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The impact of variation in a product can be determined by comparing the distribution of the parameter of interest with the specified limits to that parameter. One measure of variation is the population standard deviation (sigma), which is estimated from samples using the formula:
Using the standard deviation, one can then determine the proportion of the product which will be between the upper and lower specified limits, and thus considered acceptable. For example, if a parameter is distributed normally, 66.3% of the product will have a parameter value within plus and minus one standard deviation of the mean value of the parameter, 95.5% will measure between plus and minus two, and 99.7% will be between plus and minus three sigmas from the mean. Figure 2 illustrates this.
Figure 2 (Click to Zoom)
Comparing the specification limits to the variation in the product yields a measure of robustness. One of these measures is Process Capability, which is calculated by Equation 2. A Process Capability of 1.0 means that 99.7% of the product will be "in-spec." Anything lower is generally considered bad, and quality oriented companies aim at higher values.
One shortcoming of the Process Capability measure is that it presumes the mean of the parameter of interest in the product will be its target value, as illustrated in Figure 3. However, "real world" distributions are more likely to resemble Figure 4, where the product mean value is displaced from the target. For this reason, a measure called Process Performance, Equation 3, is often preferred.
Figure 3 (Click to Zoom)
Figure 4 (Click to Zoom)
The "Six Sigma" program formulated by Motorola aims for such low variability in the product that six sigmas will fit between the specification limits (i.e., a Process Capability of 2.0), which, presuming the mean of the product is 1.5 sigmas off target (i.e., a Process Performance of 1.5), translates to 3.4 items per million out of specified limits. By way of comparison, the average business process is a "four sigma" process which translates to 6,200 items per million "out of spec." Achieving a "six sigma" process requires the control of critical process parameters, which can be identified by the statistical design of experiments, the next topic.