ASTM E2587-16(2021)e1
(Practice)Standard Practice for Use of Control Charts in Statistical Process Control
Standard Practice for Use of Control Charts in Statistical Process Control
ABSTRACT
This guide covers fundamental concepts, applications, and mathematical relationships associated with reliability as used in industrial areas and as applied to simple components, processes, and systems or complex final products. This guide summarizes selected concepts, terminology, formulas, and methods associated with reliability and its application to products and processes.
SIGNIFICANCE AND USE
4.1 This practice describes the use of control charts as a tool for use in statistical process control (SPC). Control charts were developed by Shewhart (2)3 in the 1920s and are still in wide use today. SPC is a branch of statistical quality control (3, 4), which also encompasses process capability analysis and acceptance sampling inspection. Process capability analysis, as described in Practice E2281, requires the use of SPC in some of its procedures. Acceptance sampling inspection, described in Practices E1994, E2234, and E2762, requires the use of SPC to minimize rejection of product.
4.2 Principles of SPC—A process may be defined as a set of interrelated activities that convert inputs into outputs. SPC uses various statistical methodologies to improve the quality of a process by reducing the variability of one or more of its outputs, for example, a quality characteristic of a product or service.
4.2.1 A certain amount of variability will exist in all process outputs regardless of how well the process is designed or maintained. A process operating with only this inherent variability is said to be in a state of statistical control, with its output variability subject only to chance, or common, causes.
4.2.2 Process upsets, said to be due to assignable, or special causes, are manifested by changes in the output level, such as a spike, shift, trend, or by changes in the variability of an output. The control chart is the basic analytical tool in SPC and is used to detect the occurrence of special causes operating on the process.
4.2.3 When the control chart signals the presence of a special cause, other SPC tools, such as flow charts, brainstorming, cause-and-effect diagrams, or Pareto analysis, described in various references (4-8), are used to identify the special cause. Special causes, when identified, are either eliminated or controlled. When special cause variation is eliminated, process variability is reduced to its inherent variability, and control...
SCOPE
1.1 This practice provides guidance for the use of control charts in statistical process control programs, which improve process quality through reducing variation by identifying and eliminating the effect of special causes of variation.
1.2 Control charts are used to continually monitor product or process characteristics to determine whether or not a process is in a state of statistical control. When this state is attained, the process characteristic will, at least approximately, vary within certain limits at a given probability.
1.3 This practice applies to variables data (characteristics measured on a continuous numerical scale) and to attributes data (characteristics measured as percentages, fractions, or counts 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 as illustrations of calculation methods. The examples are not binding on products or test methods treated.
1.5 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
1.6 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations...
General Information
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Standards Content (Sample)
This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
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Designation: E2587 − 16 (Reapproved 2021) An American National Standard
Standard Practice for
Use of Control Charts in Statistical Process Control
This standard is issued under the fixed designation E2587; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Editorial changes were made throughout in July 2021.
1. Scope 2. Referenced Documents
2.1 ASTM Standards:
1.1 This practice provides guidance for the use of control
E177 Practice for Use of the Terms Precision and Bias in
charts in statistical process control programs, which improve
ASTM Test Methods
process quality through reducing variation by identifying and
E456 Terminology Relating to Quality and Statistics
eliminating the effect of special causes of variation.
E1994 Practice for Use of Process Oriented AOQL and
1.2 Control charts are used to continually monitor product
LTPD Sampling Plans
orprocesscharacteristicstodeterminewhetherornotaprocess
E2234 Practice for Sampling a Stream of Product by Attri-
isinastateofstatisticalcontrol.Whenthisstateisattained,the
butes Indexed by AQL
process characteristic will, at least approximately, vary within
E2281 Practice for Process Capability and Performance
certain limits at a given probability.
Measurement
E2762 Practice for Sampling a Stream of Product by Vari-
1.3 This practice applies to variables data (characteristics
ables Indexed by AQL
measured on a continuous numerical scale) and to attributes
data (characteristics measured as percentages, fractions, or
3. Terminology
counts of occurrences in a defined interval of time or space).
