1. 2.1 Proprietary to General Electric Company SDM-V8 (11/30/2000) Module 2 Sigma Calculation Basics It is important that the student understand the fundamental calculations of defect probability which will be used throughout the course and the different ways these may be represented and expressed. After completing this module, you will be able to: Use the normal distribution table to determine probabilities of defects Apply appropriate 1-tailed or 2-tailed defect calculations Apply the concept of process shift to defect probability predictions Convert between short-term and long-term data Perform calculations of DPU, dpmo, Z ST, Z LT 66six sigma sigma

2. 2.2 Proprietary to General Electric Company SDM-V8 (11/30/2000) Standard Normal Distribution Original Distribution Apply Transformation Standard Normal Distribution x -   Z=  -1 0 1 Z Z = + 2 is two Std. Deviations above the mean Z= - 0.5 is 1/2 Std. Deviation below the mean What is Z for a point that is 8 Std. Deviations above the mean? Area under Curve =1 Examples: X ZoZo Z Area under curve is probability, Z>Z o What is Probability that Z > 0.97? A normalized frequency distribution can be used to determine probability. Probability

3. 2.3 Proprietary to General Electric Company SDM-V8 (11/30/2000) The Standard Normal Distribution Allows You To Use Tables Instead of Integration Normal Distribution Z00.010.020.030.040.050.060.070.080.09 0.05.00E-014.96E-014.92E-014.88E-014.84E-014.80E-014.76E-014.72E-014.68E-014.64E-01 0.14.60E-014.56E-014.52E-014.48E-014.44E-014.40E-014.36E-014.33E-014.29E-014.25E-01 0.24.21E-014.17E-014.13E-014.09E-014.05E-014.01E-013.97E-013.94E-013.90E-013.86E-01 0.33.82E-013.78E-013.75E-013.71E-013.67E-013.63E-013.59E-013.56E-013.52E-013.48E-01 0.43.45E-013.41E-013.37E-013.34E-013.30E-013.26E-013.23E-013.19E-013.16E-013.12E-01 0.53.09E-013.05E-013.02E-012.98E-012.95E-012.91E-012.88E-012.84E-012.81E-012.78E-01 0.62.74E-012.71E-012.68E-012.64E-012.61E-012.58E-012.55E-012.51E-012.48E-012.45E-01 0.72.42E-012.39E-012.36E-012.33E-012.30E-012.27E-012.24E-012.21E-012.18E-012.15E-01 0.82.12E-012.09E-012.06E-012.03E-012.01E-011.98E-011.95E-011.92E-011.89E-011.87E-01 0.91.84E-011.81E-011.79E-011.76E-011.74E-011.71E-011.69E-011.66E-011.64E-011.61E-01 Z What is Probability that Z > 0.97? Area to the right of Z 0 = 1.66x10 -1 Probability = 0.166 or 16.6%

4. 2.4 Proprietary to General Electric Company SDM-V8 (11/30/2000) Z-Calculation and Defect Probability   11 11 Some Chance of Failure 33 Target 66 The higher the number (Z) in front of the sigma symbol the lower the chance of producing a defect USL 11 11 Much Less Chance of Failure

5. 2.5 Proprietary to General Electric Company SDM-V8 (11/30/2000) P(d) LSL from Z table P(d) USL from Z table P(d) Total = P(d) USL + P(d) LSL P(d) Total Z lt from Z table Z.st = Z.lt + 1.5 USLLSL Long -Term Z-Calculation (Two - Sided Distribution)

6. 2.6 Proprietary to General Electric Company SDM-V8 (11/30/2000) Short Term vs. Long Term LSLUSLT Time 1 Time 2 Time 3 Time 4 Inherent Capability of the Process Sustained Capability of the Process... also called “short-term capability”... reflects ‘within group’ variation... also called “long-term capability”... reflects ‘total process’ variation “Sigma” (Z ST ) of Process Refers to Short Term Capability Shift Happens! Z ST dpmo 63.4 5233 46,210 366,807 2308,538 Z ST dpmo 63.4 5233 46,210 366,807 2308,538

