1、Designation: E 2281 08aStandard Practice forProcess and Measurement Capability Indices1This standard is issued under the fixed designation E 2281; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number
2、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 guidance for the use of capabilityindices for evaluating process capability and performance.Process capability indices
3、compare the variability of a processquality measure against product specifications or tolerancesand assume the process is in a state of statistical control.Process performance indices are useful in situations when theprocess is not in a state of statistical control.2. Referenced Documents2.1 ASTM St
4、andards:2E 456 Terminology Relating to Quality and StatisticsE 2586 Practice for Calculating and Using Basic Statistics2.2 ISO Standard:ISO 3534-2 StatisticsVocabulary and Symbols-StatisticalQuality Control32.3 Other Document:MNL 7 Manual on Presentation of Data and Control ChartAnalysis43. Terminol
5、ogy3.1 DefinitionsUnless otherwise noted, all statisticalterms are defined in Terminology E 456.3.2 Definitions of Terms Specific to This Standard:3.2.1 average standard deviation, s, narithmetic averageof sample standard deviations.3.2.2 long term standard deviation, sLT, nsample stan-dard deviatio
6、n of all individual (observed) values taken over along period of time.3.2.2.1 DiscussionA long period of time may be definedas shifts, weeks, or months, etc.3.2.3 lower process capability index, Cpkl, nindex de-scribing process capability in relation to the lower specificationlimit.3.2.4 lower proce
7、ss performance index, Ppkl, nindex de-scribing process performance in relation to the lower specifi-cation limit.3.2.5 minimum process capability index, Cpk, nsmaller ofthe upper process capability index and the lower processcapability index.3.2.6 minimum process performance index, Ppk, nsmallerof t
8、he upper process performance index and the lower processperformance index.3.2.7 process capability, PC, nstatistical estimate of theoutcome of a characteristic from a process that has beendemonstrated to be in a state of statistical control.3.2.8 process capability index, Cp, nan index describingpro
9、cess capability in relation to specified tolerance.3.2.9 process performance, PP, nstatistical measure ofthe outcome of a characteristic from a process that may nothave been demonstrated to be in a state of statistical control.3.2.10 process performance index, Pp, nindex describingprocess performanc
10、e in relation to specified tolerance.3.2.11 range, R, nmaximum value minus the minimumvalue in a sample. E 25863.2.12 short term standard deviation, sST, nthe inherentvariation present when a process is operating in a state ofstatistical control, expressed in terms of standard deviation.3.2.12.1 Dis
11、cussionThis may also be stated as the inher-ent process variation.3.2.13 special cause, nsource of intermittent variation ina process. ISO 3534-23.2.13.1 DiscussionSometimes “special cause” is taken tobe synonymous with “assignable cause.” However a distinctionshould be recognized. A special cause i
12、s assignable only whenit is specifically identified. Also a common cause may beassignable.3.2.13.2 DiscussionA special cause arises because ofspecific circumstances which are not always present. As such,in a process subject to special causes, the magnitude of thevariation from time to time is unpred
13、ictable.1This practice is under the jurisdiction ofASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on StatisticalQuality Control.Current edition approved Oct. 1, 2008. Published October 2008. Originallyapproved in 2003. Last previous edition approv
14、ed in 2008 as E 228108.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3Available from American National Stan
15、dards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http:/www.ansi.org.4Available from ASTM Headquarters, 100 Barr Harbor Drive, W. Consho-hocken, PA 19428.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.3.2.14 stab
16、le process, nprocess in a state of statisticalcontrol; process condition when all special causes of variationhave been removed. ISO 3534-23.2.14.1 DiscussionObserved variation can then be attrib-uted to random (common) causes. Such a process will gener-ally behave as though the results are simple ra
17、ndom samplesfrom the same population.3.2.14.2 DiscussionThis state does not imply that therandom variation is large or small, within or outside ofspecification, but rather that the variation is predictable usingstatistical techniques.3.2.14.3 DiscussionThe process capability of a stableprocess is us
18、ually improved by fundamental changes thatreduce or remove some of the random causes present and/oradjusting the mean towards the preferred value.3.2.14.4 DiscussionContinual adjustment of a stable pro-cess will increase variation.3.2.15 upper process capability index, Cpku, nindex de-scribing proce