3.1 Definitions—Unlessotherwisenotedinthisstandard,all
1.4 The system of units for this practice is not specified.
terms relating to quality and statistics are defined in Terminol-
Dimensional quantities in the practice are presented only as
ogy E456.
illustrations of calculation methods. The examples are not
3.1.1 assignable cause, n—factor that contributes to varia-
binding on products or test methods treated.
tion in a process or product output that is feasible to detect and
identify (see special cause).
1.5 This standard does not purport to address all of the
3.1.1.1 Discussion—Many factors will contribute to
safety concerns, if any, associated with its use. It is the
variation, but it may not be feasible (economically or other-
responsibility of the user of this standard to establish appro-
wise) to identify some of them.
priate safety, health, and environmental practices and deter-
3.1.2 accepted reference value, ARV, n—value that serves as
mine the applicability of regulatory limitations prior to use.
an agreed-upon reference for comparison and is derived as: (1)
1.6 This international standard was developed in accor-
a theoretical or established value based on scientific principles,
dance with internationally recognized principles on standard-
(2) an assigned or certified value based on experimental work
ization established in the Decision on Principles for the
of some national or international organization, or (3) a consen-
Development of International Standards, Guides and Recom-
susorcertifiedvaluebasedoncollaborativeexperimentalwork
mendations issued by the World Trade Organization Technical
under the auspices of a scientific or engineering group. E177
Barriers to Trade (TBT) Committee.
3.1.3 attributes data, n—observed values or test results that
indicate the presence or absence of specific characteristics or
counts of occurrences of events in time or space.
This practice is under the jurisdiction ofASTM Committee E11 on Quality and
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical
Quality Control. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved July 15, 2021. Published July 2021. Originally contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
approved in 2007. Last previous edition approved in 2016 as E2587 – 16. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E2587-16R21E01. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
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E2587 − 16 (2021)
3.1.4 average run length (ARL), n—the average number of from their current estimate of the process average for time
times that a process will have been sampled and evaluated ordered observations, where the weights of past squared
before a shift in process level is signaled. deviations decrease geometrically with age.
3.1.4.1 Discussion—A long ARL is desirable for a process 3.1.15.1 Discussion—The estimate of the process average
located at its specified level (so as to minimize calling for used for the current deviation comes from a coupled EWMA
unneededinvestigationorcorrectiveaction)andashortARLis chart monitoring the same process characteristic.This estimate
desirable for a process shifted to some undesirable level (so is the EWMA from the previous time period, which is the
that corrective action will be called for promptly).ARLcurves forecast of the process average for the current time period.
are used to describe the relative quickness in detecting level
3.1.16 I chart, n—control chart that monitors the individual
shifts of various control chart systems (see 5.1.4). The average
subgroup observations.
number of units that will have been produced before a shift in
3.1.17 lower control limit (LCL), n—minimum value of the
level is signaled may also be of interest from an economic
control chart statistic that indicates statistical control.
standpoint.
3.1.18 MR chart, n—control chart that monitors the moving
3.1.5 c chart, n—control chart that monitors the count of
range of consecutive individual subgroup observations.
occurrences of an event in a defined increment of time or
3.1.19 p chart, n—control chart that monitors the fraction of
space.
occurrences of an event.
3.1.6 center line, n—line on a control chart depicting the
average level of the statistic being monitored. 3.1.20 R chart, n—control chart that monitors the range of
observations within a subgroup.
3.1.7 chance cause, n—source of inherent random variation
in a process which is predictable within statistical limits (see 3.1.21 rational subgroup, n—subgroup chosen to minimize
the variability within subgroups and maximize the variability
common cause).
between subgroups (see subgroup).
