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The Math Behind Six Sigma

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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 descriptive statistics.
Data about the entire population is not usually studied. It is more useful to study a sample of that population and infer from the analysis what the entire population most likely looks like. This is inferential statistics. The confidence in the correctness of that prediction is dependent upon the size of the sample and the behavior of the data. Some of the useful characteristics that can be calculated from the data are described below.
The average value of the data is called the mean or X-bar. The equation for the mean is (X1+X2+X3+...+XN)/N, also denoted by ΣXi/N. Another measure calculated from the data is the variability, or the degree to which the individuals cluster about the mean. The most common measure of variability is the variance. The variance is calculated by squaring and summing the deviation of the individual data points from the mean. The equation for variance is s²=Σ (X )²/(N 1). The square root of the variance provides the standard deviation of the sample, s. If you have your data in an Excel spreadsheet, you can easily calculate the mean and standard deviation by using the built-in functions, AVE and STDEV.
The sample mean and standard

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