19、ss capability in relation to the upper specificationlimit.3.2.16 upper process performance index (Ppku), nindexdescribing process performance in relation to the upper speci-fication limit.4. Significance and Use4.1 Process CapabilityProcess capability can be definedas the natural or inherent behavio
20、r of a stable process that is ina state of statistical control (1).5A “state of statistical control”is achieved when the process exhibits no detectable patterns ortrends, such that the variation seen in the data is believed to berandom and inherent to the process. Process capability is linkedto the
21、use of control charts and the state of statistical control.Aprocess must be studied to evaluate its state of control beforeevaluating process capability.4.2 Process ControlThere are many ways to implementcontrol charts, but the most popular choice is to achieve a stateof statistical control for the
22、process under study. Special causesare identified by a set of rules based on probability theory. Theprocess is investigated whenever the chart signals the occur-rence of special causes. Taking appropriate actions to eliminateidentified special causes and preventing their reappearance willultimately
23、obtain a state of statistical control. In this state, aminimum level of variation may be reached, which is referredto as common cause or inherent variation. For the purpose ofthis standard, this variation is a measure of the uniformity ofprocess output, typically a product characteristic.4.3 Process
24、 Capability IndicesThe behavior of a process(as related to inherent variability) in the state of statisticalcontrol is used to describe its capability. To compare a processwith customer requirements (or specifications), it is commonpractice to think of capability in terms of the proportion of thepro
25、cess output that is within product specifications or toler-ances. The metric of this proportion is the percentage of theprocess spread used up by the specification. This comparisonbecomes the essence of all process capability measures. Themanner in which these measures are calculated defines thediff
26、erent types of capability indices and their use. Two processcapability indices are defined in 5.2 and 5.3. In practice, theseindices are used to drive process improvement through con-tinuous improvement efforts. These indices may be used toidentify the need for management actions required to reducec
27、ommon cause variation, compare products from differentsources, and to compare processes.4.4 Process Performance IndicesWhen a process is not ina state of statistical control, the process is subject to specialcause variation, which can manifest itself in various ways onthe process variability. Specia
28、l causes can give rise to changesin the short-term variability of the process or can causelong-term shifts or drifts of the process mean. Special causescan also create transient shifts or spikes in the process mean.Even in such cases, there may be a need to assess the long-termvariability of the pro
29、cess against customer specifications usingprocess performance indices, which are defined in 6.2 and 6.3.These indices are similar to those for capability indices anddiffer only in the estimate of variability used in the calculation.This estimated variability includes additional components ofvariatio
30、n due to special causes. Since process performanceindices have additional components of variation, process per-formance usually has a wider spread than the process capabilityspread. These measures are useful in determining the role ofmeasurement and sampling variability when compared toproduct unifo
31、rmity.5. Process Capability Analysis5.1 It is common practice to define process behavior interms of its variability. Process capability, PC, is calculated as:PC 5 6sST(1)where sSTis the inherent variability of a controlled process(2,7). Since control charts can be used to achieve and verifycontrol f
32、or many different types of processes, the assumptionof a normal distribution is not necessary to affect control, butcomplete control is required to establish the capability of aprocess (2). Thus, what is required is a process in control withrespect to its measures of location and spread. Once this i
33、sachieved, the inherent variability of the process can be esti-mated from the control charts. The estimate obtained is anestimate of variability over a short time interval (minutes,hours, or a few batches). From control charts, sSTmay beestimated from the short-term variation within subgroupsdependi
34、ng on the type of control chart deployed, for example,average-range (X R) or individual-moving range (X MR).The estimate is:sST5Rd2orMRd2(2)where, Ris the average range, MRis the average movingrange, d2is a factor dependent on the subgroup size, n,ofthecontrol chart, (see ASTM MNL 7, Part 3). If an