3.1.7.1 Discussion—Chance causes may be unidentifiable,
or may have known origins that are not easily controllable or 3.1.21.1 Discussion—Variation within the subgroup is as-
sumed to be due only to common, or chance, cause variation,
cost effective to eliminate.
that is, the variation is believed to be homogeneous. If using a
3.1.8 common cause, n—(see chance cause).
range or standard deviation chart, this chart should be in
3.1.9 control chart, n—chart on which are plotted a statis-
statistical control. This implies that any assignable, or special,
ticalmeasureofasubgroupversustimeofsamplingalongwith
cause variation will show up as differences between the
limits based on the statistical distribution of that measure so as
¯
subgroups on a corresponding X chart.
to indicate how much common, or chance, cause variation is
3.1.22 s chart, n—control chart that monitors the standard
inherent in the process or product.
deviations of subgroup observations.
3.1.10 control chart factor, n—a tabulated constant, depend-
3.1.23 special cause, n—(see assignable cause).
ing on sample size, used to convert specified statistics or
parameters into a central line value or control limit appropriate
3.1.24 standardized chart, n—control chart that monitors a
to the control chart. standardized statistic.
3.1.24.1 Discussion—Astandardized statistic is equal to the
3.1.11 control limits, n—limits on a control chart that are
statistic minus its mean and divided by its standard error.
used as criteria for signaling the need for action or judging
whether a set of data does or does not indicate a state of
3.1.25 state of statistical control, n—process condition
statistical control based on a prescribed degree of risk.
when only common causes are operating on the process.
3.1.11.1 Discussion—For example, typical three-sigma lim-
3.1.25.1 Discussion—In the strict sense, a process being in
its carry a risk of 0.135 % of being out of control (on one side
astateofstatisticalcontrolimpliesthatsuccessivevaluesofthe
of the center line) when the process is actually in control and
characteristic have the statistical character of a sequence of
the statistic has a normal distribution.
observations drawn independently from a common distribu-
tion.
3.1.12 EWMA chart, n—control chart that monitors the
exponentially weighted moving averages of consecutive sub-
3.1.26 statistical process control (SPC), n—set of tech-
groups.
niques for improving the quality of process output by reducing
variability through the use of one or more control charts and a
3.1.13 EWMV chart, n—control chart that monitors the
corrective action strategy used to bring the process back into a
exponentially weighted moving variance.
state of statistical control.
3.1.14 exponentially weighted moving average (EWMA),
3.1.27 subgroup, n—set of observations on outputs sampled
n—weightedaverageoftimeordereddatawheretheweightsof
from a process at a particular time.
past observations decrease geometrically with age.
3.1.14.1 Discussion—Data used for the EWMAmay consist 3.1.28 u chart, n—control chart that monitors the count of
of individual observations, averages, fractions, numbers occurrences of an event in variable intervals of time or space,
defective, or counts. or another continuum.
3.1.15 exponentially weighted moving variance (EWMV), 3.1.29 upper control limit (UCL), n—maximum value of the
n—weighted average of squared deviations of observations control chart statistic that indicates statistical control.
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E2587 − 16 (2021)
3.1.30 variables data, n—observations or test results de- surface inspected for blemishes, the number of minor injuries
fined on a continuous scale. per month, or scratches on bearing race surfaces.
3.1.31 warning limits, n—limits on a control chart that are 3.2.14 moving range (MR), n—absolute difference between
two adjacent subgroup observations in an I chart.
two standard errors below and above the centerline.
3.1.32 X-bar chart, n—control chart that monitors the aver- 3.2.15 observation, n—asinglevalueofaprocessoutputfor
charting purposes.
age of observations within a subgroup.
3.2.15.1 Discussion—This term has a different meaning
3.2 Definitions of Terms Specific to This Standard:
than the term defined in Terminology E456, which refers there
3.2.1 allowance value, K, n—amount of process shift to be
to a component of a test result.
detected.
3.2.16 overall proportion, n—average subgroup proportion
3.2.2 allowance multiplier, k, n—multiplier of standard
calculated by dividing the total number of events by the total
deviation that defines the allowance value, K.
number of objects inspected (see average proportion).
