Non-technical Definition Of Six Sigma 

Six Sigma management is the relentless and rigorous pursuit of the reduction of variation in all critical processes to achieve continuous and breakthrough improvements that impact the bottom-line and / or top-line of the organization and increase customer satisfaction. Another common definition of Six Sigma management is that it is an organizational initiative designed to create manufacturing, service and administrative processes that produce a high rate of sustained improvement in both defect reduction and cycle time (e.g., when Motorola began their effort the rate they chose was a 10-fold reduction in defects in two years along with a 50% reduction in cycle). For example, a bank takes 60 days on average to process a loan with a 10% rework rate in 2000.  In a “Six Sigma” organization, the bank should take no longer than 30 days on average to process a loan with a 1% error rate in 2002, and no more than 15 days on average to process a loan with a 0.10% error rate by 2004.  Clearly, this requires a dramatically improved/innovated loan process.

Technical Definition of “Six Sigma” Management

The Normal Distribution.  The term “Six Sigma” is derived from the normal distribution used in statistics.  Many observable phenomena can be graphically represented as a bell-shaped curve or a normal distribution. Figure 1 shows a theoretical normal distribution (smooth blue line) with a mean (center of distribution) of zero and a standard deviation (spread of distribution) of one, as well 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 (µ = 0) and Standard Deviation
(σ =1)

 


In a normal distribution, the interval created by the mean plus or minus 2 standard deviations contains 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 area created 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 standard deviations 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.

Relationship Between Voice of the Process and Voice of the Customer. Six Sigma promotes the idea that the distribution of output for a stable normally distributed process (Voice of the Process) should be designed to take-up no more than half of the tolerance allowed by the specification limits (Voice of the Customer).  Although processes may be designed to be at their best, it is assumed that over time the variation may increase in the processes.  This increase in variation may be due to small variation with process inputs, the way the process is monitored, changing conditions, etc.  The increase in process variation is often assumed for sake of descriptive simplicity to be similar to temporary shifts in the underlying process mean.  The increase in process variation has been shown in practice to be equivalent to an average shift of 1.5 standard deviations in the mean of the originally designed and monitored process.  If a process is originally designed to be twice as good as a customer demands (i.e., the specifications representing the customer requirements are 6 standard deviations from the process target), then even with a shift 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 standard deviations there are 4.5 standard deviations between the process mean (µ + 1.50σ) and closest specification 
(
µ + 6.00σ). This results in at worst 3.4 defects per million opportunities (DPMO) at the time the process has shifted or the variation has increased to have similar impact as a 1.5 standard deviation shift. 

Three Sigma Process with 0.0 Shift in the Mean. Figure 2 shows the “Voice of the Process” for an accounting function with an average of 7 days, a standard deviation of 1 day, and a stable normal distribution.  It also shows a nominal value of 7 days, a lower specification limit of 4 days, and an upper specification limit of 10 days.  The accounting function is referred to as a 3-sigma process because the process mean plus or minus three standard deviations is equal to the specification limits, in other terms, USL= µ+3σ and LSL = µ–3σ. This scenario will yield 2,700 defects per million opportunities or one early or late monthly report in 30.86 years [(1/0.0027)/12]. 

Figure 2: Three Sigma Process with 0.0 in the Mean

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

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

Six Sigma Process with a 0.0 Shift in the Mean. Figure 4 shows the same scenario as figure 2 except the Voice of the Process only takes up half the distance between the specification limits. The process mean remains the same as in figure 2, but the process standard deviation has been reduced to one half-day through application of the DMAIC model.  In this case, the resulting output 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

Six Sigma Process with 1.5 Sigma Shift in the Mean. Figure 5 shows the same scenario as figure 4, but the process average shifts 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 0.5 days] 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 per million opportunities, or one early or late monthly report in 24,510 years [(1/.0000034/12].  This is the definition of 6 Sigma level of quality.

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

The difference between a 3 sigma process (66,807 defects per million opportunities) and a 6 sigma process (3.4 defects per million opportunities) can be seen in a service with 20 component steps.  If each of the 20 component steps has a quality level of 66,807 defects per million opportunities, assuming each step does not allow rework, then the likelihood of a defect at each step is 0.066807 (66,807/1,000,000).  By subtraction, the likelihood of a defect free step is 0.933193 (1.0 - 0.066807).   Consequently, the likelihood of delivering a defect free final service is 25.08 percent.  This is computed by multiplying 0.933193 by itself 20 times ([1.0 - 0.066807]20 = 0.2508 = 25.08%).  However, if each of the 20 component parts has a quality level of 3.4 defects per million opportunities (0.0000034), then the likelihood of delivering a defect free final service is 99.99932% ([1.0 - 0.0000034]20 = 0.9999996620 = 0.9999932 = 99.99932%).  A 3 sigma process generates 25.08% defect-free services, while a 6 sigma process generates 99.99932% defect-free services.  The difference between the three sigma process and the “Six Sigma” process is dramatic enough to certainly believe that 6 Sigma level of performance matters, especially with more complex processes with a greater number of steps or activities.