ASTM E2587-16(2021)e1

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.
´1
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
´1
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.
´1
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.
´1
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 ithsubgroup.
(7.2.2)
T = process target value for process mean (12.1.1)
3.2.28.1 Discussion—For fixed sample sizes the symbol n is
u = counts of the observed occurrences of events in the
used. i
inspection interval divided by the size of the
3.2.29 subgroupstandarddeviation(s),n—samplestandard
i
inspection interval for the ith subgroup (10.4.2)
deviation of the observations for the ith subgroup in an s chart.
V = exponentially-weighted moving variance at time
zero (13.2.1)
3.3 Symbols:
V = exponentially-weighted moving variance statistic
i
A = factor for converting the average range to three
at time i (13.1)
X = single observation in the ith subgroup for the I
standard errors for the X-bar chart (Table 1)
i
A = factor for converting the average standard devia- chart (8.2.1)
tion to three standard errors of the average for the X = thejthobservationintheithsubgroupfortheX-bar
ij
chart (6.2.1)
X-bar chart (Table 1)
¯
B,B = factors for converting the average standard devia- = average of the individual observations over k
X
3 4
subgroups for the I chart (8.2.2)
tion to three-sigma limits for the s chart (Table 1)
* *
¯
B ,B = factors for converting the initial estimate of the = average of the ith subgroup observations for the
X
5 6 i
X-bar chart (6.2.1)
variancetothree-sigmalimitsfortheEWMVchart
= average of the k subgroup averages for the X-bar
(Table 11)
X
C = cumulative sum (CUSUM) at time zero (12.2.2) chart (6.2.2)
Y = value of the statistic being monitored by an
c = factor for converting the average standard devia-
4 i
tion to an unbiased estimate of sigma (see σ) EWMA chart at time i (11.2.1)
z = the standardized statistic for the ith subgroup
(Table 1)
i
(9.4.1.3)
c = counts of the observed occurrences of events in the
i
Z = exponentially-weighted moving average at time
ith subgroup (10.2.1)
zero (11.2.1.1)
C = cumulative sum (CUSUM) at time, i (12.1)
i
c¯ = average of the k subgroup counts (10.2.1) Z = exponentially-weighted average (EWMA) statistic
i
¯
= factor for converting the average range to an at time i (11.2.1)
d
δ = relative process shift for calculation of the allow-
estimate of sigma (see σ)(Table 1)
D,D = factors for converting the average range to three-
ance multiplier, k (12.1.5.1)
3 4
λ = factor (0 < λ < 1) which determines the weighing
sigma limits for the R chart (Table 1)
D = the squared deviation of the observation at time i of data in the EWMA statistic (11.2.1)
i
minus its forecast average (13.1) σˆ = estimated common cause standard deviation of the
h = decision interval multiplier for calculation of the process (6.2.4)
decision interval, H (12.1.5) σˆ = standard error of c, the number of observed counts
c
H = decision interval for calculation of CUSUM con- (10.2.1.2)
trol limits (12.1.5) σˆ = standard error of p, the proportion of observed
p
k = number of subgroups used in calculation of control occurrences (9.2.2.4)
ν = effective degrees of freedom for the EWMV
limits (6.2.1)
k = allowancemultiplierforcalculationof K(12.1.5.1)
(13.1.2)
K = amount of process shift to detect with a CUSUM ω = factor (0 < ω < 1) which determines the weighting
chart (12.1.5) of squared deviations in the EWMVstatistic (13.1)
MR = absolute value of the difference of the observations
i
4. Significance and Use
in the (i-1)th and the ith subgroups in a MR chart
(8.2.1) 4.1 Thispracticedescribestheuseofcontrolchartsasatool
¯
= average of the subgroup moving ranges (8.2.2.1) for use in statistical process control (SPC). Control charts were
~MR!
n = subgroup size, number of observations in a sub- developed by Shewhart (2) in the 1920s and are still in wide
group (5.1.3) use today. SPC is a branch of statistical quality control (3, 4),
n = subgroup size, number of observations (objects
which also encompasses process capability analysis and accep-
i
inspected) in the ith subgroup (9.1.2)
tance sampling inspection. Process capability analysis, as
p = proportionoftheobservedoccurrencesofeventsin
described in Practice E2281, requires the use of SPC in some
i
the ith subgroup (9.2.1)
ofitsprocedures.Acceptancesamplinginspection,describedin
p¯ = average of the k subgroup proportions (9.2.1)
PracticesE1994,E2234,andE2762,requirestheuseofSPCto
R = range of the observations in the ith subgroup for
i
minimize rejection of product.
