**Statistical process control (SPC)** is the use of statistical methods to assess the stability of a process and the quality of its outputs. Proper implementation of SPC can result in reduced process cycle times and improvements in yields. SPC is invaluable to companies for both a cost-reduction and customer fulfillment.

A good example of SPC in action is a bottling plant. The entire system of production that produces filled bottles is called a *process*. The weight of the liquid added to a bottle is critical for the bottling plant's cost control and customer satisfaction. The target weight for filled bottles is 250 grams, but it is acceptable if the actual filled bottle weight is between 245 and 255 grams. SPC could be used to verify that the process is occurring as desired (i.e. filled bottle weights range between 245 and 255 grams).

SPC relies on quantitative and graphic analysis of measurements to evaluate observed variation. If the attributes of interest (filled bottle weight in this example) vary within an acceptable range, a process is said to be *in control,* *in statistical control,* or *stable.* When unacceptable variation is noted (i.e. filled bottle weight is discovered to be below 245 grams or above 255 grams), actions are typically taken to determine and correct their cause. In the bottling plant example, suppose too many bottles are filled with less than 245 grams. Checking the plant equipment reveals that one of ten filler valves is failing to work.

SPC has had a wide application in manufacturing since its introduction in the 1920s and in many other kinds of repeated activities.

Much of the power of SPC lies in the ability to examine a process using tools which give weight to objective analysis over subjective opinions and which allow the strength of each source to be determined numerically. Variations in the process, which might affect the quality of the end-product or service can be detected and corrected, thus reducing waste as well as the likelihood that problems will be passed on to the customer. With its emphasis on early detection and prevention of problems, SPC has a distinct advantage over other quality methods, such as inspection, which apply resources to detecting and correcting problems after they have occurred.

In addition to subtracting waste, SPC can lead to increased efficiency and less time required to produce the product or service from end-to-end. This is partially due to a reduced likelihood that the final product will have to be reworked, but it may also result from using SPC data to identify bottlenecks, wait times, and other sources of delays within the process.

History
-------

Statistical process control was pioneered by Walter A. Shewhart in the early 1920s. Shewhart created the basis for the control chart and the concept of a state of *statistical control* by carefully designed experiments. While Dr. Shewhart drew from pure mathematical statistical theories, he understood that data from physical processes seldom produces a "normal distribution curve" (a Gaussian distribution, also commonly referred to as a "bell curve"). He discovered how the observed variation in manufacturing data did not always behave the same way as data in nature (for example, Brownian motion of particles). Dr. Shewhart concluded that while every process displays variation, some processes display controlled variation that is natural to the process (common causes of variation), while others display uncontrolled variation that is not present in the process causal system at all times (special causes of variation). Uncontrolled variation is often associated with defective products, providing a data-driven means to identify problems and improve quality.

W. Edwards Deming later applied SPC methods in the U.S. during World War II, thereby successfully improving quality in the manufacture of munitions and other strategically important products. After the war ended, he was instrumental in introducing SPC methods to Japanese industry. Deming's approach of using SPC with related management practices came to be known as a quality management system.

Application
-----------

The following description relates to manufacturing rather than to the service industry, although the principles of SPC can be successfully applied to either. For a description and example of how SPC applies to a service environment, refer to Roberts (2005). Selden describes how to use SPC in the fields of sales, marketing, and customer service, using Deming's famous Red Bead Experiment as an easy to follow demonstration.

In mass-manufacturing, the quality of the finished article was traditionally achieved through post-manufacturing inspection of the product; accepting or rejecting each article (or samples from a production lot) based on how well it met its design specifications. In contrast, Statistical Process Control uses statistical tools to observe the performance of the production process in order to predict significant deviations that may later result in rejected product.

In SPC, monitoring means that the weight of every widget (discrete product being produced) is measured and recorded; sampling means that only a few widgets (say one in a thousand) are actually weighed and statistical analyses are performed to determine the variation of the entire lot represented by the widgets that were sampled.

