 0.4535 0.4625 0.4699 0.4761 0.4812 0.4854 0.4887 0.4913 0.4934 0.4951 0.4963 0.4973 0.4980 0.4986 0.4990 0.4993 0.4995 0.4996 0.4997

bution0.090.0359 0.0753 0.1141 0.1517 0.1879 0.2224 0.2549 0.2852 0.3133 0.3389 0.3621 0.3830 0.4015 0.4177 0.4319 0.4441 0.4545 0.4633 0.4706 0.4767 0.4817 0.4857 0.4890 0.4916 0.4936 0.4952 0.4964 0.4974 0.4981 0.4986 0.4990 0.4993 0.4995 0.4997 0.4998

1.5 Sigma Process Shift Explanation I'm not going to bore you with the hard core statistics. There's a whole statistical section dealing with this issue, and every green, black and master black belt learns the calculation process in class. If you didn't go to class (or you forgot!), the table of the standard normal distribution is used in calculating the process sigma. Most of these tables, however, end at a z value of about 3. Using this table you'll find that 6 sigma actually translates to about 2 defects per billion opportunities, and 3.4 defects per million opportunities, which we normally define as 6 sigma, really corresponds to a sigma value of 4.5. Where does this 1.5 sigma difference come from? Motorola has determined, through After a process has been improved using the Lean Six Sigma DMAIC methodology, we calculate the process standard deviation and sigma value. These are considered to be short-term values because the data only contains common cause variation -- DMAIC projects and the associated collection of process data occur over a period of months, rather than years. Long-term data, on the other hand, contains common cause variation and special (or assignable) "By offsetting normal distribution by a 1.5 standard deviation on either side, the adjustment takes into account what happens to every process over many cycles of manufacturing Simply put, accommodating shift and drift is our 'fudge factor,' or a way to allow for unexpected errors or movement over time. Using 1.5 sigma as a standard deviation gives us a strong advantage in improving Statistical Take Away: The reporting convention of Six Sigma requires the process capability to be reported in short-term sigma -- without the presence of special cause variation. Long-term sigma is determined by subtracting 1.5 sigma