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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:LightPink;">Control chart</th></tr><tr><td colspan="2" class="infobox-image"></td></tr><tr><th colspan="2" class="infobox-header" style="background:LightPink;font-size:110%;">One of the <b><a href="Seven_basic_tools_of_quality" title="Seven basic tools of quality">Seven basic tools of quality</a></b></th></tr><tr><th scope="row" class="infobox-label">First described by</th><td class="infobox-data"><a href="Walter_A._Shewhart" title="Walter A. Shewhart">Walter A. Shewhart</a></td></tr><tr><th scope="row" class="infobox-label">Purpose</th><td class="infobox-data">To determine whether a process should undergo a formal examination for quality-related problems</td></tr></tbody></table>

<p><b>Control charts</b> are <a href="Plot_(graphics)" title="Plot (graphics)">graphical plots</a> used in production control to determine whether quality and <a href="Manufacturing" title="Manufacturing">manufacturing</a> processes are being <a href="Statistical_Process_Control" class="mw-redirect" title="Statistical Process Control">controlled</a> under stable conditions. (ISO 7870-1)<sup id="cite_ref-ISO7870-1_1-0" class="reference"><a href="#cite_note-ISO7870-1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

The hourly status is arranged on the graph, and the occurrence of abnormalities is judged based on the presence of data that differs from the conventional trend or deviates from the control limit line.

Control charts are classified into <a href="Shewhart_individuals_control_chart" title="Shewhart individuals control chart">Shewhart individuals control chart</a> (ISO 7870-2)<sup id="cite_ref-ISO7870-2_2-0" class="reference"><a href="#cite_note-ISO7870-2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and <a href="CUSUM" title="CUSUM">CUSUM</a>(CUsUM)(or cumulative sum control chart)(ISO 7870-4).<sup id="cite_ref-ISO7870-4_3-0" class="reference"><a href="#cite_note-ISO7870-4-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Control charts, also known as <b>Shewhart charts</b> (after <a href="Walter_A._Shewhart" title="Walter A. Shewhart">Walter A. Shewhart</a>) or <b>process-behavior charts</b>, are a <a href="Statistical_process_control" title="Statistical process control">statistical process control</a> tool used to determine if a <a href="Manufacturing_process_management" title="Manufacturing process management">manufacturing</a> or <a href="Business_process" title="Business process">business process</a> is in a state of <a href="Statistical_control" class="mw-redirect" title="Statistical control">control</a>. It is more appropriate to say that the control charts are the graphical device for statistical process monitoring (SPM). Traditional control charts are mostly designed to monitor process parameters when the underlying form of the process distributions are known. However, more advanced techniques are available in the 21st century where incoming data streaming can-be monitored even without any knowledge of the underlying process distributions. <a href="Distribution-free_control_chart" title="Distribution-free control chart">Distribution-free control charts</a> are becoming increasingly popular.

</p>

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

<p>If analysis of the control chart indicates that the process is currently under control (i.e., is stable, with variation only coming from sources common to the process), then no corrections or changes to process control parameters are needed or desired. In addition, data from the process can be used to <a href="Predictive_analytics" title="Predictive analytics">predict</a> the future performance of the process. If the chart indicates that the monitored process is not in control, analysis of the chart can help determine the sources of <a href="Statistical_dispersion" title="Statistical dispersion">variation</a>, as this will result in degraded process performance.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A process that is stable but operating outside desired (specification) limits (e.g., scrap rates may be in statistical control but above desired limits) needs to be improved through a deliberate effort to understand the causes of current performance and fundamentally improve the process.<sup id="cite_ref-WheelerUV_5-0" class="reference"><a href="#cite_note-WheelerUV-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>The control chart is one of the <a href="Seven_basic_tools_of_quality" title="Seven basic tools of quality">seven basic tools</a> of <a href="Quality_control" title="Quality control">quality control</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Typically control charts are used for <a href="Time_series" title="Time series">time-series</a> data, also known as <a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">continuous</a> data or variable data. Although they can also be used for data that has logical comparability (i.e. you want to compare samples that were taken all at the same time, or the performance of different individuals); however the type of chart used to do this requires consideration.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The control chart was invented by <a href="Walter_A._Shewhart" title="Walter A. Shewhart">Walter A. Shewhart</a> working for <a href="Bell_Labs" title="Bell Labs">Bell Labs</a> in the 1920s.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The company's engineers had been seeking to improve the reliability of their <a href="Telephony" title="Telephony">telephony</a> transmission systems. Because <a href="Amplifier" title="Amplifier">amplifiers</a> and other equipment had to be buried underground, there was a stronger business need to reduce the frequency of failures and repairs. By 1920, the engineers had already realized the importance of reducing variation in a manufacturing process. Moreover, they had realized that continual process-adjustment in reaction to non-conformance actually increased variation and degraded quality. Shewhart framed the problem in terms of <a href="Common-_and_special-causes" class="mw-redirect" title="Common- and special-causes">common- and special-causes</a> of variation and, on May 16, 1924, wrote an internal memo introducing the control chart as a tool for distinguishing between the two. Shewhart's boss, George Edwards, recalled: "Dr. Shewhart prepared a little memorandum only about a page in length. About a third of that page was given over to a simple diagram which we would all recognize today as a schematic control chart. That diagram, and the short text which preceded and followed it set forth all of the essential principles and considerations which are involved in what we know today as process quality control."<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Shewhart stressed that bringing a production process into a state of <a href="Statistical_control" class="mw-redirect" title="Statistical control">statistical control</a>, where there is only common-cause variation, and keeping it in control, is necessary to predict future output and to manage a process economically.

