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{{redirect|SQc|other uses|SQC (disambiguation)}}

{{More citations needed section|date=March 2022}}

[[File:Example Control Chart - DSE Si Etch.jpg|alt=plot showing silicon etch rate versus date, over months, with ±5% and mean values shown.|thumb|297x297px|Simple example of a process control chart, tracking the etch (removal) rate of Silicon in an [[Inductively coupled plasma|ICP]] [[Plasma etching|Plasma Etcher]] at a [[microelectronics]] [[Wafer fabrication|waferfab]].<ref>{{Cite web |last1=Dutra |first1=Noah |last2=John |first2=Demis |title=Process Group - Process Control Data - UCSB Nanofab Wiki |url=https://wiki.nanofab.ucsb.edu/wiki/Process_Group_-_Process_Control_Data |access-date=2024-11-08 |website=UCSB NanoFab Wiki}}</ref> Time-series data shows the mean value and ±5% bars. A more sophisticated SPC chart may include "control limit" & "spec limit" % lines to indicate whether/what action should be taken.]]

'''Statistical process control''' ('''SPC''') or '''statistical quality control''' ('''SQC''') is the application of [[statistics|statistical methods]] to monitor and [[quality control|control the quality]] of a [[production (economics)|production]] process. This helps to ensure that the process operates efficiently, producing more specification-conforming products with less waste (rework or [[scrap]]). SPC can be applied to any process where the "conforming product" (product meeting specifications) output can be measured. Key tools used in SPC include [[run chart]]s, [[control chart]]s, a focus on [[Continuous Improvement Process|continuous improvement]], and [[Design of experiments|the design of experiments]]. An example of a process where SPC is applied is manufacturing lines.

SPC must be practiced in two phases: Thethe first phase is the initial establishment of the process, and the second phase is the regular production use of the process. In the second phase, a decision of the period to be examined must be made, depending upon the change in 5M&E conditions (Man, Machine, Material, Method, Movement, Environment) and wear rate of parts used in the manufacturing process (machine parts, jigs, and fixtures).

'''Statistical process control''' ('''SPC''') or '''statistical quality control''' ('''SQC''') is the application of [[statistics|statistical methods]] to monitor and [[quality control|control the quality]] of a [[production (economics)|production]] process. This helps to ensure that the process operates efficiently, producing more specification-conforming products with less waste (rework or [[scrap]]). SPC can be applied to any process where the "conforming product" (product meeting specifications) output can be measured. Key tools used in SPC include [[run chart]]s, [[control chart]]s, a focus on [[Continuous Improvement Process|continuous improvement]], and [[Design of experiments|the design of experiments]]. An example of a process where SPC is applied is manufacturing lines.

SPC must be practiced in two phases: The first phase is the initial establishment of the process, and the second phase is the regular production use of the process. In the second phase, a decision of the period to be examined must be made, depending upon the change in 5M&E conditions (Man, Machine, Material, Method, Movement, Environment) and wear rate of parts used in the manufacturing process (machine parts, jigs, and fixtures).

An advantage of SPC over other methods of quality control, such as "[[inspection]]," is that it emphasizes early detection and prevention of problems, rather than the correction of problems after they have occurred.

In addition to reducing waste, SPC can lead to a reduction in the time required to produce the product. SPC makes it less likely the finished product will need to be [[Rework (electronics)|reworked]] or scrapped.

== History ==

Statistical process control was pioneered by [[Walter A. Shewhart]] at [[Bell Laboratories]] in the early 1920s. Shewhart developed the control chart in 1924 and the concept of a state of statistical control. Statistical control is equivalent to the concept of [[exchangeability]]<ref>{{harvnb|Barlow & |Irony (|1992)}}</ref><ref>{{harvnb|Bergman (|2009)}}</ref> developed by logician [[William Ernest Johnson]] also in 1924 in his book ''Logic, Part III: The Logical Foundations of Science''.<ref>{{harvnb|Zabell (|1992)}}</ref> Along with a team at AT&T that included [[Harold F. Dodge|Harold Dodge]] and Harry Romig he worked to put [[Sampling (statistics)|sampling]] inspection on a rational statistical basis as well. Shewhart consulted with Colonel Leslie E. Simon in the application of control charts to munitions manufacture at the Army's [[Picatinny Arsenal]] in 1934. That successful application helped convince Army Ordnance to engage AT&T's [[George D. Edwards]] to consult on the use of statistical quality control among its divisions and contractors at the outbreak of World War II.

