What it does
- Calculates confidence intervals for the process capability indices Cp, CPL and CPU, and Cpk
- Calculates one-sided and two-sided tolerance factors: given the sample average x-bar and sample standard deviation s
- One-sided: 100g% confidence that 100P% of the population is >= x-bar - ks OR <= x-bar + ks
- Two-sided: 100g% confidence that 100P% of the population is >= x-bar - ks AND <= x-bar + ks
- ActiveX page with the program
How it works
- The chi square distribution is used for confidence intervals for Cp. See Levinson and Tumbelty (1997), Chapter 6.
- The noncentral t distribution is used to calculate confidence intervals for CPL/CPU, and one-sided tolerance factors.
- Levinson, W. 1997. "Exact Confidence Limits for Process Capabilities." Quality Engineering, 9(3), 521-528
- The bivariate noncentral t distribution is used for confidence intervals for Cpk, and for two-sided tolerance factors.
- Same reference (includes a bibliography of other references)
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Process Capabilities Menu:
Enter
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Statistical Utilities Menu:
Use the horizontal slider to enter degrees of freedom for cumulative probability F(x) or inverse cumulative probability (quantile) calculations. Enter the sample size for tolerance factors. Chi square: enter the argument in the box on top and press the desired button. For example, Quantile and an argument of 0.5 yields the median of the chi square distribution with 30 degrees of freedom. Noncentral t: Enter the noncentrality parameter and t' and select "Noncentral t F(t')" or "Bivariate Noncentral t F(t')" to get the cumulative distribution. Noncentral t: Enter the population percent P and the confidence level. "One-sided tolerance factor" returns the tolerance factor and the noncentral t distribution's noncentrality parameter (standard normal deviate of P times the square root of sample size n) and t' (tolerance factor times n0.5). "Two-sided tolerance factor does the same for the bivariate noncentral t distribution. This may take several seconds or more. |
Simpson's rule integration provides cumulative noncentral t and bivariate noncentral t distributions. Bisection finds the noncentrality parameter (for process capability confidence limits) or the noncentral t (for tolerance factors).
Program Tests (and disclaimer)
The Statistical Utilities menu was used mainly to test the accuracy of the routines for the noncentral t, bivariate noncentral t, and chi square distributions. The former two rely on Simpson's rule integration (currently with 200 steps), which may not deliver accurate results for extreme situations; no warranty is made regarding accuracy or fitness for use. The following tests show that the routines should be reliable for sample sizes between 10 and 50.Test of Noncentral t routine:
"Actual" is from Juran and Gryna (1988), Juran's Quality Control Handbook, Appendix II V| Sample | g=0.95, P=0.90 | g=0.95, P=0.999 | g=0.99, P=0.999 | |||
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Program
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Actual
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Program
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Actual
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Program
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Actual
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|
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3
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6.280
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6.158
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15.364
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13.857
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not given
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6
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3.0065
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3.006
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6.645
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6.612
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9.601
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9.540
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10
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2.355
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2.355
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5.206
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5.203
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6.626
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6.603
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30
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1.778
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1.778
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4.022
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4.022
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4.508
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4.508
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50
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1.646
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1.646
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3.766
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3.766
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4.097
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4.096
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Test of Bivariate Noncentral t routine:
"Actual" is from Owen (1965, "A Special Case of a Bivariate Non-central t Distribution," Biometrika, 52, 3 and 4, pp. 437-447 Table 3, "Two-sided, k1=k2, p1=p2 ", where p refers to the population fraction in each tail.) The problem for n=100 could result from the need for a higher "infinity" for the Simpson's Rule integration of the bivariate noncentral t distribution, or from the larger intervals in the integration. Good results are obtained for samples of 10 to 50. The program does not attempt the calculation for n<8, since the routine for the initial estimate of t' or the noncentrality parameter doesn't work for small samples.|
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Sample n
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Program
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Reference
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Program
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Reference
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2
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14.682
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17.574
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3
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5.448
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6.554
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4
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3.890
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4.700
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||
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5
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3.257
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3.948
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10
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2.318
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2.318
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2.841
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2.840
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25
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1.829
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1.828
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2.270
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2.270
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50
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1.641
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1.641
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2.054
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2.054
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100
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0.908
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1.524
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1.001
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1.920
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¥
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z0.90=1.282
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z0.95=1.645
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The program has been marked as safe for initialization and scripting. It does not read or write files.
Note: This program's output does not constitute consulting engineering advice. Such advice can be provided only in person by an engineer who is registered to practice in your state.