the R chart (6.2.1.2)
4.2 Principles of SPC—Aprocess may be defined as a set of
¯
= average of the k subgroup ranges (6.2.2)
R
interrelatedactivitiesthatconvertinputsintooutputs.SPCuses
s = Sample standard deviation of the observations in
i
the ith subgroup for the s chart (7.2.1)
The boldface numbers in parentheses refer to a list of references at the end of
s = standard error of the EWMA statistic (11.2.1.2)
z
this standard.
´1
E2587 − 16 (2021)
NOTE1—SomepractitionerscombineStagesAandBintoaPhaseIand
various statistical methodologies to improve the quality of a
denote Stage C as Phase II (10).
process by reducing the variability of one or more of its
outputs, for example, a quality characteristic of a product or
5. Control Chart Principles and Usage
service.
4.2.1 Acertain amount of variability will exist in all process
5.1 One or more observations of an output characteristic are
outputs regardless of how well the process is designed or
periodically sampled from a process at a defined frequency. A
maintained. A process operating with only this inherent vari-
control chart is basically a time plot summarizing these
ability is said to be in a state of statistical control, with its
observations using a sample statistic, which is a function of the
output variability subject only to chance, or common, causes.
observations. The observations sampled at a particular time
4.2.2 Process upsets, said to be due to assignable, or special
point constitute a subgroup. Control limits are plotted on the
causes, are manifested by changes in the output level, such as
chart based on the sampling distribution of the sample statistic
a spike, shift, trend, or by changes in the variability of an
being evaluated (see 5.2 for further discussion).
output.ThecontrolchartisthebasicanalyticaltoolinSPCand
NOTE 2—Subgroup statistics commonly used are the average, range,
is used to detect the occurrence of special causes operating on
standard deviation, variance, percentage or fraction of an occurrence of an
the process.
event among multiple opportunities, or the number of occurrences during
4.2.3 When the control chart signals the presence of a a defined time period or in a defined space.
special cause, other SPC tools, such as flow charts,
5.1.1 The subgroup sampling frequency is determined by
brainstorming, cause-and-effect diagrams, or Pareto analysis,
practical considerations, such as time and cost of an
described in various references (4-8), are used to identify the
observation, the process dynamics (how quickly the output
special cause. Special causes, when identified, are either
respondstoupsets),andconsequencesofnotreactingpromptly
eliminated or controlled. When special cause variation is
toaprocessupset.Rulesfornonrandomness(see5.2.2)assume
eliminated, process variability is reduced to its inherent
that the plotted points on the chart are independent of one
variability, and control charts then function as a process
another. This shall be considered when determining the sam-
monitor. Further reduction in variation would require modifi-
plingfrequencyforthecontrolchartsdiscussedinthispractice.
cation of the process itself.
NOTE3—Samplingattoohighofafrequencymayintroducecorrelation
4.3 The use of control charts to adjust one or more process
between successive subgroups. This is referred to as autocorrelation.
inputsisnotrecommended,althoughacontrolchartmaysignal
Controlchartsthatcanhandlethistypeofcorrelationareoutsidethescope
of this practice.
the need to do so. Process adjustment schemes are outside the
scopeofthispracticeandarediscussedbyBoxandLuceño (9).
5.1.2 The sampling plan for collecting subgroup observa-
tions should be designed to minimize the variation of obser-
4.4 The role of a control chart changes as the SPC program
vations within a subgroup and to maximize variation between
evolves. An SPC program can be organized into three stages
subgroups.This is termed rational subgrouping.This gives the
(10).
best chance for the within-subgroup variation to estimate only
4.4.1 Stage A, Process Evaluation—Historical data from the
the inherent, or common-cause, process variation.
process are plotted on control charts to assess the current state
of the process, and control limits from this data are calculated
NOTE 4—For example, to obtain hourly rational subgroups of size four
for further use. See Ref. (1) for a more complete discussion on
in a product-filling operation, four bottles should be sampled within a
the use of control charts for data analysis. Ideally, it is short time span, rather than sampling one bottle every 15 min. Sampling
over 1 h allows the admission of special cause variation as a component
recommended that 100 or more numeric data points be
of within-subgroup variation.