7. 2.7 Proprietary to General Electric Company SDM-V8 (11/30/2000) Long-Term / Short-Term Conversions Shift Z Z LT = Z ST -Z SH =>dpmo 4.561.53.4 3.551.5233 2.541.56,210 1.531.566,807 0.521.5308,538 Z ST dpmo 63.4 5233 46,210 366,807 2308,538 Z ST dpmo 63.4 5233 46,210 366,807 2308,538 or  ST  LT Short Term Long Term USLT Inflate  Z LT Cal Z LT

8. 2.8 Proprietary to General Electric Company SDM-V8 (11/30/2000) DPU and dpmo = 1 unit= 1 opportunity 1 unit = 4 opportunities = defect DPU = 1 dpo = 0.25 dpmo = 250,000 DPU = 2 dpo = 0.5 dpmo = 500,000 DPU = 3 dpo = 0.75 dpmo = 750,000 DPU = 1 dpo = 0.125 dpmo = 125,000 Is this unit better than the one shown above it?

9. 2.9 Proprietary to General Electric Company SDM-V8 (11/30/2000) 1.Calculate the probability of a defect for the following: 2. Calculate the probability of a defect for the following: 3.For the following continuous and attribute data, calculate the DPU, Z LT and Z ST. Assume data provided is long - term data. EXERCISE: Defect Calculations 03.5  =1.4 USL 03.5  =1.4 USL -4.2 LSL LowerUpperStd Spec LimitSpec LimitMeanDevdpmoOpportunitiesDPUZ LT Z ST a)301.2 1 b)-3301.2 1 c)-3301.22 d)124001 e)124002 P defect = ?

10. 2.10 Proprietary to General Electric Company SDM-V8 (11/30/2000) 1.Calculate the probability of a defect for the following: 2. Calculate the probability of a defect for the following: 3.For the following continuous and attribute data, calculate the DPU, Z LT and Z ST. Assume data provided is long -term data. EXERCISE: Defect Calculations 03.5  =1.4 USL 03.5  =1.4 USL -4.2 LSL LowerUpperStd Spec LimitSpec LimitMeanDevdpmoOpportunitiesDPUZ LT Z ST 301.2 62101.006212.54.0 -3301.2124201.01242.243.74 -3301.2124202.02482.243.74 124001.01242.243.74 124002.02482.243.74 P defect =.00621 P defect =.00756