</p><p>Shewhart created the basis for the control chart and the concept of a state of statistical control by carefully designed experiments. While Shewhart drew from pure mathematical statistical theories, he understood that data from physical processes typically produce a "<a href="Normal_distribution" title="Normal distribution">normal distribution</a> curve" (a <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a>, also commonly referred to as a "<a href="Normal_distribution" title="Normal distribution">bell curve</a>"). He discovered that observed variation in manufacturing data did not always behave the same way as data in nature (<a href="Brownian_motion" title="Brownian motion">Brownian motion</a> of particles). Shewhart concluded that while every process displays variation, some processes display controlled variation that is natural to the process, while others display uncontrolled variation that is not present in the process causal system at all times.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>In 1924, or 1925, Shewhart's innovation came to the attention of <a href="W._Edwards_Deming" title="W. Edwards Deming">W. Edwards Deming</a>, then working at the <a href="Hawthorne_facility" class="mw-redirect" title="Hawthorne facility">Hawthorne facility</a>. Deming later worked at the <a href="United_States_Department_of_Agriculture" title="United States Department of Agriculture">United States Department of Agriculture</a> and became the mathematical advisor to the <a href="United_States_Census_Bureau" title="United States Census Bureau">United States Census Bureau</a>. Over the next half a century, Deming became the foremost champion and proponent of Shewhart's work. After the defeat of <a href="Japan" title="Japan">Japan</a> at the close of <a href="World_War_II" title="World War II">World War II</a>, Deming served as statistical consultant to the <a href="Supreme_Commander_for_the_Allied_Powers" title="Supreme Commander for the Allied Powers">Supreme Commander for the Allied Powers</a>. His ensuing involvement in Japanese life, and long career as an industrial consultant there, spread Shewhart's thinking, and the use of the control chart, widely in Japanese manufacturing industry throughout the 1950s and 1960s.

</p><p>Bonnie Small worked in an Allentown plant in the 1950s after the <a href="Transistor" title="Transistor">transistor</a> was made. Used Shewhart's methods to improve plant performance in quality control and made up to 5000 control charts. In 1958, <i>The Western Electric Statistical Quality Control Handbook</i> had appeared from her writings and led to use at AT&T.<sup id="cite_ref-Best_&_Neuhauser_2006_11-0" class="reference"><a href="#cite_note-Best_&_Neuhauser_2006-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Chart_details">Chart details</h2></div>

<p>A control chart consists of:

</p>

<ul><li>Points representing a statistic (e.g., a <a href="Mean" title="Mean">mean</a>, range, proportion) of measurements of a quality characteristic in samples taken from the process at different times (i.e., the data)</li>

<li>The mean of this statistic using all the samples is calculated (e.g., the mean of the means, mean of the ranges, mean of the proportions) - or for a reference period against which change can be assessed. Similarly a median can be used instead.</li>

<li>A centre line is drawn at the value of the mean or median of the statistic</li>

<li>The <a href="Standard_deviation" title="Standard deviation">standard deviation</a> (e.g., sqrt(variance) of the mean) of the statistic is calculated using all the samples - or again for a reference period against which change can be assessed. in the case of XmR charts, strictly it is an approximation of standard deviation, the does not make the assumption of homogeneity of process over time that the standard deviation makes.</li>