collectedforthisstage.Forsingleobservationspersubgroupat
least 30 data points should be collected (6, 7). For attributes, a 5.1.3 The subgroup size, n, is the number of observations
total of 20 to 25 subgroups of data are recommended. At this persubgroup.Foreaseofinterpretationofthecontrolchart,the
stage, it will be difficult to find special causes, but it would be subgroup size, n, should be fixed, and this is the usual case for
useful to compile a list of possible sources for these for use in variables data (see 5.3.1). In some situations, often involving
the next stage. retrospective data, variable subgroup sizes may be
unavoidable,whichisoftenthesituationforattributesdata(see
4.4.2 Stage B, Process Improvement—Process data are col-
lected in real time and control charts, using limits calculated in 5.3.2).
StageA, are used to detect special causes for identification and 5.1.4 Subgroup Size and Average Run Length—The average
resolution. A team approach is vital for finding the sources of runlength(ARL)isameasureofhowquicklythecontrolchart
special cause variation, and process understanding will be
signals a sustained process shift of a given magnitude in the
increased. This stage is completed when further use of the output characteristic being monitored. It is defined as the
control chart indicates that a state of statistical control exists.
average number of subgroups needed to respond to a process
4.4.3 Stage C, Process Monitoring—The control chart is shift of h sigma units, where sigma is the intrinsic standard
used to monitor the process to confirm continually the state of deviation estimated by σ (see 6.2.4). The theoretical back-
statistical control and to react to new special causes entering ground for this relationship is developed in Montgomery (4),
the system or the reoccurrence of previous special causes. In andFig.1givesthecurvesrelatingARLtotheprocessshiftfor
the latter case, an out-of-control action plan (OCAP) can be selected subgroup sizes in an X-bar chart. An ARL = 1 means
developed to deal with this situation (7, 11). Update the control that the next subgroup will have a very high probability of
limits periodically or if process changes have occurred. detecting the shift.
´1
E2587 − 16 (2021)
FIG. 1 ARL for the X-Bar Chart to Detect an h-Sigma Process Shift by Subgroup Size, n
5.2 The control chart is a plot of the subgroup statistic in (2) Fifteen consecutive values are all within the 6 one-
time order. The chart also features a center line, representing
sigma limits on either side of the center line,
the time-averaged value of the statistic, and the lower and
(3) Fourteen consecutive values are alternating up and
upper control limits, that are located at 6three standard errors
down, and
of the statistic around the center line. The center line and
(4) Eight consecutive values are outside the 6 one-sigma
control limits are calculated from the process data and are not
limits.
based in any way on specification limits. The presence of a
5.2.2.3 These rules should be used judiciously since they
specialcauseisindicatedbyasubgroupstatisticfallingoutside
willincreasetheriskofafalsealarm,inwhichthecontrolchart
the control limits.
indicates lack of statistical control when only common causes
5.2.1 The use of three standard errors for control limits
are operating. The effect of using each of the rules, and groups
(so-called “three-sigma limits”) was chosen by Shewhart (2),
of these rules, on false alarm incidence is discussed by Champ
and therefore are also known as Shewhart Limits. Shewhart
and Woodall (12).
chose these limits to balance the two risks of: (1) failing to
signal the presence of a special cause when one occurs, and (2)
5.3 This practice describes the use of control charts for
occurrence of an out-of-control signal when the process is
variables and attributes data.
actually in a state of statistical control (a false alarm).
5.3.1 Variables data represent observations obtained by
5.2.2 Special cause variation may also be indicated by
observing and recording the magnitude of an output character-
certain nonrandom patterns of the plotted subgroup statistic, as
istic measured on a continuous numerical scale. Control charts
detected by using the so-called Western Electric Rules (3).To
are described for monitoring process variability and process
implement these rules, additional limits are shown on the chart
level, and these two types of charts are used as a unit for
at 6twostandarderrors(warninglimits)andat 6onestandard
process monitoring.
error (see 7.3 for example).
5.3.1.1 For multiple observations per subgroup, the sub-
5.2.2.1 Western Electric Rules—A shift in the process level
group average is the statistic for monitoring process level
is indicated if:
(X-bar chart) and either the subgroup range (R chart), or the
(1) One value falls outside either control limit,
subgroup standard deviation (s chart) is used for monitoring
(2) Two out of three consecutive values fall outside the
processvariability.Therangeiseasiertocalculateandisnearly
warning limits on the same side,
as efficient as the standard deviation for small subgroup sizes.
