Designation: D5406 − 93 (Reapproved 2016)Standard Practice forRubber—Calculation of Producer’s Process PerformanceIndexes1This standard is issued under the fixed designation D5406; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice provides a calculation procedure and aformat for reporting the process performance of a manufactur-ing operation for a rubber or rubber product.1.2 This practice is specifically designed to be used fortechnically significant properties of the final product.2. Referenced Documents2.1 Quality Assurance for the Chemical and ProcessIndustries, American Society for Quality Control, Chemicaland Process Industries Division, Chemical Interest Committee,1987.23. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 common cause variation—that residual variation in-herent in any process that (1) is operating in a state of statisticalcontrol, and (2) is operating at some recognized or ascertainedlevel of technological competence.3.1.2 lower specification limit (LSL)—the producer’s mini-mum permissible value of any relevant measured productproperty.3.1.3 Pp producer’s process performance index—the ratioof the difference (USL − LSL) to the total process variation;the index does not consider where the process is centered. (See7.4.)3.1.4 Ppk producer’s process performance index—theminimum of two ratios: (1) the ratio of the difference(USL − process mean) to one-half of the total processvariation, or (2) the ratio of the difference (processmean − LSL) to one-half of the total process variation; themagnitude of the minimum index value, used with the Pp index, indicates how well the process mean is centered. (See7.5.)3.1.5 special cause variation—that variation attributable tocertain specific or assignable sources that have been (or maybe) discovered through an investigation of the process.3.1.6 target value—the aiming point of the process; this isoften (USL + LSL) ⁄2.3.1.7 total process variation—a range, along the measuredproperty scale, defined as six times the standard deviation(determined under specified process conditions); the variationmay contain either common or combined common and specialcause sources.3.1.8 upper specification limit (USL)—the producer’s maxi-mum permissible value of any relevant measured productproperty.4. Summary of Practice4.1 During the production of any product, certain physicalor chemical properties, or both, are normally measured tocontrol the properties of the final product. When enough datahave been accumulated to form a sufficient database, it ispossible to determine the centering (mean) and variation (rangeor standard deviation) of the process property distribution.These results may be informally compared to the desired targetand specification limits to determine if the process is producingan acceptable product.4.2 On a more formal basis, the two process performanceparameters, the “process mean” and the “standard deviation,”are used to calculate two Producer’s Process PerformanceIndexes designated as Pp and Ppk . These indexes allow astandardized comparison of an actual process performance tothe general specifications, the comparison of different pro-cesses for producing the same product, or comparison of thesame process at different times.5. Significance and Use5.1 This practice is used to evaluate the conformance of aproduction process to specifications when (1) special causes ofvariation may be present, and (2) the process may not be in astate of statistical control. This evaluation may also be used tocompare different manufacturing operations for conformanceto specifications.1This practice is under the jurisdiction ofASTM Committee D11 on Rubber andis the direct responsibility of Subcommittee D11.16 on Application of StatisticalMethods.Current edition approved June 1, 2016. Published July 2016. Originally approvedin 1993. Last previous edition approved in 2011 as D5406 – 93 (2011). DOI:10.1520/D5406-93R16.2Available from American Society for Quality (ASQ), 600 N. Plankinton Ave.,Milwaukee, WI 53203, http://www.asq.org.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States16. Background and Precautions6.1 A process is said to be in a state of statistical controlwhen there are only common causes of variation present (noassignable causes).Astate of statistical control is a requirementto perform a process capability calculation which can be usedfor prediction. A state of statistical control is not required toperform the process performance calculations covered by thispractice.6.2 Although the defining calculation equations for theproducer’s process performance indexes (Pp and Ppk ) givenin 7.4 and 7.5 bear a resemblance to the defining equations forthe process capability indexes (Cp and Cpk), a legitimatecalculation of Cp and Cpk can be made only when a process isin a state of statistical control. The symbols Pp and Ppk havebeen selected to represent the producer’s process performanceindexes to distinguish them from other capability and processperformance indexes.6.3 The Pp and Ppk indexes are historical in nature. Theyindicate whether or not the process variation could have metthe specifications over the time period covered by the data.Since a state of statistical control is not required to do thesecalculations, the indexes can not be used to predict futureperformance.6.4 For the best understanding of the information presented,associated control charts, histograms, performance, and capa-bility indexes should be reviewed.6.5 The formula for calculating the standard deviation (7.3)will be used for test results which have a normal (Gaussian) ornon-normal distribution. Although possible