35、average-standard deviation (X s) chart is used, the estimate becomes:sST5sc4(3)5The boldface numbers in parentheses refer to the list of references at the end ofthis standard.E 2281 08a2where s is the average standard deviation, and c4is a factordependent on the subgroup size, n, of the control char
36、t, (seeASTM MNL 7, Part 3).5.1.1 Therefore, PC is estimated by:6 sST56Rd2or6sc4(4)5.2 Process Capability Index, CP:5.2.1 The process capability index relates the process capa-bility to the customers specification tolerance. The processcapability index, Cp, is:Cp5Specification ToleranceProcess Capabi
37、lity5USL 2 LSL6sST(5)where USL = upper specification limit and LSL = lowerspecification limit. For a process that is centered with anunderlying normal distribution, Fig. 1, Fig. 2, and Fig. 3denotes three cases where PC, the process capability, is widerthan ( Fig. 1), equal to (Fig. 2), and narrower
38、 than (Fig. 3) thespecification tolerance.5.2.2 Since the tail area of the distribution beyond specifi-cation limits measures the proportion of product defectives, alarger value of Cpis better. The relationship between Cpandthe percent defective product produced by a centered process(with a normal d
39、istribution) is:CpPercentDefectiveParts perMillionCpPercentDefectiveParts perMillion0.6 7.19 71900 1.1 0.0967 9670.7 3.57 35700 1.2 0.0320 3180.8 1.64 16400 1.3 0.0096 960.9 0.69 6900 1.33 0.00636 641.0 0.27 2700 1.67 0.00006 0.575.2.3 From these examples, one can see that any processwith a Cp1. Som
40、e industries consider Cp=1.33 (an 8sSTspecification tolerance) a minimum with a Cp=1.66 preferred (3). Improvement of Cpshould depend on acompanys quality focus, marketing plan, and their competi-tors achievements, etc.5.3 Process Capability Indices Adjusted For Process Shift,Cpk:5.3.1 The above exa
41、mples depict process capability for aprocess centered within its specification tolerance. Processcentering is not a requirement since process capability isindependent of any specifications that may be applied to it. Theamount of shift present in a process depends on how far theprocess average is fro
42、m the center of the specification spread.In the last part of the above example (Cp 1), suppose that theprocess is actually centered above the USL. The Cphas a value1, but clearly this process is not producing as much conform-ing product as it would have if it were centered on target.5.3.2 For those
43、cases where the process is not centered,deliberately run off-center for economic reasons, or only asingle specification limit is involved, Cpis not the appropriateprocess capability index. For these situations, the Cpkindex isused. Cpkis a process capability index that considers theprocess average a
44、gainst a single or double-sided specificationlimit. It measures whether the process is capable of meeting thecustomers requirements by considering:5.3.2.1 The specification limit(s),5.3.2.2 The current process average, and5.3.2.3 The current sST5.3.3 Under the assumption of normality,6Cpkis calculat
45、edas:Cpk5 minCpku, Cpkl# (6)and is estimated by:Cpk5 min Cpku, Cpkl# (7)where the estimated upper process capability index isdefined as:Cpku5USL 2 X3 sST(8)and the estimated lower process capability index is definedas:Cpkl5X2 LSL3 sST(9)5.3.4 These one-sided process capability indices (CpkuandCpkl)
46、are useful in their own right with regard to single-sided6Testing for the normality of a set of data may range from simply plotting thedata on a normal probability plot (2) to more formal tests, e.g., Anderson-Darlingtest (which can be found in many statistical software programs, for example,Minitab
47、).FIG. 1 Process Capability Wider Than Specifications, Cp1, the process is capable andperforming within the specifications,5.3.5.5 If both Cpand Cpkare 1 and Cpkis 200 or a minimumof 100) are required to estimate Cpkwith a high level ofconfidence (at least 95%).FIG. 2 Process Capability Equal to Spe
48、cification Tolerance, Cp=1FIG. 3 Process Capability Narrower Than Specifications, Cp1FIG. 4 Noncentered Process, Cp 1 and Cpku) = 100(1-a)%. Let z1-abea point on a standard normal distribution such P(Zz1-a)=100a%. For the statistic, Pp, an exact result for the lowerconfidence bound may be given (Ref
49、 9). The lower 100(1-a)%confidence bound for process capability index Ppis:Pp$Ppun21(17)The approximate standard error for the statistic Ppis:se Pp! 5Pp=2n21!(18)For the process capability index Ppk, the approximate 100(1-a)% lower confidence bound is:Ppk$Ppk z1a19 n1Ppk22n 2(19)The approximate standard error for the statistic Ppkis:se Ppk! 519 n1Ppk2n 2(20)Results (Eq 19) and (Eq 20) are approximate and useful forpractical purposes.7.4 It is sometimes desirable to ask for a combination ofsample size and
