Operating characteristics curve (OCC, for large lots)
The OC analysis of sample plans helps to provide the desired consumer and producer risk. Consumer risk is the risk of accepting low quality lots, also called Type I risk (α). Producer risk is the risk of rejecting good quality lots, also called Type II risk (β). The OCC plots the probability of acceptance for different levels of quality, and the objective is to be able to accept lots according to the desired acceptance quality level (AQL) 95% of the time.
The picture below shows an OC analysis drawn with MM4XL’s Quality Manager tool. The result can, of course, be printed in a worksheet.
The OCC behaves in compliance with the rules governing the Binomial probability distribution function. It requires large samples and assumes a low probability of occurrence for nonconformities, it works with attribute measures, and it assumes only two possible outcomes, such as good-bad, on-off, etc. A chart of the Binomial distribution is shown in the material concerning P-charts in this chapter.
Input data
The input data for the OCC does not require a worksheet range selection. Instead the user must enter the following values in the tool window:
• Sample size is the number of items in a lot.
• Low, is the lower bound of the x-axis (horizontal), cell A10 in the first table of the Output Results section.
• High, is the upper bound of the x-axis, cell A29 in the first table of the Output Results section.
• Number of classes, e.g. rows 10-29 in the first table of the Output Results section.
• Number of columns. These are the lines in the charts. The first column (line) is equal to the probability of finding zero defectives in the lot; the second column is equal to one defective in the lot and so on. The number of columns should be lower than the sample size plus one, because there cannot be more defectives than the total items in the sample.
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Output from the OCC is made up of two charts and two tables in accordance with the user’s selection in the third window.
The table below shows the Acceptance curve, which is the probability of accepting a lot according to different levels of error (columns from zero to four). We read, for instance, 64% in cell D14. This means that if 4% (cell A14) of items in the lot are defective there is a probability equal to 64% that the lot will be accepted as a good one according to our hypothesis in cell D9 that only two items are defective. The information in the table is summarized in the chart Probability of Acceptance.
Excel formula for Acceptance
=BINOMIAL(B9;[sample size];A10)
The table below shows the probability of rejecting an acceptable lot, which is found by subtracting the probability of acceptance (see the table above) from one. The information in the table is summarized in the chart Probability of Rejection.
Excel formula for Rejection
=1-BINOMIAL(I9;[sample size];H10)
7. Quality Manager
Hypergeometric operating characteristics curve (HOCC, for small lots)
The HOCC is used instead of the OCC when the number of lots is small. In the literature it is sometimes recommended that you use the OCC when the sample size exceeds 10% of the lot size. Other authors suggest using the HOCC for samples smaller than 20% of the lot size.
The picture below shows an HOCC drawn with MM4XL’s Quality Manager tool. The result can, of course, be printed in a worksheet.
The HOCC behaves in compliance with the rules governing the Hypergeometric probability distribution function. It can work with small samples, it requires attribute measures, and it assumes only two possible outcomes, such as good-bad, on-off, etc. The following is a chart of the Hypergeometric distribution.
Input data
The input data for the HOCC does not require a worksheet range selection. Instead the user must enter the following values in the tool window:
• Lot size is the number of items in a lot.
• Low, is the lower bound of the x-axis (horizontal), cell C39 in the first table of the Output Results section.
• High, is the upper bound of the x-axis, cell C61 in the first table of the Output Results section.
• Increment, is the number of defectives in a lot, range C39:C61 in the first table of the Output Results section. The value must be smaller than that of the Lot size.
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• Number of columns. These are the lines in the charts. The first column (line) is equal to the probability of finding zero defectives in the lot; the second column is equal to one defective in the lot and so on. The number of columns should be lower than the sample size plus one, because there cannot be more defectives than the total items in the sample.
Output results
The HOCC curve works the same way as the OCC, with the difference that the first column of the tables refers to the number of defective items rather than to the percentage. One table and one chart refer to the probability of acceptance of samples.
Excel formula for Acceptance
=HYPERGEO(D39;[sample size];C39;[lot size])
The table below shows the probability of rejecting an acceptable lot, found by subtracting the probability of acceptance (see the table above) from one. The information in the table is summarized in the chart Probability of Rejection.