11. 2.11 Proprietary to General Electric Company SDM-V8 (11/30/2000) Normal Distribution Z Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.00 5.00e-0014.96e-0014.92e-0014.88e-0014.84e-0014.80e-0014.76e-0014.72e-0014.68e-0014.64e-001 0.10 4.60e-0014.56e-0014.52e-0014.48e-0014.44e-0014.40e-0014.36e-0014.33e-0014.29e-0014.25e-001 0.20 4.21e-0014.17e-0014.13e-0014.09e-0014.05e-0014.01e-0013.97e-0013.94e-0013.90e-0013.86e-001 0.30 3.82e-0013.78e-0013.74e-0013.71e-0013.67e-0013.63e-0013.59e-0013.56e-0013.52e-0013.48e-001 0.40 3.45e-0013.41e-0013.37e-0013.34e-0013.30e-0013.26e-0013.23e-0013.19e-0013.16e-0013.12e-001 0.50 3.09e-0013.05e-0013.02e-0012.98e-0012.95e-0012.91e-0012.88e-0012.84e-0012.81e-0012.78e-001 0.60 2.74e-0012.71e-0012.68e-0012.64e-0012.61e-0012.58e-0012.55e-0012.51e-0012.48e-0012.45e-001 0.70 2.42e-0012.39e-0012.36e-0012.33e-0012.30e-0012.27e-0012.24e-0012.21e-0012.18e-0012.15e-001 0.80 2.12e-0012.09e-0012.06e-0012.03e-0012.00e-0011.98e-0011.95e-0011.92e-0011.89e-0011.87e-001 0.90 1.84e-0011.81e-0011.79e-0011.76e-0011.74e-0011.71e-0011.69e-0011.66e-0011.64e-0011.61e-001 1.00 1.59e-0011.56e-0011.54e-0011.52e-0011.49e-0011.47e-0011.45e-0011.42e-0011.40e-0011.38e-001 1.10 1.36e-0011.33e-0011.31e-0011.29e-0011.27e-0011.25e-0011.23e-0011.21e-0011.19e-0011.17e-001 1.20 1.15e-0011.13e-0011.11e-0011.09e-0011.07e-0011.06e-0011.04e-0011.02e-0011.00e-0019.85e-002 1.30 9.68e-0029.51e-0029.34e-0029.18e-0029.01e-0028.85e-0028.69e-0028.53e-0028.38e-0028.23e-002 1.40 8.08e-0027.93e-0027.78e-0027.64e-0027.49e-0027.35e-0027.21e-0027.08e-0026.94e-0026.81e-002 1.50 6.68e-0026.55e-0026.43e-0026.30e-0026.18e-0026.06e-0025.94e-0025.82e-0025.71e-0025.59e-002 1.60 5.48e-0025.37e-0025.26e-0025.16e-0025.05e-0024.95e-0024.85e-0024.75e-0024.65e-0024.55e-002 1.70 4.46e-0024.36e-0024.27e-0024.18e-0024.09e-0024.01e-0023.92e-0023.84e-0023.75e-0023.67e-002 1.80 3.59e-0023.51e-0023.44e-0023.36e-0023.29e-0023.22e-0023.14e-0023.07e-0023.01e-0022.94e-002 1.90 2.87e-0022.81e-0022.74e-0022.68e-0022.62e-0022.56e-0022.50e-0022.44e-0022.39e-0022.33e-002 2.00 2.28e-0022.22e-0022.17e-0022.12e-0022.07e-0022.02e-0021.97e-0021.92e-0021.88e-0021.83e-002 2.10 1.79e-0021.74e-0021.70e-0021.66e-0021.62e-0021.58e-0021.54e-0021.50e-0021.46e-0021.43e-002 2.20 1.39e-0021.36e-0021.32e-0021.29e-0021.25e-0021.22e-0021.19e-0021.16e-0021.13e-0021.10e-002 2.30 1.07e-0021.04e-0021.02e-0029.90e-0039.64e-0039.39e-0039.14e-0038.89e-0038.66e-0038.42e-003 2.40 