1、Designation: E2587 141An American National StandardStandard Practice forUse of Control Charts in Statistical Process Control1This standard is issued under the fixed designation E2587; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision,
2、 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.1NOTEEditorial corrections were made to Section 3 in February 2015.1. Scope1.1 This practice provides guidance for th
3、e use of controlcharts in statistical process control programs, which improveprocess quality through reducing variation by identifying andeliminating the effect of special causes of variation.1.2 Control charts are used to continually monitor productor process characteristics to determine whether or
4、 not a processis in a state of statistical control. When this state is attained, theprocess characteristic will, at least approximately, vary withincertain limits at a given probability.1.3 This practice applies to variables data (characteristicsmeasured on a continuous numerical scale) and to attri
5、butesdata (characteristics measured as percentages, fractions, orcounts of occurrences in a defined interval of time or space).1.4 The system of units for this practice is not specified.Dimensional quantities in the practice are presented only asillustrations of calculation methods. The examples are
6、 notbinding on products or test methods treated.1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of reg
7、ulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and StatisticsE1994 Practice for Use of Process Oriented AOQL andLTPD Sampling PlansE2234 Practice for Sampling a Stream of Product by Attri-butes Indexed by AQLE2281 Practice for Process
8、 and Measurement CapabilityIndicesE2762 Practice for Sampling a Stream of Product by Vari-ables Indexed by AQL3. Terminology3.1 Definitions:3.1.1 See Terminology E456 for a more extensive listing ofstatistical terms.3.1.2 assignable cause, nfactor that contributes to varia-tion in a process or produ
9、ct output that is feasible to detect andidentify (see special cause).3.1.2.1 DiscussionMany factors will contribute tovariation, but it may not be feasible (economically or other-wise) to identify some of them.3.1.3 attributes data, nobserved values or test results thatindicate the presence or absen
10、ce of specific characteristics orcounts of occurrences of events in time or space.3.1.4 average run length (ARL), nthe average number oftimes that a process will have been sampled and evaluatedbefore a shift in process level is signaled.3.1.4.1 DiscussionA long ARL is desirable for a processlocated
11、at its specified level (so as to minimize calling forunneeded investigation or corrective action) and a shortARL isdesirable for a process shifted to some undesirable level (sothat corrective action will be called for promptly). ARL curvesare used to describe the relative quickness in detecting leve
12、lshifts of various control chart systems (see 5.1.4). The averagenumber of units that will have been produced before a shift inlevel is signaled may also be of interest from an economicstandpoint.3.1.5 c chart, ncontrol chart that monitors the count ofoccurrences of an event in a defined increment o
13、f time orspace.3.1.6 center line, nline on a control chart depicting theaverage level of the statistic being monitored.3.1.7 chance cause, nsource of inherent random variationin a process which is predictable within statistical limits (seecommon cause).3.1.7.1 DiscussionChance causes may be unidenti
14、fiable,1This practice is under the jurisdiction of ASTM Committee E11 on Quality andStatistics and is the direct responsibility of Subcommittee E11.30 on StatisticalQuality Control.Current edition approved Oct. 1, 2014. Published November 2014. Originallyapproved in 2007. Last previous edition appro
15、ved in 2012 as E2587 12. DOI:10.1520/E2587-14E01.2For 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.Copyright AS
16、TM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1or may have known origins that are not easily controllable orcost effective to eliminate.3.1.8 common cause, n(see chance cause).3.1.9 control chart, nchart on which are plotted a statis-tical measu
17、re of a subgroup versus time of sampling along withlimits based on the statistical distribution of that measure so asto indicate how much common, or chance, cause variation isinherent in the process or product.3.1.10 control chart factor, na tabulated constant, depend-ing on sample size, used to con
18、vert specified statistics orparameters into a central line value or control limit appropriateto the control chart.3.1.11 control limits, nlimits on a control chart that areused as criteria for signaling the need for action or judgingwhether a set of data does or does not indicate a state ofstatistic
19、al control based on a prescribed degree of risk.3.1.11.1 DiscussionFor example, typical three-sigma lim-its carry a risk of 0.135 % of being out of control (on one sideof the center line) when the process is actually in control andthe statistic has a normal distribution.3.1.12 EWMA chart, ncontrol c
20、hart that monitors theexponentially weighted moving averages of consecutive sub-groups.3.1.13 EWMV chart, ncontrol chart that monitors theexponentially weighted moving variance.3.1.14 exponentially weighted moving average (EWMA),nweighted average of time ordered data where the weights ofpast observa
21、tions decrease geometrically with age.3.1.14.1 DiscussionData used for the EWMA may consistof individual observations, averages, fractions, numbersdefective, or counts.3.1.15 exponentially weighted moving variance (EWMV),nweighted average of squared deviations of observationsfrom their current estim
22、ate of the process average for timeordered observations, where the weights of past squareddeviations decrease geometrically with age.3.1.15.1 DiscussionThe estimate of the process averageused for the current deviation comes from a coupled EWMAchart monitoring the same process characteristic. This es
23、timateis the EWMA from the previous time period, which is theforecast of the process average for the current time period.3.1.16 I chart, ncontrol chart that monitors the individualsubgroup observations.3.1.17 lower control limit (LCL), nminimum value of thecontrol chart statistic that indicates stat
24、istical control.3.1.18 MR chart, ncontrol chart that monitors the movingrange of consecutive individual subgroup observations.3.1.19 p chart, ncontrol chart that monitors the fraction ofoccurrences of an event.3.1.20 R chart, ncontrol chart that monitors the range ofobservations within a subgroup.3.
