"Sigma" Analysis

     The graph above shows how, in "quality control", we take a series of measurements in a process which has random variation, and measure the quality of the process, relative to "tolerance limits". The red dots above represent measurements of 5.5, 5.7, 5.8, 6, 6.1, and 6.3; the green dots represent tolerance limits of 4.8616 and 8 (set to reproduce the resulting values in "Method 2" of "Example 11.1" of Implementing Six Sigma, 2nd Ed., by Forrest Breyfogle [2003]).¹
     Important "metrics" (numerical measurements) we can deduce from this are:
     Probable defective parts per million ² = 170
     "Sigma" rating = 5.08 -- This is the distance from the centerline of the bell curve, to the closest of the 2 tolerance limits, divided by the sigma (standard deviation) of the bell curve (about the distance from the centerline, out to an "inflection point", with the steepest slope), with a "shift" of 1.5 added in.
     "Cpk" (a "process capability" index) = 1.19 -- This is the distance from the mean of the bell curve, to the closest tolerance limit, divided by 3 times the curve's sigma value.
     Mean value, or centerline, of the curve = 5.9
     "Sigma" (standard deviation of the bell curve) = .2898
     Minimum X = 1.04 -- the distance from the curve centerline, to the closer of the 2 tolerance limits
     (These numbers are shown in more detail.)

     ¹ The PPM and Cpk values, given by the Mathematica "notebook" (algorithm) used for this, agree closely with those values in "Methods" 2 and 6 of "Example 11.1: Process Capability Indices", and the column "Defective ppm: 1.5 Sigma Shifted Distribution" of "Table S", in that book.
     ² This can be computed from the unshifted sigma rating (in this case, 3.58), using either "phi" (cumulative distribution function), or area under the bell curve (by integration).

Data for Different Scenarios Above
characteristic	smaller tolerance value	defective ppm	sigma ratio	sigma ratio + 1.5	Cpk
Breyfogle, "Method 2"	4.8616	169.952	3.58282	5.08	1.19
Breyfogle, "Method 6"	4.96882	657.065	3.21288	4.71288	1.07096
six sigma	4.595775	3.39761	4.5	6	1.5
Cpk = 1.67	4.450865	.286665	5	6.5	1.66666

source code for the graph and numbers above

Thanks to Mr. Breyfogle, of Smarter Solutions in Austin, Texas, for help via email on this.