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

Important "metrics" (numerical measurements) we can deduce from this are:

Probable defective parts per million

"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.)

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.