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The math behind Six Sigma metrics

Mon, 7 Jan 2008

By Valerie Bolhouse, Certified Six Sigma Blackbelt

(This information supports an article appearing in the January 2008 issue of Vision Systems Design, "Quality Numbers: Six Sigma.")

In nature and most manufacturing processes no two things are ever exactly the same. There exist small variations from part to part or measure to measure. If you were to acquire metrics on features of 100 "identical" parts and plot the values relative to frequency, you would be plotting a histogram. For stable processes, the curve would most likely be a normal, or bell-shaped, curve. The analysis of the data in this fashion is called ...view middle of the document...

The mean of the population is the central tendency of the data. The equation for the variance of the population is σ² = Σ(X-μ)²/ N, and the standard deviation, σ, is the square root of the variance.

The histogram can be made dimensionless for analysis by setting the mean to zero and scaling the units on the horizontal axis by dividing by the standard deviation. This is called the Z-Transform: Z = (point of interest--μ)/σ. This new plot is called a standard normal histogram. The normal distribution is completely described by its mean and standard deviation. The area under the curve represents 100% of the possible observations. The curve is symmetrical about the mean, and the tails extend to infinity. The area under the distribution curve also represents the probability. You can evaluate the probability of your manufacturing distribution falling outside of the specification limit, or predict the failure rate or yield by evaluating where the specification limit is set relative to the Z-score.

The area under the curve bounded by ±1σ contains 68.26% of the total population, while ±2σ has 95.44% and ±3σ has 99.73%. If the upper and lower specification limits were set at ±3σ, then 99.73% of the product would fall within specification and .27% falls outside of the limits. If 1,000,000 parts were manufactured with this process capability, then you would expect 2700 defective parts. This would be a 3σ process. Three sigma used to be the standard for high-quality manufacturing, but that is no longer true in today's competitive market.

Six sigma is the new standard. If you calculate the area under the curve outside of ±6σ, you would see that it produces two defects per billion—-not million! Since this is not feasible or cost justifiable in most instances, Six Sigma uses a more pragmatic definition of 3.4 defects per million. It assumes that process variability will stay constant over time at ±4.5σ, but you will see a shift in the process mean of up to ±1.5σ. This is realistic in most manufacturing processes. If you are drilling a hole, the diameter might not vary widely, but it is likely that the location of the hole might move over time due to fixturing tolerances or machine wear. Or if you are painting or printing, the color might shift over time, but within the lot, the color itself will remain stable (see figures above).

Six Sigma metrics include both short-term and long-term capability. Short-term capability is measured during runoff for machine acceptance, or during the process validation phase. A small sample of parts are run, and plus or minus six standard deviations of the measured variability should be within the specification limits. It is expected that over time,...

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