ASTM E2281-03

Standard Practice for Process and Measurement Capability Indices

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.

General Information

• Status: Historical
• Publication Date: 09-Jun-2003
• ICS: 25.040.40 - Industrial process measurement and control
• Technical Committee: E11 - Quality and Statistics
• Drafting Committee: E11.30 - Statistical Quality Control

Relations

• Replaced By: ASTM E2281-03e1 - Standard Practice for Process and Measurement Capability Indices
• Effective Date: 10-Jun-2003
• Referred By: ASTM F2554-22 - Standard Practice for Measurement of Positional Accuracy of Computer-Assisted Surgical Systems
• Effective Date: 10-Jun-2003
• Referred By: ASTM B945-21 - Standard Practice for Aluminum Alloy Extrusions Press Cooled from an Elevated Temperature Shaping Process for Production of T1, T2, T5 and T10-Type Tempers
• Effective Date: 10-Jun-2003
• Referred By: ASTM E2656-16 - Standard Practice for Real-time Release Testing of Pharmaceutical Water for the Total Organic Carbon Attribute
• Effective Date: 10-Jun-2003
• Referred By: ASTM B947-14(2020)e1 - Standard Practice for Hot Rolling Mill Solution Heat Treatment for Aluminum Alloy Plate
• Effective Date: 10-Jun-2003
• Referred By: ASTM E2475-10(2016) - Standard Guide for Process Understanding Related to Pharmaceutical Manufacture and Control
• Effective Date: 10-Jun-2003
• Referred By: ASTM E2587-16(2021)e1 - Standard Practice for Use of Control Charts in Statistical Process Control
• Effective Date: 10-Jun-2003
• Referred By: ASTM E456-13a(2022) - Standard Terminology Relating to Quality and Statistics
• Effective Date: 10-Jun-2003
• Referred By: ASTM E3239-21 - Standard Guide for Using Statistical Process Control Principles for Routine Dosimetry in Radiation Processing
• Effective Date: 10-Jun-2003
• Referred By: ASTM B807/B807M-20 - Standard Practice for Extrusion Press Solution Heat Treatment for Aluminum Alloys
• Effective Date: 10-Jun-2003
• Referred By: ASTM E3106-22 - Standard Guide for Science-Based and Risk-Based Cleaning Process Development and Validation
• Effective Date: 10-Jun-2003

