1、Designation: E2587 15 An 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.1. Scope1.1 This practice provides guidance for the use of controlcharts in statistical process control programs, whi
3、ch 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 not a processis in a state of statistical control. When this state
4、 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 attributesdata (characteristics measured as percentages, fractions, orco
5、unts 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 notbinding on products or test methods treated.1.5 This standard d
6、oes 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 regulatory limitations prior to use.2. Referenced Documents2.1 ASTM St
7、andards: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 and Measurement CapabilityIndicesE2762 Practice for Sampling a Str
8、eam 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 product output that is feasible to detect andidentify (see special cause
9、).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 absence of specific characteristics orcounts of occurrences of events in
10、 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 at its specified level (so as to minimize calling forunneeded inves
11、tigation 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 levelshifts of various control chart systems (see 5.1.4). The averagenu
12、mber 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 of time orspace.3.1.6 center line, nline on a control chart depictin
13、g 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 unidentifiable,or may have known origins that are not easily controllable o
14、rcost effective to eliminate.3.1.8 common cause, n(see chance cause).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 April 1, 2015. Published April 2
15、015. Originallyapproved in 2007. Last previous edition approved in 2014 as E2587 141. DOI:10.1520/E2587-15.2For 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 stand
16、ards Document Summary page onthe ASTM website.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.9 control chart, nchart on which are plotted a statis-tical measure of a subgroup versus time of sampling along withlimits based on the
17、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 convert specified statistics orparameters into a central line value or con
18、trol 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 ofstatistical control based on a prescribed degree of risk.3.1.11.1 DiscussionFor
19、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 chart that monitors theexponentially weighted moving averages of consecu
20、tive 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 observations decrease geometrically with age.3.1.14.1 DiscussionData used for
21、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 estimate of the process average for timeordered observations, where the weig
22、hts 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 estimateis the EWMA from the previous time period, which is theforecast o
23、f 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 statistical control.3.1.18 MR chart, ncontrol chart that monitors the movin
24、grange 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.3.1.21 rational subgroup, nsubgroup chosen to minimizethe variability wit
25、hin 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 homogeneous. If using arange or standard deviation chart, this chart s
26、hould 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 standarddeviations of subgroup observations.3.1.23 special cause, n(see assignable
27、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 statistical control, nprocess conditionwhen only common causes are operating on
28、 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 independently from a common distribu-tion.3.1.26 statistical process control (SP
29、C), 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 control.3.1.27 subgroup, nset of observations on outputs sampledfrom a pr
30、ocess 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 thecontrol chart statistic that indicates statistical control.3.1.30 variables d
31、ata, 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 aver-age of observations within a subgroup.3.2 Definitions of Terms Spec
32、ific 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 proportions.3.2.4 average range R!,narithmetic average of subgroupranges.3.2.5 a
33、verage 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 of time space or another continuum.3.2.7.1 DiscussionExamples are 10 00
34、0 metres of wireinspected for insulation defects, 100 square feet of materialsurface inspected for blemishes, the number of minor injuriesper month, or scratches on bearing race surfaces.E2587 1523.2.8 moving range (MR), nabsolute difference betweentwo adjacent subgroup observations in an I chart.3.
35、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 overall proportion, naverage subgroup proportioncalculated by dividing the
36、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 activitiesthat convert input into outputs.3.2.12 subgroup average (Xi), navera
37、ge 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 of the EWMV for theith subgroup in an EWMV chart.3.2.16 subgroup individu
38、al 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 bek-1 moving ranges.3.2.18 subgroup proportion (pi), nproportion for th
39、e 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 subgroup.3.2.20.1 DiscussionFor fixed sample sizes the symbol n isused.3
40、.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 1)A3= factor for converting the average standard devia-tion to three sta
41、ndard 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 thevariance to three-sigma limits for the EWMV chart(Table 11)c4= factor for co
42、nverting 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 converting the average range to anestimate of sigma (see )(Table 1)D3,D4= f
43、actors 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 controllimits (6.2.1)MRi= absolute value of the difference of the observations
44、in 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 (objectsinspected) in the ith subgroup (9.1.2)pi= proportion of the observed occurren
45、ces 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 deviation of the observations inthe ith subgroup for the s chart (7.2.1)sz
46、= 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 3.267 0.79793 1.023 0 2.575 1.693 1.954 0 2.568 0.88624 0.729 0 2.282 2.059
47、 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 2.970 1.032 0.239 1.761 0.969310 0.308 0.223 1.777 3.078 0.975 0.284 1.
48、716 0.9727Note: for larger numbers of n, see Ref. (1).E2587 153ui= 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= exponentially-weighted moving variance at timezero (12.2.1)Vi= exponentially-weight
49、ed 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 statistic being monitored by anEWMA chart at time i (11.2.1)zi= the standardized s