3.2.3 average count ~c¯!,n—arithmetic average of subgroup
3.2.16.1 Discussion—Thiscalculationmaybeusedforfixed
counts.
or variable sample sizes.
¯
3.2.4 average moving range ~MR!,n—arithmetic average of
3.2.17 process,n—setofinterrelatedorinteractingactivities
subgroup moving ranges.
that convert input into outputs.
3.2.5 average proportion p¯ ,n—arithmetic average of sub-
~ !
3.2.18 process target value, T, n—target value for the
group proportions.
observed process mean.
¯
3.2.6 average range ~R!,n—arithmeticaverageofsubgroup
3.2.19 relative size of process shift, δ,n—size of process
ranges.
shift to detect in standard deviation units.
¯
3.2.7 average standard deviation s¯ ,n—arithmetic average
~ !
3.2.20 subgroup average (X ), n—average for the ith sub-
i
of subgroup sample standard deviations.
group in an X-bar chart.
3.2.8 cumulative sum, CUSUM, n—cumulative sum of de-
3.2.21 subgroup count (c), n—count for the ith subgroup in
i
viations from the target value for time-ordered data.
a c chart.
3.2.8.1 Discussion—Data used for the CUSUM may consist
3.2.22 subgroup EWMA(Z), n—valueoftheEWMAforthe
i
of individual observations, subgroup averages, fractions
ith subgroup in an EWMA chart.
defective, numbers defective, or counts.
3.2.23 subgroupEWMV(V),n—valueoftheEWMVforthe
i
3.2.9 CUSUM chart, n—control chart that monitors the
ith subgroup in an EWMV chart.
cumulative sum of consecutive subgroups.
¯
3.2.24 subgroup individual observation (X ), n—valueofthe
i
3.2.10 decision interval, H, n—the distance between the
single observation for the ith subgroup in an I chart.
center line and the control limits.
3.2.25 subgroup moving range (MR), n—moving range for
i
3.2.11 decision interval multiplier, h, n—multiplier of stan-
the ith subgroup in an MR chart.
dard deviation that defines the decision interval, H.
3.2.25.1 Discussion—If there are k subgroups, there will be
k – 1 moving ranges.
3.2.12 grandaverage(X),n—averageofsubgroupaverages.
3.2.26 subgroup proportion (p), n—proportion for the ith
i
3.2.13 inspection interval, n—a subgroup size for counts of
subgroup in a p chart.
eventsinadefinedintervaloftimespaceoranothercontinuum.
3.2.13.1 Discussion—Examples are 10 000 metres of wire 3.2.27 subgrouprange(R),n—rangeoftheobservationsfor
i
inspected for insulation defects, 100 square feet of material the ith subgroup in an R chart.
TABLE 1 Control Chart Factors
for X-Bar and R Charts for X-Bar and S Charts
nA D D d A B B c
2 3 4 2 3 3 4 4
2 1.880 0 3.267 1.128 2.659 0 3.267 0.7979
3 1.023 0 2.575 1.693 1.954 0 2.568 0.8862
4 0.729 0 2.282 2.059 1.628 0 2.266 0.9213
5 0.577 0 2.114 2.326 1.427 0 2.089 0.9400
6 0.483 0 2.004 2.534 1.287 0.030 1.970 0.9515
7 0.419 0.076 1.924 2.704 1.182 0.118 1.882 0.9594
8 0.373 0.136 1.864 2.847 1.099 0.185 1.815 0.9650
9 0.337 0.184 1.816 2.970 1.032 0.239 1.761 0.9693
10 0.308 0.223 1.777 3.078 0.975 0.284 1.716 0.9727
A
Note: for larger numbers of n, see Ref. (1).
A
The boldface numbers in parentheses refer to a list of references at the end of this standard.
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E2587 − 16 (2021)
3.2.28 subgroup size (n), n—the number of observations,
s¯ = average of the k subgroup standard deviations
i
objectsinspected,ortheinspectionintervalinthe
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