<li>Upper and lower <a href="Control_limits" class="mw-redirect" title="Control limits">control limits</a> (sometimes called "natural process limits") that indicate the threshold at which the process output is considered statistically 'unlikely' and are drawn typically at 3 standard deviations from the center line</li></ul>

<p>The chart may have other optional features, including:

</p>

<ul><li>More restrictive upper and lower warning or control limits, drawn as separate lines, typically two standard deviations above and below the center line. This is regularly used when a process needs tighter controls on variability.</li>

<li>Division into zones, with the addition of rules governing frequencies of observations in each zone</li>

<li>Annotation with events of interest, as determined by the Quality Engineer in charge of the process' quality</li>

<li>Action on special causes</li></ul>

<p>(n.b., there are several rule sets for detection of signal; this is just one set. The rule set should be clearly stated.)

</p>

<ol><li>Any point outside the control limits</li>

<li>A Run of 7 Points all above or all below the central line - Stop the production

<ul><li>Quarantine and 100% check</li>

<li>Adjust Process.</li>

<li>Check 5 Consecutive samples</li>

<li>Continue The Process.</li></ul></li>

<li>A Run of 7 Point Up or Down - Instruction as above</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Chart_usage">Chart usage</h3></div>

<p>If the process is in control (and the process statistic is normal), 99.7300% of all the points will fall between the control limits. Any observations outside the limits, or systematic patterns within, suggest the introduction of a new (and likely unanticipated) source of variation, known as a <a href="Common-_and_special-causes" class="mw-redirect" title="Common- and special-causes">special-cause</a> variation. Since increased variation means increased <a href="Quality_costs" title="Quality costs">quality costs</a>, a control chart "signaling" the presence of a special-cause requires immediate investigation.

</p><p>This makes the control limits very important decision aids. The control limits provide information about the process behavior and have no intrinsic relationship to any <a href="Specification" class="mw-redirect" title="Specification">specification</a> targets or <a href="Engineering_tolerance" title="Engineering tolerance">engineering tolerance</a>. In practice, the process mean (and hence the centre line) may not coincide with the specified value (or target) of the quality characteristic because the process design simply cannot deliver the process characteristic at the desired level.

</p><p>Control charts limit <a href="Specification_(technical_standard)" title="Specification (technical standard)">specification limits</a> or targets because of the tendency of those involved with the process (e.g., machine operators) to focus on performing to specification when in fact the least-cost course of action is to keep process variation as low as possible. Attempting to make a process whose natural centre is not the same as the target perform to target specification increases process variability and increases costs significantly and is the cause of much inefficiency in operations. <a href="Process_capability" title="Process capability">Process capability</a> studies do examine the relationship between the natural process limits (the control limits) and specifications, however.

</p><p>The purpose of control charts is to allow simple detection of events that are indicative of an increase in process variability.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This simple decision can be difficult where the process characteristic is continuously varying; the control chart provides statistically objective criteria of change. When change is detected and considered good its cause should be identified and possibly become the new way of working, where the change is bad then its cause should be identified and eliminated.

</p><p>The purpose in adding warning limits or subdividing the control chart into zones is to provide early notification if something is amiss. Instead of immediately launching a process improvement effort to determine whether special causes are present, the Quality Engineer may temporarily increase the rate at which samples are taken from the process output until it is clear that the process is truly in control. Note that with three-sigma limits, <a href="Common-_and_special-causes" class="mw-redirect" title="Common- and special-causes">common-cause</a> variations result in signals less than once out of every twenty-two points for skewed processes and about once out of every three hundred seventy (1/370.4) points for normally distributed processes.<sup id="cite_ref-Wheeler112010_13-0" class="reference"><a href="#cite_note-Wheeler112010-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The two-sigma warning levels will be reached about once for every twenty-two (1/21.98) plotted points in normally distributed data. (For example, the means of sufficiently large samples drawn from practically any underlying distribution whose variance exists are normally distributed, according to the Central Limit Theorem.)

</p>

<div class="mw-heading mw-heading3"><h3 id="Choice_of_limits">Choice of limits</h3></div>

<p>Shewhart set <i>3-sigma</i> (3-standard deviation) limits on the following basis.

