ASTM E2281-08a(2012)e1
(Practice)Standard Practice for Process and Measurement Capability Indices
Standard Practice for Process and Measurement Capability Indices
SIGNIFICANCE AND USE
4.1 Process Capability—Process capability can be defined as the natural or inherent behavior of a stable process that is in a state of statistical control (1).5 A “state of statistical control” is achieved when the process exhibits no detectable patterns or trends, such that the variation seen in the data is believed to be random and inherent to the process. Process capability is linked to the use of control charts and the state of statistical control. A process must be studied to evaluate its state of control before evaluating process capability.
4.2 Process Control—There are many ways to implement control charts, but the most popular choice is to achieve a state of statistical control for the process under study. Special causes are identified by a set of rules based on probability theory. The process is investigated whenever the chart signals the occurrence of special causes. Taking appropriate actions to eliminate identified special causes and preventing their reappearance will ultimately obtain a state of statistical control. In this state, a minimum level of variation may be reached, which is referred to as common cause or inherent variation. For the purpose of this standard, this variation is a measure of the uniformity of process output, typically a product characteristic.
4.3 Process Capability Indices—The behavior of a process (as related to inherent variability) in the state of statistical control is used to describe its capability. To compare a process with customer requirements (or specifications), it is common practice to think of capability in terms of the proportion of the process output that is within product specifications or tolerances. The metric of this proportion is the percentage of the process spread used up by the specification. This comparison becomes the essence of all process capability measures. The manner in which these measures are calculated defines the different types of capability indices and their use. Two process capability i...
SCOPE
1.1 This practice provides guidance for the use of capability indices for evaluating process capability and performance. Process capability indices compare the variability of a process quality measure against product specifications or tolerances and assume the process is in a state of statistical control. Process performance indices are useful in situations when the process is not in a state of statistical control.
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Standards Content (Sample)
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Designation: E2281 − 08a(Reapproved 2012) An American National Standard
Standard Practice for
Process and Measurement Capability Indices
This standard is issued under the fixed designation E2281; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Editorial corrections were made to Equation 20 in October 2012.
1. Scope 3.2.2.1 Discussion—A long period of time may be defined
as shifts, weeks, or months, etc.
1.1 Thispracticeprovidesguidancefortheuseofcapability
indices for evaluating process capability and performance. 3.2.3 lower process capability index, C ,n—indexdescrib-
pkl
Process capability indices compare the variability of a process ing process capability in relation to the lower specification
limit.
quality measure against product specifications or tolerances
and assume the process is in a state of statistical control.
3.2.4 lower process performance index, P ,n—index de-
pkl
Process performance indices are useful in situations when the
scribing process performance in relation to the lower specifi-
process is not in a state of statistical control.
cation limit.
3.2.5 minimum process capability index, C ,n—smaller of
2. Referenced Documents
pk
the upper process capability index and the lower process
2.1 ASTM Standards:
capability index.
E456Terminology Relating to Quality and Statistics
3.2.6 minimum process performance index, P ,n—smaller
E2586Practice for Calculating and Using Basic Statistics
pk
of the upper process performance index and the lower process
2.2 ISO Standard:
performance index.
ISO 3534-2Statistics—Vocabulary and Symbols-Statistical
Quality Control
3.2.7 process capability, PC, n—statistical estimate of the
2.3 Other Document:
outcome of a characteristic from a process that has been
MNL 7Manual on Presentation of Data and Control Chart
demonstrated to be in a state of statistical control.
Analysis
3.2.8 process capability index, C,n—an index describing
p
process capability in relation to specified tolerance.
3. Terminology
3.2.9 process performance, PP, n—statisticalmeasureofthe
3.1 Definitions—Unlessotherwisenoted,allstatisticalterms
outcome of a characteristic from a process that may not have
are defined in Terminology E456.
been demonstrated to be in a state of statistical control.
3.2 Definitions of Terms Specific to This Standard:
3.2.10 process performance index, P,n—index describing
p
3.2.1 average standard deviation, s¯, n—arithmetic average
process performance in relation to specified tolerance.
of sample standard deviations.
3.2.11 range, R, n—maximum value minus the minimum
3.2.2 long term standard deviation, σ ,n—samplestandard
LT
value in a sample. E2586
deviation of all individual (observed) values taken over a long
period of time.
3.2.12 short term standard deviation, σ ,n—the inherent
ST
variation present when a process is operating in a state of
statistical control, expressed in terms of standard deviation.
ThispracticeisunderthejurisdictionofASTMCommitteeE11onQualityand
Statistics and is the direct responsibility of Subcommittee E11.30 on Statistical 3.2.12.1 Discussion—This may also be stated as the inher-
Quality Control.
ent process variation.
