Husaria wingLevinson Productivity Systems, P.C.
William A. Levinson, P.E.  Principal
570-824-1986
TheBoss at ct-yankee.com
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The Process Acceptance Chart (SPC chart with center band)

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SPC for nonnormal distributions

Animated control chart simulator

Services available: industrial statistics

  1. SPC for nonnormal distributions.
    • The problem: what if it's not a bell curve?
    • The solution (and why the central limit theorem doesn't help!)
      • Treatment of the gamma distribution, including sample averages (x-bar chart for a gamma distribution)
    • Process capability indices for nonnormal distributions
  2. Process capability indices
    • Many customers demand process capability index calculations (Cp, Cpk). They are very straightforward as long as the process follows a normal distribution (bell curve).
    • Techniques exist for calculating indices for nonnormal distributions and distributions with nested normal variation sources.
    • Process capability indices are subject to fairly wide confidence intervals unless you have plenty of data.
  3. Consulting
    • Statistical process control
    • Design of Experiments: including nonparametric methods for when the traditional ones (ANOVA, factorial, regression) don't work.
  4. Training for your workforce
    • Statistical process control (SPC) for operators, engineers, and technicians. See Levinson and Tumbelty, SPC Essentials and Productivity Improvement: A Manufacturing Approach from ASQ Quality Press. (Disclosure of my relationship to this product: my former employer receives all royalties.)
      • Understanding variation and accuracy: animated targets, histograms, and control charts show how SPC works.
      •  Animated control chart simulator designed to teach your production staff how to read and interpret control charts in less than an hour.
    • Design of Experiments (DOE)
      • Currently available: a six-hour overview for people with no statistical background. It does not make them into industrial statisticians (this requires several college-level or even graduate-level courses) but it equips them to know what methods are available and when they should be applied.
The Process Acceptance Chart (x-bar chart with center band)
The assumption of a typical x-bar chart is that any process shift is bad, and requires corrective action. There are often practical situations, though, in which process adjustment in response to small shifts can actually aggravate the process shift, e.g. if extremely fine adjustments are not possible. If the process is highly capable, it is often acceptable for the process mean to vary within a certain range. The shop floor solution is the process acceptance chart, or an x-bar chart with a center band. Feigenbaum (Total Quality Management, 1991, 426-428) provides more details.


The process acceptance chart's center band is defined by the lower and upper acceptable process levels [LAPL, UAPL] between which any process mean is acceptable. Note that there is a penalty for using this approach; the center band crowds out some process capability as shown by the equation for Cpk.



Downloadable table of control chart factors, sample sizes of 2 through 25 (Excel spreadsheet)
Table of control chart factors
n A A2 A3 c4 B3 B4 B5 B6 d2 1/d2 d3 D1 D2 D3 D4
2 2.121 1.880 2.659 0.7979 0.000 3.267 0.000 2.606 1.128 0.8862 0.853 0.000 3.686 0.000 3.267
3 1.732 1.023 1.954 0.8862 0.000 2.568 0.000 2.276 1.693 0.5908 0.888 0.000 4.358 0.000 2.575
4 1.500 0.729 1.628 0.9213 0.000 2.266 0.000 2.088 2.059 0.4857 0.880 0.000 4.698 0.000 2.282
5 1.342 0.577 1.427 0.9400 0.000 2.089 0.000 1.964 2.326 0.4299 0.864 0.000 4.918 0.000 2.114
6 1.225 0.483 1.287 0.9515 0.030 1.970 0.029 1.874 2.534 0.3946 0.848 0.000 5.079 0.000 2.004
7 1.134 0.419 1.182 0.9594 0.118 1.882 0.113 1.806 2.704 0.3698 0.833 0.205 5.204 0.076 1.924
8 1.061 0.373 1.099 0.9650 0.185 1.815 0.179 1.751 2.847 0.3512 0.820 0.388 5.307 0.136 1.864
9 1.000 0.337 1.032 0.9693 0.239 1.761 0.232 1.707 2.970 0.3367 0.808 0.547 5.394 0.184 1.816
10 0.949 0.308 0.975 0.9727 0.284 1.716 0.276 1.669 3.078 0.3249 0.797 0.686 5.469 0.223 1.777
11 0.905 0.285 0.927 0.9754 0.321 1.679 0.313 1.637 3.173 0.3152 0.787 0.811 5.535 0.256 1.744
12 0.866 0.266 0.886 0.9776 0.354 1.646 0.346 1.610 3.258 0.3069 0.778 0.923 5.594 0.283 1.717
13 0.832 0.249 0.850 0.9794 0.382 1.618 0.374 1.585 3.336 0.2998 0.770 1.025 5.647 0.307 1.693
14 0.802 0.235 0.817 0.9810 0.406 1.594 0.399 1.563 3.407 0.2935 0.763 1.118 5.696 0.328 1.672
15 0.775 0.223 0.789 0.9823 0.428 1.572 0.421 1.544 3.472 0.2880 0.756 1.203 5.740 0.347 1.653
16 0.750 0.212 0.763 0.9835 0.448 1.552 0.440 1.526 3.532 0.2831 0.750 1.282 5.782 0.363 1.637
17 0.728 0.203 0.739 0.9845 0.466 1.534 0.458 1.511 3.588 0.2787 0.744 1.356 5.820 0.378 1.622
18 0.707 0.194 0.718 0.9854 0.482 1.518 0.475 1.496 3.640 0.2747 0.739 1.424 5.856 0.391 1.609
19 0.688 0.187 0.698 0.9862 0.497 1.503 0.490 1.483 3.689 0.2711 0.733 1.489 5.889 0.404 1.596
20 0.671 0.180 0.680 0.9869 0.510 1.490 0.504 1.470 3.735 0.2677 0.729 1.549 5.921 0.415 1.585
21 0.655 0.173 0.663 0.9876 0.523 1.477 0.516 1.459 3.778 0.2647 0.724 1.606 5.951 0.425 1.575
22 0.640 0.167 0.647 0.9882 0.534 1.466 0.528 1.448 3.819 0.2618 0.720 1.660 5.979 0.435 1.565
23 0.626 0.162 0.633 0.9887 0.545 1.455 0.539 1.438 3.858 0.2592 0.716 1.711 6.006 0.443 1.557
24 0.612 0.157 0.619 0.9892 0.555 1.445 0.549 1.429 3.895 0.2567 0.712 1.759 6.032 0.452 1.548
25 0.600 0.153 0.606 0.9896 0.565 1.435 0.559 1.420 3.931 0.2544 0.708 1.805 6.056 0.459 1.541



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