(3) Four out of five consecutive values fall outside the 6
The X-bar, R chart combination is discussed in Section 6. The
one-sigma limits on the same side, and
X-bar, s chart combination is discussed in Section 7.
(4) Eight consecutive values either fall above or fall below
the center line.
NOTE 5—For processes producing discrete items, a subgroup usually
5.2.2.2 Other Western Electric rules indicate less common
consists of multiple observations. The subgroup size is often five or less,
situations of nonrandom behavior:
but larger subgroup sizes may be used if measurement ease and cost are
(1) Six consecutive values in a row are steadily increasing
low. The larger the subgroup size, the more sensitive the control chart is
or decreasing (trend), to smaller shifts in the process level (see 5.1.4).
´1
E2587 − 16 (2021)
5.3.1.2 For single observations per subgroup, the subgroup 5.3.4 The CUSUM chart plots the accumulated total value
individual observation is the statistic for monitoring process of differences between the measured values or monitored
level (I chart) and the subgroup moving range is used for statistics and the predefined target or reference value as
monitoring process variability (MR chart). The I, MR chart described in Montgomery (4). The CUSUM may be calculated
combination is discussed in Section 8. for variables data using individual observations or subgroup
averages and attributes data using percent defectives or counts
NOTE 6—For batch or continuous processes producing bulk material,
of occurrences over time or space. The calculations for the
often only a single observation is taken per subgroup, as multiple
CUSUM chart are defined and discussed in Section 12.
observations would only reflect measurement variation.
5.3.4.1 The CUSUM chart is used when smaller process
5.3.2 Attributes data consist of two types: (1) observations
shifts (1 to 1.5 sigma) are of interest. The CUSUM chart
representing the frequency of occurrence of an event in the
effectively detects a sustained small shift in the process mean
subgroup, for example, the number or percentage of defective
or a slow process drift or trend. The CUSUM chart can also be
units in a subgroup of inspected units, or (2) observations
used to evaluate the direction and the magnitude of the drift
representing the count of occurrences of an event in a defined
from the process target or reference value.
interval of time or unit of space, for example, numbers of auto
5.3.5 The CUSUM chart is not very effective in detecting
accidents per month in a given region. For attributes data, the
large process shifts.Therefore, it is often used as a supplemen-
standard error of the mean is a function of the process average,
tary chart to an I chart or an X-bar chart. In this case, either the
so that only a single control chart is needed.
I chart or the X-bar chart detects larger process shifts. The
CUSUM chart detects smaller shifts (1 to 1.5 sigma) in
NOTE 7—The subgroup size for attributes data, because of their lower
process.
cost and quicker measurement, is usually much greater than for numeric
observations. Additionally, variables data contain more information than
5.4 The EWMV chart is useful for monitoring the variance
attributes data, thus requiring a smaller subgroup size.
of a process characteristic from a continuous process where
5.3.2.1 For monitoring the frequency of occurrences of an
single measurements have been taken at each time point (see
event with fixed subgroup size, the statistic is the proportion or
5.3.1.2), and the EWMV chart may be considered as an
fraction of objects having the attribute (p chart). An alternate
alternative or companion to the Moving Range chart. The
statistic is the number of occurrences for a given subgroup size
EWMV chart is based on the squared deviation of the current
(np chart) and these charts are described in Section 9. For
process observation from an estimate of the current process
monitoring with variable subgroup sizes, a modified p chart
average, which is obtained from a companion EWMA chart.
with variable control limits or a standardized control chart is
The calculations for the EWMV chart are defined and dis-
used, and these charts are also described in Section 9.
cussed in Section 13.
5.3.2.2 For monitoring the count of occurrences over a
defined time or space interval, termed the inspection interval,
6. Control Charts for Multiple Numerical Measurements
the statistic depends on whether or not the inspection interval
per Subgroup (X-Bar, R Charts)
is fixed or variable over subgroups. For a fixed inspection
6.1 Control Chart Usage—These control charts are used for
interval for all subgroups the statistic is the count (c chart); for
subgroupsconsistingofmultiplenumericalmeasurements.The
variableinspectionunitsthestatisticisthecountperinspection
X-bar chart is used for monitoring the process level, and the R
interval (u chart). Both charts are described in Section 10.
chart is used for monitoring the short-term variability. The two
5.3.3 The EWMA chart plots the exponentially weighted
charts use the same subgroup data and are used as a unit for
moving average statistic which is described by Hunter (13).