8.20e-0037.98e-0037.76e-0037.55e-0037.34e-0037.14e-0036.95e-0036.76e-0036.57e-0036.39e-003 2.50 6.21e-0036.04e-0035.87e-0035.70e-0035.54e-0035.39e-0035.23e-0035.08e-0034.94e-0034.80e-003 2.60 4.66e-0034.53e-0034.40e-0034.27e-0034.15e-0034.02e-0033.91e-0033.79e-0033.68e-0033.57e-003 2.70 3.47e-0033.36e-0033.26e-0033.17e-0033.07e-0032.98e-0032.89e-0032.80e-0032.72e-0032.64e-003 2.80 2.56e-0032.48e-0032.40e-0032.33e-0032.26e-0032.19e-0032.12e-0032.05e-0031.99e-0031.93e-003 2.90 1.87e-0031.81e-0031.75e-0031.69e-0031.64e-0031.59e-0031.54e-0031.49e-0031.44e-0031.39e-003 3.00 1.35e-0031.31e-0031.26e-0031.22e-0031.18e-0031.14e-0031.11e-0031.07e-0031.04e-0031.00e-003 3.10 9.68e-0049.35e-0049.04e-0048.74e-0048.45e-0048.16e-0047.89e-0047.62e-0047.36e-0047.11e-004 3.20 6.87e-0046.64e-0046.41e-0046.19e-0045.98e-0045.77e-0045.57e-0045.38e-0045.19e-0045.01e-004 3.30 4.83e-0044.66e-0044.50e-0044.34e-0044.19e-0044.04e-0043.90e-0043.76e-0043.62e-0043.49e-004 3.40 3.37e-0043.25e-0043.13e-0043.02e-0042.91e-0042.80e-0042.70e-0042.60e-0042.51e-0042.42e-004 3.50 2.33e-0042.24e-0042.16e-0042.08e-0042.00e-0041.93e-0041.85e-0041.78e-0041.72e-0041.65e-004 3.60 1.59e-0041.53e-0041.47e-0041.42e-0041.36e-0041.31e-0041.26e-0041.21e-0041.17e-0041.12e-004 3.70 1.08e-0041.04e-0049.96e-0059.57e-0059.20e-0058.84e-0058.50e-0058.16e-0057.84e-0057.53e-005 3.80 7.23e-0056.95e-0056.67e-0056.41e-0056.15e-0055.91e-0055.67e-0055.44e-0055.22e-0055.01e-005 3.90 4.81e-0054.61e-0054.43e-0054.25e-0054.07e-0053.91e-0053.75e-0053.59e-0053.45e-0053.30e-005 4.00 3.17e-0053.04e-0052.91e-0052.79e-0052.67e-0052.56e-0052.45e-0052.35e-0052.25e-0052.16e-005 4.10 2.07e-0051.98e-0051.89e-0051.81e-0051.74e-0051.66e-0051.59e-0051.52e-0051.46e-0051.39e-005 4.20 1.33e-0051.28e-0051.22e-0051.17e-0051.12e-0051.07e-0051.02e-0059.77e-0069.34e-0068.93e-006 4.30 8.54e-0068.16e-0067.80e-0067.46e-0067.12e-0066.81e-0066.50e-0066.21e-0065.93e-0065.67e-006 4.40 5.41e-0065.17e-0064.94e-0064.71e-0064.50e-0064.29e-0064.10e-0063.91e-0063.73e-0063.56e-006 4.50 3.40e-0063.24e-0063.09e-0062.95e-0062.81e-0062.68e-0062.56e-0062.44e-0062.32e-0062.22e-006 4.60 2.11e-0062.01e-0061.92e-0061.83e-0061.74e-0061.66e-0061.58e-0061.51e-0061.43e-0061.37e-006 4.70 1.30e-0061.24e-0061.18e-0061.12e-0061.07e-0061.02e-0069.68e-0079.21e-0078.76e-0078.34e-007 4.80 7.93e-0077.55e-0077.18e-0076.83e-0076.49e-0076.17e-0075.87e-0075.58e-0075.30e-0075.04e-007 4.90 4.79e-0074.55e-0074.33e-0074.11e-0073.91e-0073.71e-0073.52e-0073.35e-0073.18e-0073.02e-007 Z LT, 1-Tailed