Current edition approved Oct. 1, 2012. Published October 2012. Originally
3.2.13 special cause, n—sourceofintermittentvariationina
approved in 2003. Last previous edition approved in 2008 as E2281–08a. DOI:
10.1520/E2281-08AR12E01.
process. ISO 3534-2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
3.2.13.1 Discussion—Sometimes“specialcause”istakento
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
besynonymouswith“assignablecause.”Howeveradistinction
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
should be recognized.Aspecial cause is assignable only when
Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
it is specifically identified. Also a common cause may be
4th Floor, New York, NY 10036, http://www.ansi.org.
4 assignable.
Available from ASTM Headquarters, 100 Barr Harbor Drive, W.
Conshohocken, PA 19428. 3.2.13.2 Discussion—A special cause arises because of
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
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E2281 − 08a (2012)
specific circumstances which are not always present.As such, process output that is within product specifications or toler-
in a process subject to special causes, the magnitude of the ances. The metric of this proportion is the percentage of the
variation from time to time is unpredictable. process spread used up by the specification. This comparison
becomes the essence of all process capability measures. The
3.2.14 stable process, n—process in a state of statistical
manner in which these measures are calculated defines the
control; process condition when all special causes of variation
different types of capability indices and their use.Two process
have been removed. ISO 3534-2
capability indices are defined in 5.2 and 5.3. In practice, these
3.2.14.1 Discussion—Observed variation can then be attrib-
indices are used to drive process improvement through con-
uted to random (common) causes. Such a process will gener-
tinuous improvement efforts. These indices may be used to
ally behave as though the results are simple random samples
identify the need for management actions required to reduce
from the same population.
common cause variation, compare products from different
3.2.14.2 Discussion—This state does not imply that the
sources, and to compare processes.
random variation is large or small, within or outside of
specification, but rather that the variation is predictable using
4.4 Process Performance Indices—Whenaprocessisnotin
statistical techniques. a state of statistical control, the process is subject to special
3.2.14.3 Discussion—The process capability of a stable cause variation, which can manifest itself in various ways on
process is usually improved by fundamental changes that
the process variability. Special causes can give rise to changes
reduce or remove some of the random causes present or in the short-term variability of the process or can cause
adjusting the mean towards the preferred value, or both.
long-term shifts or drifts of the process mean. Special causes
3.2.14.4 Discussion—Continual adjustment of a stable pro- can also create transient shifts or spikes in the process mean.
cess will increase variation.
Eveninsuchcases,theremaybeaneedtoassessthelong-term
variability of the process against customer specifications using
3.2.15 upper process capability index, C ,n—index de-
pku
process performance indices, which are defined in 6.2 and 6.3.
scribingprocesscapabilityinrelationtotheupperspecification
These indices are similar to those for capability indices and
limit.
differonlyintheestimateofvariabilityusedinthecalculation.
3.2.16 upper process performance index (P ), n—index
pku
This estimated variability includes additional components of
describing process performance in relation to the upper speci-
variation due to special causes. Since process performance
fication limit.
indices have additional components of variation, process per-
formanceusuallyhasawiderspreadthantheprocesscapability
4. Significance and Use
spread. These measures are useful in determining the role of
4.1 Process Capability—Process capability can be defined
measurement and sampling variability when compared to
as the natural or inherent behavior of a stable process that is in
product uniformity.
a state of statistical control (1). A“state of statistical control”
is achieved when the process exhibits no detectable patterns or 5. Process Capability Analysis
trends, such that the variation seen in the data is believed to be
5.1 It is common practice to define process behavior in
randomandinherenttotheprocess.Processcapabilityislinked
terms of its variability. Process capability, PC, is calculated as:
to the use of control charts and the state of statistical control.
PC 56σ (1)
ST
Aprocessmustbestudiedtoevaluateitsstateofcontrolbefore
evaluating process capability. where σ is the inherent variability of a controlled process
ST
(2, 3). Since control charts can be used to achieve and verify
4.2 Process Control—There are many ways to implement
control for many different types of processes, the assumption
controlcharts,butthemostpopularchoiceistoachieveastate
of a normal distribution is not necessary to affect control, but
ofstatisticalcontrolfortheprocessunderstudy.Specialcauses
complete control is required to establish the capability of a
are identified by a set of rules based on probability theory.The
process (2). Thus, what is required is a process in control with
process is investigated whenever the chart signals the occur-
respect to its measures of location and spread. Once this is
renceofspecialcauses.Takingappropriateactionstoeliminate
achieved, the inherent variability of the process can be esti-
identifiedspecialcausesandpreventingtheirreappearancewill
mated from the control charts. The estimate obtained is an
ultimately obtain a state of statistical control. In this state, a
estimate of variability over a short time interval (minutes,
minimum level of variation may be reached, which is referred
hours, or a few batches). From control charts, σ may be
ST
to as common cause or inherent variation. For the purpose of
estimated from the short-term variation within subgroups
this standard, this variation is a measure of the uniformity of
depending on the type of control chart deployed, for example,
process output, typically a product characteristic.