[[W. Edwards Deming]] invited Shewhart to speak at the Graduate School of the U.S. Department of Agriculture and served as the editor of Shewhart's book ''Statistical Method from the Viewpoint of Quality Control'' (1939), which was the result of that lecture. Deming was an important architect of the quality control short courses that trained American industry in the new techniques during WWII. The graduates of these wartime courses formed a new professional society in 1945, the [[American Society for Quality Control]], which elected Edwards as its first president. Deming travelled to Japan during the Allied Occupation and met with the Union of Japanese Scientists and Engineers (JUSE) in an effort to introduce SPC methods to Japanese industry .<ref>{{cite book |author-link=W. Edwards Deming, |first=W. Edwards, |last=Deming |title=Lectures on statistical control of quality., |publisher=Nippon Kagaku Gijutsu Remmei, 1950|edition=Rev. 2nd |date=1952 |oclc=2518026 }}</ref><ref>Deming, W. Edwards and Dowd S. John (translator) Lecture to Japanese Management, Deming Electronic Network Web Site, 1950 (from a Japanese transcript of a lecture by Deming to "80% of Japanese top management" given at the Hotel de Yama at Mr. Hakone in August 1950)</ref>

==='Common' and 'special' sources of variation===

{{Main|Common cause and special cause (statistics)}}

Shewhart read the new statistical theories coming out of Britain, especially the work of [[William Sealy Gosset]], [[Karl Pearson]], and [[Ronald Fisher]]. However, he understood that data from physical processes seldom produced a [[normal distribution]] curve (that is, a [[Gaussian distribution]] or '[[Normal distribution|bell curve]]'). He discovered that data from measurements of variation in manufacturing did not always behave the same way as data from measurements of natural phenomena (for example, [[Brownian motion]] of particles). Shewhart concluded that while every process displays variation, some processes display variation that is natural to the process ("''common''" sources of variation); these processes he described as being ''in (statistical) control''. Other processes additionally display variation that is not present in the causal system of the process at all times ("''special''" sources of variation), which Shewhart described as ''not in control''.<ref>{{cite book |title=Why SPC? |agency=British Deming Association |publisher=SPC Press, Inc.|isbn=978-0-945320-17-3 |year=1992}}</ref>

===Application to non-manufacturing processes===

Statistical process control is appropriate to support any repetitive process, and has been implemented in many settings where for example [[ISO 9000]] quality management systems are used, including financial auditing and accounting, IT operations, health care processes, and clerical processes such as loan arrangement and administration, customer billing etc. Despite criticism of its use in design and development, it is well-placed to manage semi-automated data governance of high-volume data processing operations, for example in an enterprise data warehouse, or an enterprise data quality management system. <ref>{{cite book |first=Larry |last=English |title=Improving Data Warehouse and Business Information Quality : Methods for Reducing Costs and Increasing Profits |publisher=Wiley |date=1999 |isbn=978-0-471-25383-9 }}</ref>

In the 1988 [[Capability Maturity Model]] (CMM) the [[Software Engineering Institute]] suggested that SPC could be applied to software engineering processes. The Level 4 and Level 5 practices of the Capability Maturity Model Integration ([[CMMI]]) use this concept.

The application of SPC to non-repetitive, knowledge-intensive processes, such as research and development or systems engineering, has encountered skepticism and remains controversial.<ref>{{cite journal |first1=Bob |last1=Raczynski and [[|author2-link=Dr Bill Curtis |first2=Bill |last2=Curtis]] (2008)|title=Point/Counterpoint: Counterpoint Argument: Software Data Violate SPC's Underlying Assumptions, |journal=IEEE Software, May/June|volume=25 2008,|issue=3 Vol.|pages=49–51 25,|date=May–June No. 3,2008 pp|doi=10.1109/MS.2008.68 49-51}}</ref><ref>{{cite journal |first=Robert V. |last=Binder (1997) |title=Can a Manufacturing Quality Model Work for Software?, |journal=IEEE Software, September/October|volume=14 |issue=5 |pages=101–5 |date=September–October 1997, pp|doi=10.1109/52.605937 101-105|s2cid=40550515 }}</ref><ref>{{Cite web|last=Raczynski|first=Bob|date=February 20, 2009|title=Is Statistical Process Control Applicable to Software Development Processes?|url=https://www.stickyminds.com/article/statistical-process-control-applicable-software-development-processes|website=StickyMinds|language=en}}</ref>