Excel formula for Rejection
=1-HYPERGEO(D39;[sample size];C39;[lot size])
7. Quality Manager
Average outgoing quality (AOQ)
Outgoing quality (OQ) is the quality of a lot after it has been inspected, and it is a function of incoming quality and the sampling plan. Incoming quality is the quality of material when it comes into the plant. Low incoming quality will never produce perfect outgoing quality. However, an accurate sampling plan may help to increase the level of outgoing quality when the incoming quality is low. The AOQ is found by multiplying the incoming quality by the probability of acceptance.
The picture below shows an AOQ analysis drawn with MM4XL’s Quality Manager tool. The result can, of course, be printed in a worksheet.
Input data
The input data for the AOQ does not require a worksheet range selection. Instead the user must enter the following values in the tool window:
• Sample size is the number of items in a lot.
• Low, is the lower bound of the x-axis (horizontal), cell C39 in the table of the Output Results section.
• High, is the upper bound of the x-axis, cell C58 in the table of the Output Results section.
• Number of classes, e.g. rows 39-58 in the tables of the Output Results section.
• Number of columns. These are the lines in the charts.
Output results
The diagonal line in the chart below shows the maximum possible outgoing quality, and the curves show the outgoing quality level for different levels of defectives in the sample. For instance, 58% stands for the level of outgoing quality when the incoming quality is roughly 60%. This means that incoming quality equals outgoing quality, so that management could decide to maintain the status quo or improve outgoing quality.
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Average Outgoing Quality
7%
16%
27%
41%
58%
0%
20%
40%
60%
80%
0.0 0.2 0.4 0.6 0.8
Incoming Quality
Outgoing Quality
The information in the chart is detailed in the following table that accompanies the AOQ analysis.
Formula
Average Outgoing Quality =Max([col]39:[col]58)
Diagonal =[previous class]+Increment
Increment =(High-Low)/Num. of Classes
Curves = BINOMIAL(D38;[sample size];C39)
7. Quality Manager
Technicalities
Input requirements
Valid user data selections require at least five cells of values. When this minimum is not reached, Quality Manager allows you to use only the Acceptance Sampling family of tools. These are: Operating characteristics curve (OCC, for large lots), Hypergeometric operating characteristics curve (HOCC, for small lots), Average outgoing quality (AOQ).
Blank and missing input
Quality Manager automatically sets blank and missing values to zero values. This may affect the end results of your analysis.
Constants
X charts made with Quality Manager use constant values in accordance with the guidelines suggested in the Manual on Presentation of Data and Control Chart Analysis published by ASTM (table 16, page 77, 7th edition).
Large input series
Very large input series, say over 500 data points on a Pentium 4 PC, may impose a long waiting time before the results can be shown in the preview window. The waiting time may be even longer when simulated data are produced. To shorten this time you can uncheck the option Automatic update of charts in the first window of Quality Manager. The Recalculate button in the second window will then allow you to update the preview after your selection is made.
At the lower left side of the window, the status bar displays the rank number of the data point loading. This feature runs too fast to be seen with short data series.
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References
Michael R. Beauregard, Raymond J. Mikaluk, Barbara A. Olson A Practical Guide to Statistical Quality Improvement
Van Nostrand Reinhold, New York, 1992 Committee E-11 on Quality and Statistics
Manual on Presentation of Data and Control Chart Analysis ASTM International, 2002
Gopal K. Kanji 100 Statistical Tests Sage Publications, 1993 Douglas C. Montgomery
Introduction to Statistical Quality Control John Wiley & Sons, New York, 1991 Walter A. Shewhart
Statistical Methods From the Viewpoint of Quality Control Edward Deming Editor, 1939
G. Barrie Wetherill, Don W. Brown Statistical Process Control
Chapman and Hall, London, 1991
Steven M. Zimmermann, Marjorie L. Icenogle Statistical Quality Control Using Excel
American Society for Quality (ASQ), Milwaukee, WI, 1999 W. Edwards Deming
Quality Productivity and Competitive Position Massachusetts Institute of Technology, 1982
7. Quality Manager
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