SPC purposes.
The EWMAmay be calculated for individual observations and
6.2 Control Chart Design and Calculations:
averages of multiple observations of variables data, and for
6.2.1 Denoteanobservation X ,asthe jthobservation, j=1,
ij
percent defective, or counts of occurrences over time or space
…, n,inthe ith subgroup i=1, …, k. For each of the k
for attributes data. The calculations for the EWMA chart are
subgroups, calculate the ith subgroup average,
defined and discussed in Section 11.
n
5.3.3.1 The EWMA chart is also a useful supplementary
¯
X 5 X /n 5 X 1X …1X /n (1)
~ !
i ( ij i1 i2 in
control chart to the previously discussed charts in SPC, and is j5l
a particularly good companion chart to the I chart for indi-
6.2.1.1 Averages may be rounded to one more significant
vidual observations. The EWMA reacts more quickly to
figure than the data.
smaller shifts in the process characteristic, on the order of 1.5
6.2.1.2 For each of the k subgroups, calculate the ith
standard errors or less, whereas the Shewhart-based charts are
subgroup range, the difference between the largest and the
moresensitivetolargershifts.ExamplesoftheEWMAchartas
smallest observation in the subgroup.
a supplementary chart are given in 11.4 and Appendix X1.
R 5 Max X , … , X 2 Min X , …, X (2)
~ ! ~ !
i i1 in i1 in
5.3.3.2 The EMWAchart is also used in process adjustment
schemes where the EWMA statistic is used to locate the local 6.2.1.3 The averages and ranges are plotted as dots on the
mean of a non-stationary process and as a forecast of the next X-bar chart and the R chart, respectively. The dots may be
observation from the process. This usage is beyond the scope connected by lines, if desired.
ofthispracticebutisdiscussedbyBoxandPaniagua-Quiñones 6.2.2 Calculate the grand average and the average range
(14) and by Lucas and Saccucci (15). over all k subgroups:
´1
E2587 − 16 (2021)
k
6.3 Example—Liquid Product Filling into Bottles—At a
% ¯ ¯ ¯ ¯
X 5 X /k 5 ~X 1X 1…1X !/k (3)
( i 1 2 k
frequency of 30 min, four consecutive bottles are pulled from
i5l
the filling line and weighed. The observations, subgroup
k
¯
R 5 R /k 5 ~R 1R 1…1R !/k (4) averages, and subgroup ranges are listed in Table 2, and the
( i 1 2 k
i5l
grand average and average range are calculated at the bottom
6.2.2.1 These values are used for the center lines on the
of the table.
control chart, which are usually depicted as solid lines on the
6.3.1 The control limits are calculated as follows:
control chart, and may be rounded to one more significant
6.3.1.1 X-Bar Chart:
figure than the data.
LCL 5 246.44 2 ~0.729!~5.92! 5 242.12
6.2.3 UsingthecontrolchartfactorsinTable1,calculatethe
UCL 5 246.441~0.729!~5.92! 5 250.76
lower control limits (LCL) and upper control limits (UCL) for
the two charts.
6.3.1.2 R Chart:
6.2.3.1 For the X-Bar Chart:
LCL 5 0 5.92 5 0
~ !~ !
% ¯
LCL 5 X 2 A R (5)
UCL 5 2.282 5.92 5 13.51
~ !~ !
% ¯
UCL 5 X1A R (6)
6.3.1.3 Estimate of inherent standard deviation:
6.2.3.2 For the R Chart:
σ 5 5.92/2.059 5 2.87
¯
LCL 5 D R (7)
3 6.3.1.4 The control charts are shown in Fig. 2 and Fig. 3
Both charts indicate that the filling weights are in statistical
¯
UCL 5 D R (8)
control.
6.2.3.3 The control limits are usually depicted as dashed
lines on the control chart.