In ''No Silver Bullet'', [[Fred Brooks]] points out that the complexity, conformance requirements, changeability, and invisibility of software<ref>{{Cite journal |author-link=Fred Brooks | last1 = Brooks, Jr. | first1 = F. P., J.| doi = 10.1109/MC.1987.1663532 | title = No Silver Bullet—Essence and Accidents of Software Engineering | journal = Computer | volume = 20 | issue = 4 | pages = 10–19 | year = 1987 | url = http://faculty.salisbury.edu/~xswang/Research/Papers/SERelated/no-silver-bullet.pdf| citeseerx = 10.1.1.117.315}}</ref><ref name="Brooks, Proc. IFIP" >{{cite conference |first=Fred P. |last=Brooks (1986) |title=No Silver Bullet&nbsp;— Essence and Accident in Software Engineering, Proceedings|book-title=Information processing 86: proceedings of the IFIP Tenth10th World ComputingComputer ConferenceCongress|publisher=North-Holland |date=1986, pp.|isbn=978-0-444-70077-3 1069–1076|pages=1069–76 }}</ref> results in inherent and essential variation that cannot be removed. This implies that SPC is less effective in the software development than in, e.g., manufacturing.

==Variation in manufacturing==

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From an SPC perspective, if the weight of each cereal box varies randomly, some higher and some lower, always within an acceptable range, then the process is considered stable. If the cams and pulleys of the machinery start to wear out, the weights of the cereal box might not be random. The degraded functionality of the cams and pulleys may lead to a non-random linear pattern of increasing cereal box weights. We call this common cause variation. If, however, all the cereal boxes suddenly weighed much more than average because of an unexpected malfunction of the cams and pulleys, this would be considered a special cause variation.

==Industry 4.0 and Artificial Intelligence==

The advent of Industry 4.0 has broadened the scope of statistical process control from traditional manufacturing processes to modern cyber-physical and data-driven systems. The review article of Colosimo et al. (2024)<ref>{{cite journal

|last1=Colosimo

|first1=Bianca M.

|last2=Jones-Farmer

|first2=L. Allison

|last3=Megahed

|first3=Fadel M.

|last4=Paynabar

|first4=Kamran

|last5=Ranjan

|first5=Chetan

|last6=Woodall

|first6=William H.

|title=Statistical process monitoring from Industry 2.0 to Industry 4.0: Insights into research and practice

|journal=Technometrics

|date=October 2024

|volume=66

|issue=4

|pages=507–530

|doi=10.1080/00401706.2024.2327341

|doi-access=free}}</ref> note that SPC now plays a role in monitoring complex, high-dimensional, and often automated processes that characterise Industry 4.0 environments, including the use of machine learning and artificial intelligence (AI) models in production settings.

One emerging line of research applies SPC techniques to artificial neural networks and other machine learning models. Instead of directly monitoring product quality, the focus is on the detection of unreliable behavior of AI systems. For example, nonparametric multivariate control charts have been proposed to track shifts in the distribution of neural network embeddings, allowing detection of nonstationarity and concept drift without requiring labelled data. This enables real-time monitoring of deployed AI systems in industrial contexts<ref>{{cite journal

|last1=Malinovskaya

|first1=Anna

|last2=Mozharovskyi

|first2=Pavlo

|last3=Otto

|first3=Philipp

|title=Statistical process monitoring of artificial neural networks

|journal=Technometrics

|date=January 2024

|volume=66

|issue=1

|pages=104–117

|doi=10.1080/00401706.2023.2239886

|doi-access=free}}</ref>.

==Application==

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#Eliminating assignable (special) sources of variation, so that the process is stable.

#Monitoring the ongoing production process, assisted by the use of control charts, to detect significant changes of mean or variation.

The proper implementation of SPC has been limited, in part due to a lack of statistical expertise at many organizations.<ref>{{cite journal |last1=Zwetsloot |first1=Inez M. |last2=Jones-Farmer |first2=L. Allison |last3=Woodall |first3=William H. |title=Monitoring univariate processes using control charts: Some practical issues and advice |journal=Quality Engineering |date=2 July 2024 |volume=36 |issue=3 |pages=487–499 |doi=10.1080/08982112.2023.2238049|quote=There are few areas of statistical application with a wider gap between methodological development and application than is seen in SPC (statistical process control). Many organizations in dire need of SPC are not using it at all, while most of the remainder are using methods essentially exactly as Shewhart proposed them early this century. The reasons for this are varied. One that cannot be overlooked is Deming’s observation that any procedure which requires regular intervention by an expert statistician to work properly will not be implemented.|doi-access=free }}</ref>