Two kinds of variation can occur in all manufacturing processes: both these types of process variations cause subsequent variation in the final product. The first is known as natural or common cause variation and consists of the variation inherent in the process as it is designed. Common cause variation may include variations in temperature, properties of raw materials, strength of an electrical current etc. The second kind of variation is known as special cause variation, or assignable-cause variation, and happens less frequently than the first. With sufficient investigation, a specific cause, such as abnormal raw material or incorrect set-up parameters, can be found for special cause variations.

For example, a breakfast cereal packaging line may be designed to fill each cereal box with 500 grams of product, but some boxes will have slightly more than 500 grams, and some will have slightly less, in accordance with a normal distribution of net weights. If the production process, its inputs, or its environment changes (for example, the machines doing the manufacture begin to wear) this distribution can change. In this example, as its cams and pulleys wear out, the cereal filling machine may start putting more cereal into each box than specified. If this change is allowed to continue unchecked, more and more products will be produced that fall outside of the tolerances of the manufacturer or consumer, resulting in waste. While in this case, the waste is in the form of "free" product for the consumer, typically waste consists of rework or scrap.

By watching at the right time what happened in the process that led to a change, the quality engineer or any member of the team responsible for the production line can troubleshoot the root cause of the variation that has crept in to the process and correct the problem.

SPC indicates when an action should be taken in a process, but it also indicates when NO action should be taken. An example is a person who would like to maintain a constant body weight and takes weight measurements weekly. A person who does not understand SPC concepts might start dieting every time his or her weight increased, or eat more every time his or her weight decreased. This type of action could be harmful and possibly generate even more variation in body weight. SPC would account for normal weight variation and better indicate when the person is in fact gaining or losing weight.

Basic Steps of SPC
------------------

Statistical Process Control may be broken down into three sets of activities without the minor details: understanding the process; understanding the causes of variation; and elimination of the sources of special cause variation. Key tools in SPC are control charts, a focus on continuous improvement and designed experiments.

In understanding a process, the process is typically shown (with being monitored) control charts. Control charts are used to identify variation that may be due to special causes, and to free the user from concern over variation due to common causes. This is a continuous, ongoing activity. When a process is stable and does not trigger any of the detection rules for a control chart, a process-capability analysis may also be performed to predict the ability of the current process to produce conforming (i.e. within specification) product in the future.

When excessive variation is identified by the control-chart detection rules, or the process capability is found lacking, additional effort is exerted to determine causes of that variance. The tools used include Ishikawa diagrams, designed experiments and Pareto charts. Designed experiments are critical to this phase of SPC, as they are the only means of objectively quantifying the relative importance of the many potential causes of variation.

Once the causes of variation have been quantified, effort is spent in eliminating those causes that are both statistically and practically significant (i.e. a cause that has only a small but statistically significant effect may not be considered cost-effective to fix; however, a cause that is not statistically significant can never be considered practically significant). Generally, this includes development of standard operating procedures, error-proofing, and employee training. Additional process changes may be required to reduce variation or align the process with the desired target, especially if there is a problem with process capability.

SPC and Software Quality
------------------------

In 1989, the Software Engineering Institute introduced the notion that SPC can be usefully applied to non-manufacturing processes, such as software-engineering processes, in the Capability Maturity Model (CMM). This idea exists today within the Level 4 and Level 5 practices of the Capability Maturity Model Integration (CMMI). However, this notion that SPC is a useful tool when applied to non-repetitive, knowledge-intensive processes, such as engineering processes, has encountered much skepticism, and remains controversial today. The problem lies in numerous areas of software which are not repetitive, but instead are one-time or one-off aspects of quality, rather than being observed for repeated performance over a long-term view.