7. Control Charts for Multiple Numerical Measurements
6.2.4 An estimate of the inherent (common cause) standard
per Subgroup (X-Bar, s Charts)
deviation may be calculated as follows:
7.1 Control Chart Usage—These control charts are used for
¯
σˆ 5 R/d (9) subgroups consisting of multiple numerical measurements, the
X-bar chart for monitoring the process level, and the s chart for
6.2.4.1 This estimate is useful in process capability studies
monitoring the short-term variability. The two charts use the
(see Practice E2281).
same subgroup data and are used as a unit for SPC purposes.
6.2.5 Subgroupstatisticsfallingoutsidethecontrollimitson
7.2 Control Chart Design and Calculations:
the X-bar chart or the R chart indicate the presence of a special
cause. The Western Electric Rules may also be applied to the 7.2.1 Denoteanobservation X ,asthe jthobservation, j=1,
ij
X-bar and R chart (see 5.2.2). …, n,inthe ith subgroup, i=1, …, k. For each of the k
TABLE 2 Example of X-Bar, R Chart for Bottle-Filling Operation
Subgroup Bottle 1 Bottle 2 Bottle 3 Bottle 4 Average Range
1 246.5 250.7 246.1 250.2 248.38 4.6
2 246.5 243.7 241.7 248.0 244.98 6.3
3 246.5 243.3 250.1 243.5 245.85 6.8
4 246.5 248.5 250.5 242.0 246.88 8.5
5 246.5 242.9 248.0 249.4 246.70 6.5
6 246.7 250.6 246.0 246.1 247.35 4.6
7 246.6 247.3 251.6 248.8 248.58 5.0
8 246.5 249.6 246.6 243.6 246.58 6.0
9 246.4 251.1 247.7 245.5 247.68 5.6
10 246.4 245.7 245.8 247.0 246.23 1.3
11 246.5 242.6 241.5 248.3 244.73 6.8
12 246.4 247.3 244.1 243.3 245.28 4.0
13 246.4 250.1 249.0 245.3 247.70 4.8
14 246.3 247.8 239.4 245.7 244.80 8.4
15 246.6 242.7 244.1 249.7 245.78 7.0
16 246.6 248.4 246.8 251.0 248.20 4.4
17 246.4 246.0 250.3 246.2 247.23 4.3
18 246.5 250.2 243.2 246.9 246.70 7.0
19 246.4 247.5 246.6 244.8 246.33 2.7
20 246.3 248.4 244.6 244.9 246.05 3.8
21 246.5 244.7 243.0 248.0 245.55 5.0
22 246.6 249.2 250.5 242.6 247.23 7.9
23 246.5 249.7 240.7 246.7 245.90 9.0
24 246.6 244.0 238.5 243.0 243.03 8.1
25 246.5 251.5 248.9 242.0 247.23 9.5
Grand average 246.44
Average range 5.92
´1
E2587 − 16 (2021)
FIG. 2 X-Bar Chart for Filling Line 3
FIG. 3 Range Chart for Filling Line 3
subgroups, calculate the ith subgroup average and the ith 7.2.2.1 These values are used for the center lines on the
subgroup standard deviation:
control chart, usually depicted as solid lines, and may be
n
rounded to the same number of significant figures as the
¯
X 5 X /n 5 ~X 1X 1…1X !/n (10)
subgroup statistics.
i ( ij i1 i2 in
j5l
7.2.3 UsingthecontrolchartfactorsinTable1,calculatethe
n
¯ LCL and UCL for the two charts.
s 5 ~X 2 X ! / n 2 1 (11)
Π~ !
i ij i
(
j5l
7.2.3.1 For the X-Bar Chart:
7.2.1.1 Averages may be rounded to one more significant
%
LCL 5 X 2 A s¯ (14)
figure than the data.
7.2.1.2 Sample standard deviations may be rounded to two %
UCL 5 X1A s¯ (15)
or three significant figures.
7.2.3.2 For the s Chart:
7.2.1.3 The averages and standard deviations are plotted as
dots on the X-bar chart and the s chart, respectively. The dots
LCL 5 B s¯ (16)
may be connected by lines, if desired.
UCL 5 B s¯ (17)
7.2.2 Calculate the grand average and the average standard
deviation over all k subgroups:
7.2.3.3 The control limits are usually depicted by dashed
k
lines on the control charts.
% ¯ ¯ ¯ ¯
~ !