===Control charts===

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When monitoring many processes with control charts, it is sometimes useful to calculate quantitative measures of the stability of the processes. These metrics can then be used to identify/prioritize the processes that are most in need of corrective actions. These metrics can also be viewed as supplementing the traditional [[process capability]] metrics. Several metrics have been proposed, as described in Ramirez and Runger.<ref name="Ramarez2006">{{cite journal

| author1last1 = Ramirez, |first1=B.

| author2last2 = Runger, |first2=G.

|title= Quantitative Techniques to Evaluate Process Stability

|journal= Quality Engineering

|volume=18

|issue=1

|atpages= pp. 53–68

|year=2006 | doi = 10.1080/08982110500403581

|s2cid=109601393

}}</ref>

They are (1) a Stability Ratio which compares the long-term variability to the short-term variability, (2) an ANOVA Test which compares the within-subgroup variation to the between-subgroup variation, and (3) an Instability Ratio which compares the number of subgroups that have one or more violations of the [[Western Electric rules]] to the total number of subgroups.

==Mathematics of control charts==

Digital control charts use logic-based rules that determine "derived values" which signal the need for correction. For example,

Control charts are based on a time-ordered sequence of observations <math>X_1, X_2, \dots, X_t</math> of a process characteristic. The monitored characteristic can be single observations, averages of samples or batches, ranges, variances, or residuals from a fitted model, depending on the application.

:derived value = last value + [[Mean absolute difference|average absolute difference]] between the last N numbers.

A typical chart consists of:

* a center line (CL) representing the in-control mean, often estimated as

<math>\text{CL} = \bar{X} = \tfrac{1}{n}\sum_{i=1}^n X_i ,</math>

* control limits, usually defined as

<math>\text{UCL} = \mu_0 + k\sigma, \quad \text{LCL} = \mu_0 - k\sigma ,</math>

where <math>\mu_0</math> and <math>\sigma</math> denote the in-control mean and standard deviation, and <math>k</math> is commonly chosen as 3 (the "three-sigma rule").

An observation <math>X_t</math> falling outside the interval <math>[\text{LCL}, \text{UCL}]</math> signals a potential out-of-control condition. Variants such as the cumulative sum ([[CUSUM]]) chart and the exponentially weighted moving average charts ([[EWMA chart]]) are used to improve sensitivity to small or persistent shifts.

In many applications, however, the assumption of independent observations is violated, for example in autocorrelated time series. In such cases, the conventional control limits may produce excessive false alarms. A common solution is to fit a time series model (e.g., ARIMA) and construct a residual control chart, where the model residuals

<math>\hat{\varepsilon}_t = X_t - \hat{X}_t</math>

are monitored instead, or to adjust the control limits accordingly. Because the residuals are designed to be approximately independent and identically distributed, standard control chart theory can be applied to them. Adjusted control limits or model-based approaches are therefore required when processes exhibit dependence.

==See also==

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==Bibliography==

{{refbegin|30em}}

*Barlow, R. E. & Irony, T. Z. (1992) "Foundations of statistical quality control" in Ghosh, M. & Pathak, P.K. (eds.) ''Current Issues in Statistical Inference: Essays in Honor of D. Basu'', Hayward, CA: Institute of Mathematical Statistics, 99–112.

*{{cite book |last1=Barlow |first1=R.E. |last2=Irony |first2=T.Z. |editor-last=Ghosh |editor-first=M. |editor2-last=Pathak |editor2-first=P.K. |chapter=Foundations of statistical quality control |chapter-url={{GBurl|qddbFWqbl4YC|p=99}} |title=Current Issues in Statistical Inference: Essays in Honor of D. Basu |publisher=Institute of Mathematical Statistics |___location=Hayward, CA |date=1992 |isbn=978-0-940600-24-9 |pages=99–112 }}

*Bergman, B. (2009) "Conceptualistic Pragmatism: A framework for Bayesian analysis?", ''IIE Transactions'', '''41''', 86–93

*{{cite journal |first=B. |last=Bergman |title=Conceptualistic Pragmatism: A framework for Bayesian analysis? |journal=IIE Transactions |volume=41 |issue= |pages=86–93 |date=2009 |doi=10.1080/07408170802322713 |s2cid=119485220 }}

*Deming, W E (1975) "On probability as a basis for action", ''The American Statistician'', '''29(4)''', 146–152

*{{cite journal |author-link=W. Edwards Deming |first=W. E. |last=Deming |title=On probability as a basis for action |journal=The American Statistician |volume=29 |issue=4 |pages=146–152 |date=1975 |doi=10.1080/00031305.1975.10477402 |pmid=1078437 |s2cid=21043630 }}

