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TECHNICAL DEFINITION OF SIX SIGMA MANAGEMENT

The Normal Distribution. The term Six Sigma is derived from the normal distribution used intatistics. Many observable phenomena can be graphically represented as a bell-shaped curver a normal distribution. Figure 1 shows a theoretical normal distribution (smooth blue line) with amean (center of distribution) of zero and a standard deviation (spread of distribution) of one, aswell as a random sample of 10,000 normally distributed data points (histogram) with a mean of

zero and a standard deviation of one.

Figure 1: Normal Distribution with Mean ( = 0) and Standard Deviation

( =1)

In a normal distribution, the interval created by the mean plus or minus 2 standard deviationscontains 95.44% of the data points, or 45,600 data points per million are outside of the area

created by the mean plus or minus 2 standard deviations [(1.00 - .9544 = .0456) x 1,000,000 =45,600]. In a normal distribution the interval created by the mean plus or minus 3 standard

deviations contains 99.73% of the data, or 2, 700 data points per million are outside of the areacreated by the mean plus or minus 3 standard deviations [(1.00 - .9973 = .0027) x 1,000,000 =

2,700]. In a normal distribution the interval created by the mean plus or minus 6 standarddeviations contains 99.9999998% of the data, or 2 data points per billion data points outside of

the area created by the mean plus or minus 6 standard deviations.

elationship Between Voice of the Process and Voice of the Customer.Six Sigma promotes the idea that the distribution of

tput for a stable normally distributed process (Voice of the Process) should be designed to take-up no more than half of theerance allowed by the specification limits (Voice of the Customer). Although processes may be designed to be at their best, itassumed that over time the variation may increase in the processes. This increase in variation may be due to small variationth process inputs, the way the process is monitored, changing conditions, etc. The increase in process variation is oftensumed for sake of descriptive simplicity to be similar to temporary shifts in the underlying process mean. The increase inocess variation has been shown in practice to be equivalent to an average shift of 1.5 standard deviations in the mean of theginally designed and monitored process. If a process is originally designed to be twice as good as a customer demands (i.e.,e specifications representing the customer requirements are 6 standard deviations from the process target), then even with aift in the distribution of output the customer demands are likely to be met. In fact, even if the process shifted off target by 1.5

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andard deviations there are 4.5 standard deviations between the process mean ( + 1.50) and closest specification+ 6.00). This results in at worst 3.4 defects per million opportunities (DPMO) at the time the process has shifted or theriation has increased to have similar impact as a 1.5 standard deviation shift.

hree Sigma Process with 0.0 Shift in the Mean. Figure 2 shows the Voice of the Process for an accounting function with anerage of 7 days, a standard deviation of 1 day, and a stable normal distribution. It also shows a nominal value of 7 days, a

wer specification limit of 4 days, and an upper specification limit of 10 days. The accounting function is referred to as a 3-sigmaocess because the process mean plus or minus three standard deviations is equal to the specification limits, in other terms,SL= +3 and LSL = 3. This scenario will yield 2,700 defects per million opportunities or one early or late monthly report in.86 years [(1/0.0027)/12].

Figure 2: Three Sigma Process with 0.0 in the Mean

hree Sigma Process with a 1.5 Sigma Shift in the Mean. Figure 3 shows the same scenario as figure 2, but the process meanifts by 1.5 standard deviations (the process average is shifted down or up by 1.5 standard deviations [or 1.5 days] from 7.0ys to 5.5 days or 8.5 days) over time. This is not an uncommon phenomenon. The 1.5 standard deviation shift in the meansults in 66,807 defects per million opportunities at the nearest specification limit, or one early or late monthly report in 1.25ars [(1/.066807)/12].

Figure 3: Three Sigma Process with a 1.5 Sigma Shift in the Mean

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x Sigma Process with a 0.0 Shift in the Mean. Figure 4 shows the same scenario as figure 2 except the Voice of the Processly takes up half the distance between the specification limits. The process mean remains the same as in figure 2, but theocess standard deviation has been reduced to one half-day through application of the DMAIC model. In this case, the resultingtput will exhibit 2 defects per billion opportunities, or one early or late monthly report in 41,666,667 years [(1/.000000002)/12].

Figure 4: Six Sigma Process with a 0.0 Shift in the Mean

x Sigma Process with 1.5 Sigma Shift in the Mean. Figure 5 shows the same scenario as figure 4, but the process average

ifts by 1.5 standard deviations (the process average is shifted down or up by 1.5 standard deviations [or 0.75 days = 1.5 x 0.5ys] from 7.0 days to 6.25 days or 7.75 days) over time. The 1.5 standard deviation shift in the mean results in 3.4 defects perllion opportunities, or one early or late monthly report in 24,510 years [(1/.0000034/12]. This is the definition of 6 Sigma level oality.

Figure 5: Six Sigma Process with 1.5 Sigma Shift in the Mean

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he difference between a 3 sigma process (66,807 defects per million opportunities) and a 6 sigma process (3.4 defects perllion opportunities) can be seen in a service with 20 component steps.

each of the 20 component steps has a quality level of 66,807 defects per million opportunities, assuming each stepoes not allow rework, then the likelihood of a defect at each step is 0.066807 (66,807/1,000,000). By subtraction, thekelihood of a defect free step is 0.933193 (1.0 - 0.066807). Consequently, the likelihood of delivering a defect free finalervice is 25.08 percent. This is computed by multiplying 0.933193 by itself 20 times ([1.0 - 0.066807]20 = 0.2508 =5.08%). However, if each of the 20 component parts has a quality level of 3.4 defects per million opportunities.0000034), then the likelihood of delivering a defect free final service is 99.99932% ([1.0 - 0.0000034]20 = 0.9999996620 =9999932 = 99.99932%). A 3 sigma process generates 25.08% defect-free services, while a 6 sigma process generates.99932% defect-free services. The difference between the three sigma process and the Six Sigma process isamatic enough to certainly believe that 6 Sigma level of performance matters, especially with more complex processesth a greater number of steps or activities.