12. 2.12 Proprietary to General Electric Company SDM-V8 (11/30/2000) Normal Distribution (cont.) Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 5.00 2.87e-0072.72e-0072.58e-0072.45e-0072.33e-0072.21e-0072.10e-0071.99e-0071.89e-0071.79e-007 5.10 1.70e-0071.61e-0071.53e-0071.45e-0071.37e-0071.30e-0071.23e-0071.17e-0071.11e-0071.05e-007 5.20 9.96e-0089.44e-0088.95e-0088.48e-0088.03e-0087.60e-0087.20e-0086.82e-0086.46e-0086.12e-008 5.30 5.79e-0085.48e-0085.19e-0084.91e-0084.65e-0084.40e-0084.16e-0083.94e-0083.72e-0083.52e-008 5.40 3.33e-0083.15e-0082.98e-0082.82e-0082.66e-0082.52e-0082.38e-0082.25e-0082.13e-0082.01e-008 5.50 1.90e-0081.79e-0081.69e-0081.60e-0081.51e-0081.43e-0081.35e-0081.27e-0081.20e-0081.14e-008 5.60 1.07e-0081.01e-0089.55e-0099.01e-0098.50e-0098.02e-0097.57e-0097.14e-0096.73e-0096.35e-009 5.70 5.99e-0095.65e-0095.33e-0095.02e-0094.73e-0094.46e-0094.21e-0093.96e-0093.74e-0093.52e-009 5.80 3.32e-0093.12e-0092.94e-0092.77e-0092.61e-0092.46e-0092.31e-0092.18e-0092.05e-0091.93e-009 5.90 1.82e-0091.71e-0091.61e-0091.51e-0091.43e-0091.34e-0091.26e-0091.19e-0091.12e-0091.05e-009 6.00 9.87e-0109.28e-0108.72e-0108.20e-0107.71e-0107.24e-0106.81e-0106.40e-0106.01e-0105.65e-010 6.10 5.30e-0104.98e-0104.68e-0104.39e-0104.13e-0103.87e-0103.64e-0103.41e-0103.21e-0103.01e-010 6.20 2.82e-0102.65e-0102.49e-0102.33e-0102.19e-0102.05e-0101.92e-0101.81e-0101.69e-0101.59e-010 6.30 1.49e-0101.40e-0101.31e-0101.23e-0101.15e-0101.08e-0101.01e-0109.45e-0118.85e-0118.29e-011 6.40 7.77e-0117.28e-0116.81e-0116.38e-0115.97e-0115.59e-0115.24e-0114.90e-0114.59e-0114.29e-011 6.50 4.02e-0113.76e-0113.52e-0113.29e-0113.08e-0112.88e-0112.69e-0112.52e-0112.35e-0112.20e-011 6.60 2.06e-0111.92e-0111.80e-0111.68e-0111.57e-0111.47e-0111.37e-0111.28e-0111.19e-0111.12e-011 6.70 1.04e-0119.73e-0129.09e-0128.48e-0127.92e-0127.39e-0126.90e-0126.44e-0126.01e-0125.61e-012 6.80 5.23e-0124.88e-0124.55e-0124.25e-0123.96e-0123.69e-0123.44e-0123.21e-0122.99e-0122.79e-012 6.90 2.60e-0122.42e-0122.26e-0122.10e-0121.96e-0121.83e-0121.70e-0121.58e-0121.48e-0121.37e-012 7.00 1.28e-0121.19e-0121.11e-0121.03e-0129.61e-0138.95e-0138.33e-0137.75e-0137.21e-0136.71e-013 7.10 6.24e-0135.80e-0135.40e-0135.02e-0134.67e-0134.34e-0134.03e-0133.75e-0133.49e-0133.24e-013 7.20 3.01e-0132.80e-0132.60e-0132.41e-0132.24e-0132.08e-0131.94e-0131.80e-0131.67e-0131.55e-013 7.30 1.44e-0131.34e-0131.24e-0131.15e-0131.07e-0139.91e-0149.20e-0148.53e-0147.91e-0147.34e-014 7.40 6.81e-0146.31e-0145.86e-0145.43e-0145.03e-0144.67e-0144.33e-0144.01e-0143.72e-0143.44e-014 