</p>

<ul><li>The coarse result of <a href="Chebyshev's_inequality" title="Chebyshev's inequality">Chebyshev's inequality</a> that, for any <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, the <a href="Probability" title="Probability">probability</a> of an outcome greater than <i>k</i> <a href="Standard_deviation" title="Standard deviation">standard deviations</a> from the <a href="Mean" title="Mean">mean</a> is at most 1/<i>k</i><sup>2</sup>.</li>

<li>The finer result of the <a href="Vysochanskii%E2%80%93Petunin_inequality" class="mw-redirect" title="Vysochanskii–Petunin inequality">Vysochanskii–Petunin inequality</a>, that for any <a href="Unimodal_probability_distribution" class="mw-redirect" title="Unimodal probability distribution">unimodal probability distribution</a>, the <a href="Probability" title="Probability">probability</a> of an outcome greater than <i>k</i> <a href="Standard_deviation" title="Standard deviation">standard deviations</a> from the <a href="Mean" title="Mean">mean</a> is at most 4/(9<i>k</i><sup>2</sup>).</li>

<li>In the <a href="Normal_distribution" title="Normal distribution">Normal distribution</a>, a very common <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, 99.7% of the observations occur within three <a href="Standard_deviation" title="Standard deviation">standard deviations</a> of the <a href="Mean" title="Mean">mean</a> (see <a href="Normal_distribution#Standard_deviation_and_coverage" title="Normal distribution">Normal distribution</a>).</li></ul>

<p>Shewhart summarized the conclusions by saying:

</p>

<p><i>... the fact that the criterion which we happen to use has a fine ancestry in highbrow statistical theorems does not justify its use. Such justification must come from empirical evidence that it works. As the practical engineer might say, the proof of the pudding is in the eating.</i><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p>

</blockquote>

<p>Although he initially experimented with limits based on <a href="Probability_distribution" title="Probability distribution">probability distributions</a>, Shewhart ultimately wrote:

</p>

<p><i>Some of the earliest attempts to characterize a state of statistical control were inspired by the belief that there existed a special form of frequency function</i> f <i>and it was early argued that the normal law characterized such a state. When the normal law was found to be inadequate, then generalized functional forms were tried. Today, however, all hopes of finding a unique functional form</i> f <i>are blasted.</i><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

</blockquote>

<p>The control chart is intended as a <a href="Heuristic" title="Heuristic">heuristic</a>. <a href="W._Edwards_Deming" title="W. Edwards Deming">Deming</a> insisted that it is not a <a href="Hypothesis_test" class="mw-redirect" title="Hypothesis test">hypothesis test</a> and is not motivated by the <a href="Neyman%E2%80%93Pearson_lemma" title="Neyman–Pearson lemma">Neyman–Pearson lemma</a>. He contended that the disjoint nature of <a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">population</a> and <a href="Sampling_frame" title="Sampling frame">sampling frame</a> in most industrial situations compromised the use of conventional statistical techniques. <a href="W._Edwards_Deming" title="W. Edwards Deming">Deming</a>'s intention was to seek insights into the <a href="Cause_system" class="mw-redirect" title="Cause system">cause system</a> of a process <i>...under a wide range of unknowable circumstances, future and past....</i> He claimed that, under such conditions, <i>3-sigma</i> limits provided <i>... a rational and economic guide to minimum economic loss...</i> from the two errors:

</p>

<ol><li><i>Ascribe a variation or a mistake to a special cause (assignable cause) when in fact the cause belongs to the system (common cause).</i> (Also known as a <a href="Type_I_and_type_II_errors" title="Type I and type II errors">Type I error</a> or False Positive)</li>

<li><i>Ascribe a variation or a mistake to the system (common causes) when in fact the cause was a special cause (assignable cause).</i> (Also known as a <a href="Type_I_and_type_II_errors" title="Type I and type II errors">Type II error</a> or False Negative)</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Calculation_of_standard_deviation">Calculation of standard deviation</h3></div>

<p>As for the calculation of control limits, the <a href="Standard_deviation" title="Standard deviation">standard deviation</a> (error) required is that of the <a href="Common-_and_special-causes" class="mw-redirect" title="Common- and special-causes">common-cause</a> variation in the process. Hence, the usual <a href="Estimator" title="Estimator">estimator</a>, in terms of sample variance, is not used as this estimates the total squared-error loss from both <a href="Common-_and_special-causes" class="mw-redirect" title="Common- and special-causes">common- and special-causes</a> of variation.