*—{{cite (1982)book ''|author-mask=1 |first=W.E. |last=Deming |title=Out of the Crisis: Quality, Productivity and Competitive Position'' {{ISBN|date=1982 |isbn=0-521-30553-5 }}

*Grant,{{cite book |first=E. L. (|last=Grant |title=Statistical quality control |publisher= McGraw-Hill|___location= |date=1946) [|isbn=0-07-100447-5 |pages= |url=https://catalog.hathitrust.org/Record/003053820?type%5B%5D=all&lookfor%5B%5D=Statistical%20quality%20control%201946&filter%5B%5D=authorStr%3AGrant%2C%20Eugene%20Lodewick%2C%20b.%201897&ft= Statistical quality control] {{ISBN|0071004475}}

*Oakland, J (2002) ''Statistical Process Control'' {{ISBN|0-7506-5766-9}}

*Salacinski,{{cite Tbook (2015)|first=J. ''SPC -|last=Oakland |title=Statistical Process Control''. The|publisher= Warsaw|___location= University|date=2002 of Technology Publishing House. {{ISBN|978isbn=0-837506-78145766-319-29 }}

*{{cite book |first=T. |last=Salacinski |title=SPC — Statistical Process Control |publisher=The Warsaw University of Technology Publishing House |date=2015 |isbn=978-83-7814-319-2 }}

*Shewhart, W A (1931) ''Economic Control of Quality of Manufactured Product'' {{ISBN|0-87389-076-0}}

*{{cite book |first=W.A. |last=Shewhart |title=Economic Control of Quality of Manufactured Product |publisher= American Society for Quality Control|___location= |date=1931 |isbn=0-87389-076-0 }}

*— (1939) ''Statistical Method from the Viewpoint of Quality Control'' {{ISBN|0-486-65232-7}}

*{{Citecite book |urlauthor-mask=https://www1 |first=W.aiagA.org/quality/automotive-core-tools/spc |last=Shewhart |title=Statistical ProcessMethod Controlfrom (SPC)the ReferenceViewpoint Manualof Quality Control |publisher=Automotive Industry Action Group (AIAG)Courier Corporation|others___location= |yeardate=20051939 |editionisbn=20-486-65232-7 }}

*{{Cite book |url=https://www.aiag.org/quality/automotive-core-tools/spc |title=Statistical Process Control (SPC) Reference Manual |publisher=Automotive Industry Action Group (AIAG) |year=2005 |edition=2}}

*Wheeler, D J (2000) ''Normality and the Process-Behaviour Chart'' {{ISBN|0-945320-56-6}}

*{{cite book |first=D.J. |last=Wheeler |title=Normality and the Process-Behaviour Chart |publisher= SPC Press|___location= |date=2000 |isbn=0-945320-56-6 }}

*Wheeler, D J & Chambers, D S (1992) ''Understanding Statistical Process Control'' {{ISBN|0-945320-13-2}}

*Wheeler,{{cite Donaldbook |last1=Wheeler |first1=D.J. (1999).|last2=Chambers ''|first2=D.S.|title=Understanding Variation:Statistical TheProcess KeyControl to Managing Chaos - 2nd Edition''.|publisher= SPC Press,|___location= Inc.|date=1992 {{ISBN|isbn=0-945320-5313-12 }}.

*{{cite book |first=Donald J. |last=Wheeler |title=Understanding Variation: The Key to Managing Chaos |publisher=SPC Press |edition=2nd |date=1999 |isbn=0-945320-53-1 }}

*{{cite book |last1=Wise, |first1=Stephen A. & |last2=Fair, |first2=Douglas C (1998). ''|title=Innovative Control Charting: Practical SPC Solutions for Today's Manufacturing Environment.'' |publisher=ASQ Quality Press. {{ISBN|date=1998 |isbn=0-87389-385-9 }}

*{{ cite journal | authorlast=Zabell, |first=S. L. | title=Predicting the unpredictable | year=1992 | journal=Synthese | volume=90 | issue=2 | page=205 | doi=10.1007/bf00485351|s2cid=9416747 }}

{{refend}}

==External links==

{{Commons category|Statistical process control}}

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*[http://ocw.mit.edu/courses/mechanical-engineering/2-830j-control-of-manufacturing-processes-sma-6303-spring-2008/ MIT Course - Control of Manufacturing Processes]

*[{{cite web |url=http://www.itl.nist.gov/div898/handbook/index2.htm |title=NIST/SEMATECH Engineeringe-Handbook Statisticsof Handbook]Statistical Methods |date=2012 |doi=10.18434/M32189 |last1=Guthrie |first1=William F. |publisher=National Institute of Standards and Technology }}

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