presence of specialcause variation could also cause the results to be non-normallydistributed, the calculation of Pp and Ppk indexes from suchnon-normally distributed test results is useful for the purposesof this standard (5.1).6.6 The selection of the time period covered by the data setis critical if the process is known, or suspected, to havetime–dependent cycles or trends. The data should include aperiod long enough to encompass these cycles or trends. If thedata period does not include these cycles or trends, the reportedresults will understate the true process variability.6.6.1 The time period covered in the report should beestablished by mutual agreement between the producer andconsumer subject to the precaution given in 6.6. Typically, thisperiod would be for three months, but not less than 30 datapoints.6.7 Individual test results as defined in the appropriate testmethod shall be used for these calculations. Sample averagestend to have a normal distribution even when they are takenfrom non-normal populations. The use of averages in thecalculations will hide the true individual data distribution. Thestandard deviation of the individual values will be greater thanthe standard deviation of the sample averages. They are relatedby the square root of the number of values averaged:Si5 SA=n (1)where:Si= standard deviation of individual values,SA= standard deviation of averaged values, andn = number of individual values averaged.6.8 For asymmetric two-sided specifications, each portionof the specification range from the target to the limit must becalculated separately as if it were for a one-sided specification.6.9 For one-sided specifications, only the Ppk index isapplicable; however, if there is no target value because the goalis to be as low as possible (for example, impurities) or as highas possible (for example, strength), interpretation of the Ppk index should not be used to permit intentional contamination ordegradation of the product. See 2.1.6.10 The Ppk index is inherently less than or equal to thePp index. Maximum performance is achieved when the pro-cess is perfectly centered on the target and the Ppk is equal tothe Pp . A Pp or a Ppk index that steadily increases over timedemonstrates an improvement in the control of, or eliminationof, special causes of variation in the process.6.11 A Pp or Ppk index of less than 1.0 indicates that theprocess did not meet the specifications during the periodcovered by the data. When these indexes are less than 1.0, theprocess or the specification limits, or both, need to be studied.6.12 Nothing in the calculation of these indexes requires orindicates that the process is in a state of statistical control or ispredictable. This practice is not meant to replace statisticalprocess control (SPC) charts or any other statistical tool aimedat controlling or improving the process.7. Calculation7.1 Data used in the statistical calculation shall cover allproducts produced while operating to a single aiming point (ortarget) that went into a finished product area for shipment to acustomer. Thus, transition product made while switching fromone product or subproduct to another product or subproductmay be excluded. Any off-specification or out-of-control prod-uct made while producing to a common aiming point must beincluded.7.2 The process mean, x¯, is calculated as follows:X¯5(i51nxin(2)where:n = number of test results, andxi= individual test result.7.3 The calculation of the process standard deviation shouldbe based on at least 30 test results (see 6.6). The samplestandard deviation, s, is calculated as follows:s 5 ŒS1n 2 1D(i51n~xi2 x¯!2(3)where:s = sample standard deviation,n = number of test results,xi= individual test result, andx¯ = process mean.D5406 − 93 (2016)27.4 The total process variation is six times the processstandard deviation; that is, 6s. The producer’s process perfor-mance index, Pp , is calculated as follows:Pp 5USL 2 LSL6s(4)where:USL = producer’s upper specification limit, andLSL = producer’s lower specification limit.7.5 The Ppk index, when used with the Pp index, revealshow well the production process was centered in the specifi-cation range. The minimum value of either (5) or (6) is used fortwo-sided specifications. With one-sided specifications, either(5) or (6) is used, as appropriate.Ppk 5 ~USL 2 x¯ !/3 s (5)orPpk 5 ~x¯ 2 LSL!/3s (6)where:x¯ = mean value of process property for the period asspecified in 6.6.7.6 If the process center is outside the specification limits,negative values of Ppk will result.7.7 A histogram prepared from the individual results willincrease understanding of the nature of the data distribution.increase understanding of the nature of the data distribution.8. Report8.1 The report shall include the following as a minimum:8.1.1 A description of the product and the property beingreported.8.1.2 Time period of production chosen for analysis.8.1.3 Pp (for two-sided specifications only) and Ppk in-dexes.8.1.4 Producer’s USL, LSL, and target value.8.1.5 Mean, standard deviation and number of test results.8.2 If a histogram was prepared, it may be included in thereport.9. Keywords9.1 performance index; Pp ; Ppk ; producer’s process per-formanceASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentionedin this standard. Users of this standard are expressly advised that determination of the validity of any such patent rights, and the riskof infringement of such rights, are entirely their own responsibility.This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years andif not revised, either reapproved or withdrawn. 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