Standards Content (Sample)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn. Please
contact ASTM International (www.astm.org) for the latest information.
An American National Standard
Designation: E 2281 – 03
Standard Practice for
Process and Measurement Capability Indices
This standard is issued under the fixed designation E 2281; 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 (e) indicates an editorial change since the last revision or reapproval.
1. Scope 3.2.4 lower process performance index, P , n—index de-
pkl
scribing process performance in relation to the lower specifi-
1.1 This practice provides guidance for the use of capability
cation limit.
indices for evaluating process capability and performance.
3.2.5 minimum process capability index, C , n—smaller of
Process capability indices compare the variability of a process pk
the upper process capability index and the lower process
quality measure against product specifications or tolerances
capability index.
and assume the process is in a state of statistical control.
3.2.6 minimum process performance index, P , n—smaller
pk
Process performance indices are useful in situations when the
of the upper process performance index and the lower process
process is not in a state of statistical control.
performance index.
2. Referenced Documents 3.2.7 process capability, PC, n—statistical estimate of the
outcome of a characteristic from a process that has been
2.1 ASTM Standards:
demonstrated to be in a state of statistical control.
E 456 Terminology Relating to Quality and Statistics
3.2.8 process capability index, C , n—an index describing
2.2 ISO Standard: p
process capability in relation to specified tolerance.
ISO 3534-2 Statistics—Vocabulary and Symbols-Statistical
3.2.9 process performance, PP, n—statistical measure of
Quality Control
the outcome of a characteristic from a process that may not
2.3 Other Document:
have been demonstrated to be in a state of statistical control.
MNL 7 Manual on Presentation of Data and Control Chart
3.2.10 process performance index, P , n—index describing
p
Analysis
process performance in relation to specified tolerance.
3. Terminology
3.2.11 range, R, n—the largest observation minus the small-
est observation in a set of values or observations.
3.1 Definitions—Unless otherwise noted, all statistical
3.2.12 short term standard deviation, s , n—the inherent
terms are defined in Terminology E 456. ST
variation present when a process is operating in a state of
3.2 Definitions of Terms Specific to This Standard:
statistical control, expressed in terms of standard deviation.
3.2.1 average standard deviation, s¯, n—arithmetic average
3.2.12.1 Discussion—This may also be stated as the inher-
of sample standard deviations.
ent process variation.
3.2.2 long term standard deviation, s , n—sample stan-
LT
3.2.13 special cause, n—source of intermittent variation in
dard deviation of all individual (observed) values taken over a
a process. ISO 3534-2
long period of time.
3.2.13.1 Discussion—Sometimes “special cause” is taken to
3.2.2.1 Discussion—A long period of time may be defined
be synonymous with “assignable cause.” However a distinction
as shifts, weeks, or months, etc.
should be recognized. A special cause is assignable only when
3.2.3 lower process capability index, C , n—index de-
pkl
it is specifically identified. Also a common cause may be
scribing process capability in relation to the lower specification
assignable.
limit.
3.2.13.2 Discussion—A special cause arises because of
specific circumstances which are not always present. As such,
This practice is under the jurisdiction of ASTM Committee E11 on Quality and
in a process subject to special causes, the magnitude of the
Statistics and is the direct responsibility of Subcommittee E11.30 on Data Analysis.
variation from time to time is unpredictable.
Current edition approved June 10, 2003. Published July 2003.
3.2.14 stable process, n—process in a state of statistical
Annual Book of ASTM Standards, Vol 14.02.
Available from American National Standards Institute, 11 W. 42nd St., 13th control; process condition when all special causes of variation
Floor, New York, NY 10036.
have been removed. ISO 3534-2
Available from ASTM Headquarters, 100 Barr Harbor Drive, W. Consho-
hocken, PA 19428.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn. Please
contact ASTM International (www.astm.org) for the latest information.
E2281–03
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 4.4 Process Performance Indices—When a process is not in
specification, but rather that the variation is predictable using
a state of statistical control, the process is subject to special
statistical techniques. cause variation, which can manifest itself in various ways on
3.2.14.3 Discussion—The process capability of a stable
the process variability. Special causes can give rise to changes
process is usually improved by fundamental changes that in the short-term variability of the process or can cause
reduce or remove some of the random causes present and/or
long-term shifts or drifts of the process mean. Special causes
adjusting the mean towards the preferred value. can also create transient shifts or spikes in the process mean.
3.2.14.4 Discussion—Continual adjustment of a stable pro-
Even in such cases, there may be a need to assess the long-term
cess will increase variation. variability of the process against customer specifications using
3.2.15 upper process capability index, C , n—index de-
process performance indices, which are defined in 6.2 and 6.3.
pku
scribing process capability in relation to the upper specification
These indices are similar to those for capability indices and
limit.
differ only in the estimate of variability used in the calculation.
3.2.16 upper process performance index (P ), n—index
This estimated variability includes additional components of
pku
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-
formance usually has a wider spread than the process capability
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
random and inherent to the process. Process capability is linked
terms of its variability. Process capability, PC, is calculated as:
to the use of control charts and the state of statistical control.
PC 5 6s (1)
ST