25、1.21 rational subgroup, nsubgroup chosen to minimizethe variability within subgroups and maximize the variabilitybetween subgroups (see subgroup).3.1.21.1 DiscussionVariation within the subgroup is as-sumed to be due only to common, or chance, cause variation,that is, the variation is believed to be
26、 homogeneous. If using arange or standard deviation chart, this chart should be instatistical control. This implies that any assignable, or special,cause variation will show up as differences between thesubgroups on a corresponding Xchart.3.1.22 s chart, ncontrol chart that monitors the standarddevi
27、ations of subgroup observations.3.1.23 special cause, n(see assignable cause).3.1.24 standardized chart, ncontrol chart that monitors astandardized statistic.3.1.24.1 DiscussionA standardized statistic is equal to thestatistic minus its mean and divided by its standard error.3.1.25 state of statisti
28、cal control, nprocess conditionwhen only common causes are operating on the process.3.1.25.1 DiscussionIn the strict sense, a process being ina state of statistical control implies that successive values of thecharacteristic have the statistical character of a sequence ofobservations drawn independe
29、ntly from a common distribu-tion.3.1.26 statistical process control (SPC), nset of tech-niques for improving the quality of process output by reducingvariability through the use of one or more control charts and acorrective action strategy used to bring the process back into astate of statistical co
30、ntrol.3.1.27 subgroup, nset of observations on outputs sampledfrom a process at a particular time.3.1.28 u chart, ncontrol chart that monitors the count ofoccurrences of an event in variable intervals of time or space,or another continuum.3.1.29 upper control limit (UCL), nmaximum value of thecontro
31、l chart statistic that indicates statistical control.3.1.30 variables data, nobservations or test results de-fined on a continuous scale.3.1.31 warning limits, nlimits on a control chart that aretwo standard errors below and above the centerline.3.1.32 X-bar chart, ncontrol chart that monitors the a
32、ver-age of observations within a subgroup.3.2 Definitions of Terms Specific to This Standard:3.2.1 average count c!,narithmetic average of subgroupcounts.3.2.2 average moving range MR!,narithmetic average ofsubgroup moving ranges.3.2.3 average proportion p!,narithmetic average of sub-group proportio
33、ns.3.2.4 average range R!,narithmetic average of subgroupranges.3.2.5 average standard deviation s!,narithmetic averageof subgroup sample standard deviations.3.2.6 grand average (X5), naverage of subgroup averages.3.2.7 inspection interval, na subgroup size for counts ofevents in a defined interval
34、of time space or another continuum.3.2.7.1 DiscussionExamples are 10 000 metres of wireE2587 1412inspected for insulation defects, 100 square feet of materialsurface inspected for blemishes, the number of minor injuriesper month, or scratches on bearing race surfaces.3.2.8 moving range (MR), nabsolu
35、te difference betweentwo adjacent subgroup observations in an I chart.3.2.9 observation, na single value of a process output forcharting purposes.3.2.9.1 DiscussionThis term has a different meaning thanthe term defined in Terminology E456, which refers there to acomponent of a test result.3.2.10 ove
36、rall proportion, naverage subgroup proportioncalculated by dividing the total number of events by the totalnumber of objects inspected (see average proportion).3.2.10.1 DiscussionThis calculation may be used for fixedor variable sample sizes.3.2.11 process, nset of interrelated or interacting activi
37、tiesthat convert input into outputs.3.2.12 subgroup average (Xi), naverage for the ith sub-group in an X-bar chart.3.2.13 subgroup count (ci), ncount for the ith subgroup ina c chart.3.2.14 subgroup EWMA (Zi), nvalue of the EWMAfor theith subgroup in an EWMA chart.3.2.15 subgroup EWMV (Vi), nvalue o