Related pages
-------------

* Quality assurance
* Quality control
* ANOVA
* Sampling (statistics)
* Six sigma

Bibliography
------------

* Deming, W E (1975) On probability as a basis for action, *The American Statistician*, 29(4), pp146–152
* Deming, W E (1982) *Out of the Crisis: Quality, Productivity and Competitive Position* ISBN 0-521-30553-5
* Oakland, J (2002) *Statistical Process Control* ISBN 0-7506-5766-9
* Shewhart, W A (1931) *Economic Control of Quality of Manufactured Product* ISBN 0-87389-076-0
* Shewhart, W A (1939) *Statistical Method from the Viewpoint of Quality Control* ISBN 0-486-65232-7
* Wheeler, D J (2000) *Normality and the Process-Behaviour Chart* ISBN 0-945320-56-6
* Wheeler, D J & Chambers, D S (1992) *Understanding Statistical Process Control* ISBN 0-945320-13-2
* Wheeler, Donald J. (1999). *Understanding Variation: The Key to Managing Chaos - 2nd Edition*. SPC Press, Inc. ISBN 0-945320-53-1.
* Wise, Stephen A. & Fair, Douglas C (1998). *Innovative Control Charting: Practical SPC Solutions for Today's Manufacturing Environment.* ASQ Quality Press. ISBN 0-87389-385-9

Other websites
--------------

*Note: Before adding your company's link, please read WP:Spam#Other website spamming.*
* Working example Archived 2010-05-12 at the Wayback Machine of Deming's Red Bead Experiment
* Statistical Process Control
* Manufacturing Systems - Statistical Process Control Archived 2010-02-22 at the Wayback Machine
* MIT Course - Control of Manufacturing Processes
* NIST Engineering Statistics Handbook

| * v
* t
* e
Statistics |
| --- |
| 
* Outline
* Index

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| 

| Descriptive statistics |
| --- |
| 

|  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Continuous data | 

|  |  |
| --- | --- |
| Center | 
* Mean
	+ arithmetic
	+ geometric
	+ harmonic
	+ cubic
	+ generalized/power
* Median
* Mode

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| Dispersion | 
* Variance
* Standard deviation
* Average absolute deviation
* Coefficient of variation
* Percentile
* Range
* Interquartile range

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| Shape | 
* Central limit theorem
* Moments
	+ Skewness
	+ Kurtosis
	+ L-moments

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| Count data | 
* Index of dispersion

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| Summary tables | 
* Grouped data
* Frequency distribution
* Contingency table

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| Dependence | 
* Pearson product-moment correlation
* Rank correlation
	+ Spearman's ρ
	+ Kendall's τ
* Partial correlation
* Scatter plot

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| Graphics | 
* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Run chart
* Scatter plot
* Stem-and-leaf display
* Radar chart
* Violin plot

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| 

| Data collection |
| --- |
| 

|  |  |
| --- | --- |
| Study design | 
* Population
* Statistic
* Effect size
* Statistical power
* Optimal design
* Sample size determination
* Replication
* Missing data

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| Survey methodology | 
* Sampling
	+ stratified
	+ cluster
* Standard error
* Opinion poll
* Questionnaire

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| Controlled experiments | 
* Scientific control
* Randomized experiment
* Randomized controlled trial
* Random assignment
* Blocking
* Interaction
* Factorial experiment

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| Adaptive Designs | 
* Adaptive clinical trial
* Up-and-Down Designs
* Stochastic approximation

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| Observational Studies | 
* Cross-sectional study
* Cohort study
* Natural experiment
* Quasi-experiment

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| 

| Statistical inference |
| --- |
| 

|  |  |
| --- | --- |
| Statistical theory | 
* Population
* Statistic
* Probability distribution
* Sampling distribution
	+ Order statistic
* Empirical distribution
	+ Density estimation
* Statistical model
	+ Model specification
	+ L*p* space
* Parameter
	+ location
	+ scale
	+ shape
* Parametric family
	+ Likelihood (monotone)
	+ Location–scale family
	+ Exponential family
* Completeness
* Sufficiency
* Statistical functional
	+ Bootstrap
	+ U
	+ V
* Optimal decision
	+ loss function
* Efficiency
* Statistical distance
	+ divergence
* Asymptotics
* Robustness

 |
| Frequentist inference | 

|  |  |
| --- | --- |
| Point estimation | 
* Estimating equations
	+ Maximum likelihood
	+ Method of moments
	+ M-estimator
	+ Minimum distance
* Unbiased estimators
	+ Mean-unbiased minimum-variance
		- Rao–Blackwellization
		- Lehmann–Scheffé theorem
	+ Median unbiased
* Plug-in

 |
| Interval estimation | 
* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
	+ Bootstrap
	+ Jackknife