A process must be studied to evaluate its state of control before
evaluating process capability. where s is the inherent variability of a controlled process
ST
(2,7). Since control charts can be used to achieve and verify
4.2 Process Control—There are many ways to implement
control charts, but the most popular choice is to achieve a state control for many different types of processes, the assumption
of a normal distribution is not necessary to affect control, but
of statistical control for the process under study. Special causes
are identified by a set of rules based on probability theory. The complete control is required to establish the capability of a
process (2). Thus, what is required is a process in control with
process is investigated whenever the chart signals the occur-
rence of special causes. Taking appropriate actions to eliminate respect to its measures of location and spread. Once this is
identified special causes and preventing their reappearance will achieved, the inherent variability of the process can be esti-
ultimately obtain a state of statistical control. In this state, a mated from the control charts. The estimate obtained is an
minimum level of variation may be reached, which is referred estimate of variability over a short time interval (minutes,
to as common cause or inherent variation. For the purpose of hours, or a few batches). From control charts, s may be
ST
this standard, this variation is a measure of the uniformity of estimated from the short-term variation within subgroups
process output, typically a product characteristic. depending on the type of control chart deployed, for example,
¯ ¯
4.3 Process Capability Indices—The behavior of a process average-range (X − R) or individual-moving range (X − MR).
The estimate is:
(as related to inherent variability) in the state of statistical
control is used to describe its capability. To compare a process
R MR
sˆ 5 or (2)
with customer requirements (or specifications), it is common
ST
d d
2 2
practice to think of capability in terms of the proportion of the
—
¯
process output that is within product specifications or toler- where, R is the average range, MR is the average moving
ances. The metric of this proportion is the percentage of the range, d is a factor dependent on the subgroup size, n,ofthe
process spread used up by the specification. This comparison control chart, (see ASTM MNL 7, Part 3). If an average-
¯
becomes the essence of all process capability measures. The standard deviation (X − s) chart is used, the estimate becomes:
manner in which these measures are calculated defines the
s
different types of capability indices and their use. Two process sˆ 5 (3)
ST
c
capability indices are defined in 5.2 and 5.3. In practice, these
where s¯ is the average standard deviation, and c is a factor
dependent on the subgroup size, n, of the control chart, (see
5 ASTM MNL 7, Part 3).
The boldface numbers in parentheses refer to the list of references at the end of
this standard. 5.1.1 Therefore, PC is estimated by:
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn. Please
contact ASTM International (www.astm.org) for the latest information.
E2281–03
process average is from the center of the specification spread.
6R 6s
6 sˆ 5 or (4)
ST
d c In the last part of the above example (C > 1), suppose that the
2 4 p
process is actually centered above the USL. The C has a value
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
Specification Tolerance USL 2 LSL
single specification limit is involved, C is not the appropriate
C 5 5 (5)
p
p
Process Capability 6s
ST
process capability index. For these situations, the C index is
pk
where USL = upper specification limit and LSL = lower
used. C is a process capability index that considers the
pk
specification limit. For a process that is centered with an
process average against a single or double-sided specification
underlying normal distribution, Fig. 1, Fig. 2, and Fig. 3
limit. It measures whether the process is capable of meeting the
denotes three cases where PC, the process capability, is wider
customer’s requirements by considering:
than ( Fig. 1), equal to (Fig. 2), and narrower than (Fig. 3) the
5.3.2.1 The specification limit(s),
specification tolerance.
5.3.2.2 The current process average, and
5.2.2 Since the tail area of the distribution beyond specifi-
5.3.2.3 The current sˆ
ST
cation limits measures the proportion of product defectives, a
5.3.3 Under the assumption of normality, C is calculated
pk
larger value of C is better. The relationship between C and
as:
p p
the percent defective product produced by a centered process
C 5 min@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 5 min 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
C 5 (8)
pku
3 sˆ
5.2.3 From these examples, one can see that any process
ST
with a C < 1 is not as capable of meeting customer require-
p and the estimated lower process capability index is defined
ments (as indicated by % defectives) as a process with values
as:
of C $ 1. Values of C progressively greater than 1 indicate
p p
¯
X 2 LSL
more capable processes. The current focus of modern quality is ˆ
C 5 (9)
pkl
3 sˆ
ST
on process improvement with a goal of increasing product
5.3.4 These one-sided process capability indices (C and
uniformity about a target. The implementation of this focus is pku
C ) are useful in their own right with regard to single-sided
to create processes with C > 1. Some industries consider C =
pkl
p p
specification limits. Examples of this type of use would apply
1.33 (an 8s specification tolerance) a minimum with a C =
ST p
to impurities, by-products, bursting strength of bottles, etc.
1.66 preferred (3). Improvement of C should depend on a
p
Once again, the meaning of C is best viewed pictorially in
company’s quality focus, marketing plan, and their competi- pk
Fig. 4.
tor’s achievements, etc.
5.3.5 The relationship between C and C can be summa-
5.3 Process Capability Indices Adjusted For Process Shift, p pk
rized (2) as:
C :
pk
5.3.1 The above examples depict process capability for a
process centered within its specification tolerance. Process
Testing for the normality of a set of data may range from simply plotting the
centering is not a requirement since process capability is
data on a normal probability plot (2) to more forma
...