¯ ¯
average-range (X − R) or individual-moving range (X − MR).
4.3 Process Capability Indices—The behavior of a process
The estimate is:
(as related to inherent variability) in the state of statistical
¯ ¯
R MR
control is used to describe its capability.To compare a process
σˆ 5 or (2)
ST
d d
with customer requirements (or specifications), it is common
2 2
practice to think of capability in terms of the proportion of the
¯
where, R is the average range, MR— is the average moving
range, d is a factor dependent on the subgroup size, n,ofthe
control chart (see ASTM MNL 7, Part 3). If an average-
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
¯
this standard. standard deviation (X − s) chart is used, the estimate becomes:
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E2281 − 08a (2012)
s¯ 5.3 Process Capability Indices Adjusted For Process Shift,
σˆ 5 (3)
ST
c C :
4 pk
5.3.1 The above examples depict process capability for a
where s¯ is the average standard deviation, and c is a factor
process centered within its specification tolerance. Process
dependent on the subgroup size, n, of the control chart (see
centering is not a requirement since process capability is
ASTM MNL 7, Part 3).
independentofanyspecificationsthatmaybeappliedtoit.The
5.1.1 Therefore, PC is estimated by:
amount of shift present in a process depends on how far the
¯
6R 6s¯ process average is from the center of the specification spread.
6 σˆ 5 or (4)
ST
Inthelastpartoftheaboveexample(C >1),supposethatthe
d c
p
2 4
processisactuallycenteredabovetheUSL.TheC hasavalue
p
5.2 Process Capability Index, C :
P
>1, but clearly this process is not producing as much conform-
5.2.1 The process capability index relates the process capa-
ing product as it would have if it were centered on target.
bility to the customer’s specification tolerance. The process
5.3.2 For those cases where the process is not centered,
capability index, C , is:
p
deliberately run off-center for economic reasons, or only a
SpecificationTolerance USL 2 LSL
single specification limit is involved, C is not the appropriate
p
C 5 5 (5)
p
ProcessCapability 6σ
ST
process capability index. For these situations, the C index is
pk
used. C is a process capability index that considers the
where USL = upper specification limit and LSL = lower pk
process average against a single or double-sided specification
specification limit. For a process that is centered with an
limit.Itmeasureswhethertheprocessiscapableofmeetingthe
underlying normal distribution, Fig. 1, Fig. 2, and Fig. 3
customer’s requirements by considering:
denotes three cases where PC, the process capability, is wider
5.3.2.1 The specification limit(s),
than (Fig. 1), equal to (Fig. 2), and narrower than (Fig. 3) the
5.3.2.2 The current process average, and
specification tolerance.
5.3.2.3 The current σˆ .
5.2.2 Since the tail area of the distribution beyond specifi- ST
5.3.3 Under the assumption of normality, C is calculated
cation limits measures the proportion of product defectives, a
pk
as:
larger value of C is better. The relationship between C and
p p
the percent defective product produced by a centered process
C 5min@C , C # (6)
pk pku pkl
(with a normal distribution) is:
and is estimated by:
Percent Parts per Percent Parts per
C C
p p
Defective Million Defective Million ˆ ˆ ˆ
C 5min@ C , C # (7)
pk pku pkl
0.6 7.19 71900 1.1 0.0967 967
0.7 3.57 35700 1.2 0.0320 318
where the estimated upper process capability index is
0.8 1.64 16400 1.3 0.0096 96
defined as:
0.9 0.69 6900 1.33 0.00636 64
1.0 0.27 2700 1.67 0.00006 0.57
¯
USL 2 X
ˆ
5.2.3 From these examples, one can see that any process C 5 (8)
pku
3 σˆ
ST
with a C < 1 is not as capable of meeting customer
p
and the estimated lower process capability index is defined
requirements (as indicated by % defectives) as a process with
as:
values of C ≥ 1. Values of C progressively greater than 1
p p
indicate more capable processes. The current focus of modern
¯
X 2 LSL
ˆ
quality is on process improvement with a goal of increasing C 5 (9)
pkl
3 σˆ
ST
product uniformity about a target. The implementation of this
focus is to create processes with C > 1. Some industries
p
consider C = 1.33 (an 8σ specification tolerance) a mini-
p ST
Testing for the normality of a set of data may range from simply plotting the
mumwithaC =1.66preferred (4).ImprovementofC should
p p
data on a normal probability plot (2) to more formal tests, for example,Anderson-
dependonacompany’squalityfocus,marketingplan,andtheir
Darlingtest(whichcanbefoundinmanystatisticalsoftwareprograms,forexample,
competitor’s achievements, etc. Minitab).
FIG. 1 Process Capability Wider Than Specifications, C <1
p
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E2281 − 08a (2012)
FIG. 2 Process Capability Equal to Specification Tolerance, C =1
p
FIG. 3 Process Capability Narrower Than Specifi
...
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