38、f the EWMV for theith subgroup in an EWMV chart.3.2.16 subgroup individual observation (Xi), nvalue of thesingle observation for the ith subgroup in an I chart.3.2.17 subgroup moving range (MRi), nmoving range forthe ith subgroup in an MR chart.3.2.17.1 DiscussionIf there are k subgroups, there will
39、 bek-1 moving ranges.3.2.18 subgroup proportion (pi), nproportion for the ithsubgroup in a p chart.3.2.19 subgroup range (Ri), nrange of the observations forthe ith subgroup in an R chart.3.2.20 subgroup size (ni), nthe number of observations,objects inspected, or the inspection interval in the ith
40、subgroup.3.2.20.1 DiscussionFor fixed sample sizes the symbol n isused.3.2.21 subgroup standard deviation (si), nsample standarddeviation of the observations for the ith subgroup in an s chart.3.3 Symbols:A2= factor for converting the average range to threestandard errors for the X-bar chart (Table
41、1)A3= factor for converting the average standard devia-tion to three standard errors of the average for theX-bar chart (Table 1)B3,B4= factors for converting the average standard devia-tion to three-sigma limits for the s chart (Table 1)B5*,B6*= factors for converting the initial estimate of thevari
42、ance to three-sigma limits for the EWMV chart(Table 11)c4= factor for converting the average standard devia-tion to an unbiased estimate of sigma (see )(Table 1)ci= counts of the observed occurrences of events in theith subgroup (10.2.1)c = average of the k subgroup counts (10.2.1)d2= factor for con
43、verting the average range to anestimate of sigma (see )(Table 1)D3,D4= factors for converting the average range to three-sigma limits for the R chart (Table 1)Di2= the squared deviation of the observation at time iminus its forecast average (12.1)k = number of subgroups used in calculation of contro
44、llimits (6.2.1)MRi= absolute value of the difference of the observationsin the (i-1)th and the ith subgroups in a MR chart(8.2.1)MR!= average of the subgroup moving ranges (8.2.2.1)n = subgroup size, number of observations in a sub-group (5.1.3)ni= subgroup size, number of observations (objectsinspe
45、cted) in the ith subgroup (9.1.2)pi= proportion of the observed occurrences of events inthe ith subgroup (9.2.1)p = average of the k subgroup proportions (9.2.1)Ri= range of the observations in the ith subgroup forthe R chart (6.2.1.2)R= average of the k subgroup ranges (6.2.2)si= Sample standard de
46、viation of the observations inthe ith subgroup for the s chart (7.2.1)sz= standard error of the EWMA statistic (11.2.1.2)s = average of the k subgroup standard deviations(7.2.2)TABLE 1 Control Chart Factorsfor X-Bar and RCharts for X-Bar and S ChartsnA2D3D4d2A3B3B4c42 1.880 0 3.267 1.128 2.659 0 3.2
47、67 0.79793 1.023 0 2.575 1.693 1.954 0 2.568 0.88624 0.729 0 2.282 2.059 1.628 0 2.266 0.92135 0.577 0 2.114 2.326 1.427 0 2.089 0.94006 0.483 0 2.004 2.534 1.287 0.030 1.970 0.95157 0.419 0.076 1.924 2.704 1.182 0.118 1.882 0.95948 0.373 0.136 1.864 2.847 1.099 0.185 1.815 0.96509 0.337 0.184 1.816
48、 2.970 1.032 0.239 1.761 0.969310 0.308 0.223 1.777 3.078 0.975 0.284 1.716 0.9727Note: for larger numbers of n, see Ref. (1).E2587 1413ui= counts of the observed occurrences of events in theinspection interval divided by the size of theinspection interval for the ith subgroup (10.4.2)V0= exponentia
49、lly-weighted moving variance at timezero (12.2.1)Vi= exponentially-weighted moving variance statisticat time i (12.1)Xi= single observation in the ith subgroup for the Ichart (8.2.1)Xij= the jth observation in the ith subgroup for the X-barchart (6.2.1)X= average of the individual observations over ksubgroups for the I chart (8.2.2)Xi= average of the ith subgroup observations for theX-bar chart (6.2.1)X5= average of the k subgroup averages for the X-barchart (6.2.2)Yi= value of the stati