X 5 X /k 5 X 1X 1…1X /k (12)
( i 1 2 k
i51 7.2.4 An estimate of the inherent (common cause) standard
k
deviation may be calculated as follows:
s¯ 5 s /k 5 ~s 1s 1…1s !/k (13)
( i 1 2 k
i51 σˆ 5 s¯/c (18)
´1
E2587 − 16 (2021)
7.2.5 Subgroupstatisticsfallingoutsidethecontrollimitson 7.3.4.3 Subgroups 6, 7, and 8—End points of six points in a
the X-bar chart or the s chart indicate the presence of a special row steadily increasing.
cause.
7.3.4.4 Subgroup 10—Four out of five points on the same
side of the upper one-sigma limits.
7.3 Example—Vitamin tablets are compressed from blended
7.3.5 It appears that the process level has been steadily
granulated powder and tablet hardness is measured on ten
increasing during the run. Some possible special causes are
tablets each hour. The observations, subgroup averages, and
particle segregation in the feed hopper or a drift in the press
subgroup standard deviations are listed in Table 3.
settings.
7.3.1 The control limits are calculated as follows:
7.3.1.1 X-Bar Chart:
8. Control Charts for Single Numerical Measurements
LCL 5 24.141 2 0.975 1.352 5 22.823
~ !~ !
per Subgroup (I, MR Charts)
UCL 5 24.1411 0.975 1.352 5 25.459
~ !~ !
8.1 Control Chart Usage—These control charts are used for
7.3.1.2 The s Chart:
subgroups consisting of a single numerical measurement. The
LCL 5 0.284 1.352 5 0.384 I chart is used for monitoring the process level and the MR
~ !~ !
chart is used for monitoring the short-term variability. The two
UCL 5 1.716 1.352 5 2.320
~ !~ !
charts are used as a unit for SPC purposes, although some
7.3.2 The two-sigma warning limits and the one-sigma
practitioners state that the MR chart does not add value and
limitsarealsocalculatedforthe X-barcharttoillustratetheuse
recommend against its use other than calculating the control
of the Western Electric Rules.
limits for the I chart (16).
7.3.3 The warning limits and one-sigma limits for the X-bar
8.2 Control Chart Design and Calculations:
chart were calculated as follows.
8.2.1 Denote the observation, X, as the individual observa-
7.3.3.1 Warning Limits: i
tion in the ith subgroup, i=1, 2,…, k.
LCL 5 24.141 2 2~0.975!~1.352!/3 5 23.262
8.2.1.1 Note that the first subgroup will not have a moving
UCL 5 24.14112 0.975 1.352 /3 5 25.020
~ !~ !
range. For the k–1 subgroups, i = 2, …, k calculate the moving
range, the absolute value of the difference between two
7.3.3.2 One-Sigma Limits:
successive values:
LCL 5 24.141 2 ~0.975!~1.352!/3 5 23.702
MR 5 X 2 X (19)
i ? i i21?
UCL 5 24.1411 0.975 1.352 /3 5 24.580
~ !~ !
8.2.1.2 The individual values and moving ranges are plotted
7.3.3.3 Estimate of Inherent Standard Deviation:
as dots on the I chart and the MR chart, respectively. The dots
σˆ 5 1.352/0.9727 5 1.39
may be connected by lines, if desired.
8.2.2 Calculate the average of the observations over all k
7.3.3.4 The control charts are shown in Fig. 4 and Fig. 5.
The s chart indicates statistical control in the process variation. subgroups:
7.3.4 The X-Bar Chart Gives Several Out-of-Control Sig-
k
¯
X 5 X /k 5 X 1X 1…1X /k (20)
nals: ~ !
i 1 2 k
(
i51
7.3.4.1 Subgroup 1—Below the LCL.
7.3.4.2 Subgroups 2 and 3—Two points outside the warning Also calculate the average moving range for the k–1 sub-
limit on the same side. groups:
TABLE 3 Example of X-Bar, S Chart for Tablet Hardness
Subgroup T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Avg Std
1 21.3 19.5 21.3 23.1 22.4 24.6 23.4 22.4 21.4 22.9 22.23 1.419
2 21.4 22.2 22.1 23.3 23.9 22.9 21.6 24.6 25.7 24.1 23.18 1.399
3 23.9 24.2 22.8 22.9 25.9 21.4 23.1 20.5 23.6 23.8 23.21 1.494
4 23.4 26.3 24.4 25.3 22.0 25.8 22.7 26.5 21.6 25.0 24.30 1.781
5 25.6 22.9 24.6 23.8 23.6
...