</p><p>An alternative method is to use the relationship between the <a href="Range_(statistics)" title="Range (statistics)">range</a> of a sample and its <a href="Standard_deviation" title="Standard deviation">standard deviation</a> derived by <a href="Leonard_Henry_Caleb_Tippett" class="mw-redirect" title="Leonard Henry Caleb Tippett">Leonard H. C. Tippett</a>, as an estimator which tends to be less influenced by the extreme observations which typify <a href="Common-_and_special-causes" class="mw-redirect" title="Common- and special-causes">special-causes</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Rules_for_detecting_signals">Rules for detecting signals</h2></div>

<p>The most common sets are:

</p>

<ul><li>The <a href="Western_Electric_rules" title="Western Electric rules">Western Electric rules</a></li>

<li>The <a href="Donald_J._Wheeler" title="Donald J. Wheeler">Wheeler</a> rules (equivalent to the Western Electric zone tests<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>)</li>

<li>The <a href="Nelson_rules" title="Nelson rules">Nelson rules</a></li></ul>

<p>There has been particular controversy as to how long a run of observations, all on the same side of the centre line, should count as a signal, with 6, 7, 8 and 9 all being advocated by various writers.

</p><p>The most important principle for choosing a set of rules is that the choice be made before the data is inspected. Choosing rules once the data have been seen tends to increase the <a href="Type_I_error" class="mw-redirect" title="Type I error">Type I error</a> rate owing to <a href="Testing_hypotheses_suggested_by_the_data" title="Testing hypotheses suggested by the data">testing effects suggested by the data</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Alternative_bases">Alternative bases</h2></div>

<p>In 1935, the <a href="British_Standard" class="mw-redirect" title="British Standard">British Standards Institution</a>, under the influence of <a href="Egon_Pearson" title="Egon Pearson">Egon Pearson</a> and against Shewhart's spirit, adopted control charts, replacing <i>3-sigma</i> limits with limits based on <a href="Percentile" title="Percentile">percentiles</a> of the <a href="Normal_distribution" title="Normal distribution">normal distribution</a>. This move continues to be represented by John Oakland and others but has been widely deprecated by writers in the Shewhart–Deming tradition.

</p>

<div class="mw-heading mw-heading2"><h2 id="Performance_of_control_charts">Performance of control charts</h2></div>

<p>When a point falls outside the limits established for a given control chart, those responsible for the underlying process are expected to determine whether a special cause has occurred. If one has, it is appropriate to determine if the results with the special cause are better than or worse than results from common causes alone. If worse, then that cause should be eliminated if possible. If better, it may be appropriate to intentionally retain the special cause within the system producing the results.

</p><p>Even when a process is <i>in control</i> (that is, no special causes are present in the system), there is approximately a 0.27% probability of a point exceeding <i>3-sigma</i> control limits. So, even an in control process plotted on a properly constructed control chart will eventually signal the possible presence of a special cause, even though one may not have actually occurred. For a Shewhart control chart using <i>3-sigma</i> limits, this <i>false alarm</i> occurs on average once every 1/0.0027 or 370.4 observations. Therefore, the <i>in-control average run length</i> (or in-control ARL) of a Shewhart chart is 370.4.

</p><p>Meanwhile, if a special cause does occur, it may not be of sufficient magnitude for the chart to produce an immediate <i>alarm condition</i>. If a special cause occurs, one can describe that cause by measuring the change in the mean and/or variance of the process in question. When those changes are quantified, it is possible to determine the out-of-control ARL for the chart.

</p><p>It turns out that Shewhart charts are quite good at detecting large changes in the process mean or variance, as their out-of-control ARLs are fairly short in these cases. However, for smaller changes (such as a <i>1-</i> or <i>2-sigma</i> change in the mean), the Shewhart chart does not detect these changes efficiently. Other types of control charts have been developed, such as the <a href="EWMA_chart" title="EWMA chart">EWMA chart</a>, the <a href="CUSUM" title="CUSUM">CUSUM</a> chart and the real-time contrasts chart, which detect smaller changes more efficiently by making use of information from observations collected prior to the most recent data point.<sup id="cite_ref-Deng2012_17-0" class="reference"><a href="#cite_note-Deng2012-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p><p>Many control charts work best for numeric data with Gaussian assumptions. The real-time contrasts chart was proposed to monitor process with complex characteristics, e.g. high-dimensional, mix numerical and categorical, missing-valued, non-Gaussian, non-linear relationship.<sup id="cite_ref-Deng2012_17-1" class="reference"><a href="#cite_note-Deng2012-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Criticisms">Criticisms</h2></div>