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| Testing hypotheses | 
* 1- & 2-tails
* Power
	+ Uniformly most powerful test
* Permutation test
	+ Randomization test
* Multiple comparisons

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| Parametric tests | 
* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

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 |
| Specific tests | 

|  |
| --- |
| 
* *Z*-test (normal)
* Student's *t*-test
* *F*-test

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| Goodness of fit | 
* Chi-squared
* *G*-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
	+ Cross validation
	+ AIC
	+ BIC

 |
| Rank statistics | 
* Sign
	+ Sample median
* Signed rank (Wilcoxon)
	+ Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
	+ 1-way (Kruskal–Wallis)
	+ 2-way (Friedman)
	+ Ordered alternative (Jonckheere–Terpstra)

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| Bayesian inference | 
* Bayesian probability
	+ prior
	+ posterior
* Credible interval
* Bayes factor
* Bayesian estimator
	+ Maximum posterior estimator

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| 

| * Correlation
* Regression analysis
 |
| --- |
| 

|  |  |
| --- | --- |
| Correlation | 
* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

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| Regression analysis | 
* Errors and residuals
* Regression validation
* Mixed effects models
* Simultaneous equations models
* Multivariate adaptive regression splines (MARS)

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| Linear regression | 
* Simple linear regression
* Ordinary least squares
* General linear model
* Bayesian regression

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| Non-standard predictors | 
* Nonlinear regression
* Nonparametric
* Semiparametric
* Isotonic
* Robust
* Heteroscedasticity
* Homoscedasticity

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| Generalized linear model | 
* Exponential families
* Logistic (Bernoulli) / Binomial / Poisson regressions

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| Partition of variance | 
* Analysis of variance (ANOVA, anova)
* Analysis of covariance
* Multivariate ANOVA
* Degrees of freedom

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| 

| Categorical / Multivariate / Time-series / Survival analysis |
| --- |
| 

|  |  |
| --- | --- |
| Categorical | 
* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test
* Cochran-Mantel-Haenszel statistics

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| Multivariate | 
* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
	+ Factor analysis
* Multivariate distributions
	+ Elliptical distributions
		- Normal

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| Time-series | 

|  |  |
| --- | --- |
| General | 
* Decomposition
* Trend
* Stationarity
* Seasonal adjustment
* Exponential smoothing
* Cointegration
* Structural break
* Granger causality

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| Specific tests | 
* Dickey–Fuller
* Johansen
* Q-statistic (Ljung–Box)
* Durbin–Watson
* Breusch–Godfrey

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| Time domain | 
* Autocorrelation (ACF)
	+ partial (PACF)
* Cross-correlation (XCF)
* ARMA model
* ARIMA model (Box–Jenkins)
* Autoregressive conditional heteroskedasticity (ARCH)
* Vector autoregression (VAR)

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| Frequency domain | 
* Spectral density estimation
* Fourier analysis
* Wavelet
* Whittle likelihood

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| Survival | 

|  |  |
| --- | --- |
| Survival function | 
* Kaplan–Meier estimator (product limit)
* Proportional hazards models
* Accelerated failure time (AFT) model
* First hitting time

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| Hazard function | 
* Nelson–Aalen estimator

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| Test | 
* Log-rank test

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| 

| Applications |
| --- |
| 

|  |  |
| --- | --- |
| Biostatistics | 
* Bioinformatics
* Clinical trials / studies
* Epidemiology
* Medical statistics

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| Engineering statistics | 
* Chemometrics
* Methods engineering
* Probabilistic design
* Process / quality control
* Reliability
* System identification

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| Social statistics | 
* Actuarial science
* Census
* Crime statistics
* Demography
* Econometrics
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

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| Spatial statistics | 
* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

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