7.50 3.19e-0142.96e-0142.74e-0142.54e-0142.35e-0142.18e-0142.02e-0141.87e-0141.73e-0141.60e-014 7.60 1.48e-0141.37e-0141.27e-0141.17e-0141.09e-0141.00e-0149.30e-0158.60e-0157.95e-0157.36e-015 7.70 6.80e-0156.29e-0155.82e-0155.38e-0154.97e-0154.59e-0154.25e-0153.92e-0153.63e-0153.35e-015 7.80 3.10e-0152.86e-0152.64e-0152.44e-0152.25e-0152.08e-0151.92e-0151.77e-0151.64e-0151.51e-015 7.90 1.39e-0151.29e-0151.19e-0151.10e-0151.01e-0159.33e-0168.60e-0167.93e-0167.32e-0166.75e-016 8.00 6.22e-0165.74e-0165.29e-0164.87e-0164.49e-0164.14e-0163.81e-0163.51e-0163.24e-0162.98e-016 8.10 2.75e-0162.53e-0162.33e-0162.15e-0161.98e-0161.82e-0161.68e-0161.54e-0161.42e-0161.31e-016 8.20 1.20e-0161.11e-0161.02e-0169.36e-0178.61e-0177.92e-0177.28e-0176.70e-0176.16e-0175.66e-017 8.30 5.21e-0174.79e-0174.40e-0174.04e-0173.71e-0173.41e-0173.14e-0172.88e-0172.65e-0172.43e-017 8.40 2.23e-0172.05e-0171.88e-0171.73e-0171.59e-0171.46e-0171.34e-0171.23e-0171.13e-0171.03e-017 8.50 9.48e-0188.70e-0187.98e-0187.32e-0186.71e-0186.15e-0185.64e-0185.17e-0184.74e-0184.35e-018 8.60 3.99e-0183.65e-0183.35e-0183.07e-0182.81e-0182.57e-0182.36e-0182.16e-0181.98e-0181.81e-018 8.70 1.66e-0181.52e-0181.39e-0181.27e-0181.17e-0181.07e-0189.76e-0198.93e-0198.17e-0197.48e-019 8.80 6.84e-0196.26e-0195.72e-0195.23e-0194.79e-0194.38e-0194.00e-0193.66e-0193.34e-0193.06e-019 8.90 2.79e-0192.55e-0192.33e-0192.13e-0191.95e-0191.78e-0191.62e-0191.48e-0191.35e-0191.24e-019 9.00 1.13e-0191.03e-0199.40e-0208.58e-0207.83e-0207.15e-0206.52e-0205.95e-0205.43e-0204.95e-020 9.10 4.52e-0204.12e-0203.76e-0203.42e-0203.12e-0202.85e-0202.59e-0202.37e-0202.16e-0201.96e-020 9.20 1.79e-0201.63e-0201.49e-0201.35e-0201.23e-0201.12e-0201.02e-0209.31e-0218.47e-0217.71e-021 9.30 7.02e-0216.39e-0215.82e-0215.29e-0214.82e-0214.38e-0213.99e-0213.63e-0213.30e-0213.00e-021 9.40 2.73e-0212.48e-0212.26e-0212.05e-0211.86e-0211.69e-0211.54e-0211.40e-0211.27e-0211.16e-021 9.50 1.05e-0219.53e-0228.66e-0227.86e-0227.14e-0226.48e-0225.89e-0225.35e-0224.85e-0224.40e-022 9.60 4.00e-0223.63e-0223.29e-0222.99e-0222.71e-0222.46e-0222.23e-0222.02e-0221.83e-0221.66e-022 9.70 1.51e-0221.37e-0221.24e-0221.12e-0221.02e-0229.22e-0238.36e-0237.57e-0236.86e-0236.21e-023 9.80 5.63e-0235.10e-0234.62e-0234.18e-0233.79e-0233.43e-0233.10e-0232.81e-0232.54e-0232.30e-023 9.90 2.08e-0231.88e-0231.70e-0231.54e-0231.39e-0231.26e-0231.14e-0231.03e-0239.32e-0248.43e-024 10.00 7.62e-0246.89e-0246.23e-0245.63e-0245.08e-0244.59e-0244.15e-0243.75e-0243.39e-0243.06e-024 Z LT, 1-Tailed