<p>Several authors have criticised the control chart on the grounds that it violates the <a href="Likelihood_principle" title="Likelihood principle">likelihood principle</a>. However, the principle is itself controversial and supporters of control charts further argue that, in general, it is impossible to specify a <a href="Likelihood_function" title="Likelihood function">likelihood function</a> for a process not in statistical control, especially where knowledge about the <a href="Cause_system" class="mw-redirect" title="Cause system">cause system</a> of the process is weak.

</p><p>Some authors have criticised the use of average run lengths (ARLs) for comparing control chart performance, because that average usually follows a <a href="Geometric_distribution" title="Geometric distribution">geometric distribution</a>, which has high variability and difficulties.

</p><p>Some authors have criticized that most control charts focus on numeric data. Nowadays, process data can be much more complex, e.g. non-Gaussian, mix numerical and categorical, or be missing-valued.<sup id="cite_ref-Deng2012_17-2" class="reference"><a href="#cite_note-Deng2012-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_charts">Types of charts</h2></div>

<th>Chart

</th>

<th>Process observation

</th>

<th>Process observations relationships

</th>

<th>Process observations type

</th>

<th>Size of shift to detect

</th></tr>

<tr>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation>

</semantics>

</math></span><img src="./466e03e1c9533b4dab1b9949dad393883f385d80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}" loading="lazy"></span> and R chart</a>

</td>

<td>Quality characteristic measurement within one subgroup

</td>

<td>Independent

</td>

<td>Variables

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation>

</semantics>

</td>

<td>Quality characteristic measurement within one subgroup

</td>

<td>Independent

</td>

<td>Variables

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td><a href="Shewhart_individuals_control_chart" title="Shewhart individuals control chart">Shewhart individuals control chart</a> (ImR chart or XmR chart)

</td>

<td>Quality characteristic measurement for one observation

</td>

<td>Independent

</td>

<td>Variables<sup>†</sup>

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td>Three-way chart

</td>

<td>Quality characteristic measurement within one subgroup

</td>

<td>Independent

</td>

<td>Variables

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td><a href="P-chart" title="P-chart">p-chart</a>

</td>

<td>Fraction nonconforming within one subgroup

</td>

<td>Independent

</td>

<td>Attributes<sup>†</sup>

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td><a href="Np-chart" title="Np-chart">np-chart</a>

</td>

<td>Number nonconforming within one subgroup

</td>

<td>Independent

</td>

<td>Attributes<sup>†</sup>

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td><a href="C-chart" title="C-chart">c-chart</a>

</td>

<td>Number of nonconformances within one subgroup

</td>

<td>Independent

</td>

<td>Attributes<sup>†</sup>

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td><a href="U-chart" title="U-chart">u-chart</a>

</td>

<td>Nonconformances per unit within one subgroup

</td>

<td>Independent

</td>

<td>Attributes<sup>†</sup>

</td>

<td>Large (≥ 1.5σ)

</td></tr>

<tr>

<td><a href="EWMA_chart" title="EWMA chart">EWMA chart</a>

</td>

<td><a href="Exponentially_weighted_moving_average" class="mw-redirect" title="Exponentially weighted moving average">Exponentially weighted moving average</a> of quality characteristic measurement within one subgroup

</td>

<td>Independent

</td>

<td>Attributes or variables

</td>

<td>Small (< 1.5σ)

</td></tr>

<tr>

<td><a href="CUSUM" title="CUSUM">CUSUM</a> chart

</td>

<td>Cumulative sum of quality characteristic measurement within one subgroup

</td>

<td>Independent

</td>

<td>Attributes or variables

</td>

<td>Small (< 1.5σ)

</td></tr>

<tr>

<td><a href="Time_series" title="Time series">Time series</a> model

</td>

<td>Quality characteristic measurement within one subgroup

</td>

<td>Autocorrelated

</td>

<td>Attributes or variables

</td>

<td>N/A

</td></tr>

<tr>

<td><a href="Regression_control_chart" title="Regression control chart">Regression control chart</a>

</td>

<td>Quality characteristic measurement within one subgroup

</td>

<td>Dependent of process control variables

</td>

<td>Variables

</td>

<td>Large (≥ 1.5σ)

</td></tr></tbody></table>

<p><sup>†</sup>Some practitioners also recommend the use of Individuals charts for attribute data, particularly when the assumptions of either binomially distributed data (p- and np-charts) or Poisson-distributed data (u- and c-charts) are violated.<sup id="cite_ref-Wheeler2000_18-0" class="reference"><a href="#cite_note-Wheeler2000-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Two primary justifications are given for this practice. First, normality is not necessary for statistical control, so the Individuals chart may be used with non-normal data.<sup id="cite_ref-Staufer2010a_19-0" class="reference"><a href="#cite_note-Staufer2010a-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Second, attribute charts derive the measure of dispersion directly from the mean proportion (by assuming a probability distribution), while Individuals charts derive the measure of dispersion from the data, independent of the mean, making Individuals charts more robust than attributes charts to violations of the assumptions about the distribution of the underlying population.<sup id="cite_ref-WheelerSPC_20-0" class="reference"><a href="#cite_note-WheelerSPC-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> It is sometimes noted that the substitution of the Individuals chart works best for large counts, when the binomial and <a href="Poisson_distribution" title="Poisson distribution">Poisson distributions</a> approximate a normal distribution. i.e. when the number of trials <span class="texhtml"><var>n</var> > 1000</span> for p- and np-charts or <span class="texhtml"><var>λ</var> > 500</span> for u- and c-charts.

</p><p>Critics of this approach argue that control charts should not be used when their underlying assumptions are violated, such as when process data is neither normally distributed nor binomially (or Poisson) distributed. Such processes are not in control and should be improved before the application of control charts. Additionally, application of the charts in the presence of such deviations increases the <a href="Type_I_and_type_II_errors" title="Type I and type II errors">type I and type II error</a> rates of the control charts, and may make the chart of little practical use.

</p>

<ul><li><a href="Analytic_and_enumerative_statistical_studies" title="Analytic and enumerative statistical studies">Analytic and enumerative statistical studies</a></li>

<li><a href="Common_cause_and_special_cause" class="mw-redirect" title="Common cause and special cause">Common cause and special cause</a></li>

<li><a href="Process_capability" title="Process capability">Process capability</a></li>

<li><a href="Seven_Basic_Tools_of_Quality" class="mw-redirect" title="Seven Basic Tools of Quality">Seven Basic Tools of Quality</a></li>

<li><a href="Six_Sigma" title="Six Sigma">Six Sigma</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Statistical process control</a></li>

<li><a href="Total_quality_management" title="Total quality management">Total quality management</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFWheelerChambers1992" class="citation book cs1"><a href="Donald_J._Wheeler" title="Donald J. Wheeler">Wheeler, Donald J.</a>; Chambers, David S. (1992). <a href="Statistical_process_control" title="Statistical process control"><i>Understanding statistical process control</i></a> (2 ed.). <a href="Knoxville%2C_Tennessee" title="Knoxville, Tennessee">Knoxville, Tennessee</a>: SPC Press. p. 96. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-945320-13-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/27187772">27187772</a>.</cite></span>

</li>

<li id="cite_note-Deng2012-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Deng2012_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Deng2012_17-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Deng2012_17-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFDeng,_H.Runger,_G.Tuv,_E.2012" class="citation journal cs1">Deng, H.; Runger, G.; Tuv, E. (2012). "System monitoring with real-time contrasts". <i>Journal of Quality Technology</i>. <b>44</b> (1). pp. 9–27. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00224065.2012.11917878">10.1080/00224065.2012.11917878</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119835984">119835984</a>.</cite></span>

</li>

<li id="cite_note-Wheeler2000-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wheeler2000_18-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWheeler2000" class="citation book cs1">Wheeler, Donald J. (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/understandingvar00dona/page/140"><i>Understanding Variation: the key to managing chaos</i></a>. SPC Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/understandingvar00dona/page/140">140</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-945320-53-1</bdi>.</cite></span>

</li>

<li id="cite_note-Staufer2010a-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Staufer2010a_19-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFStaufer" class="citation web cs1">Staufer, Rip. <a rel="nofollow" class="external text" href="http://www.qualitydigest.com/inside/quality-insider-article/some-problems-attribute-charts.html">"Some Problems with Attribute Charts"</a>. Quality Digest<span class="reference-accessdate">. Retrieved <span class="nowrap">2 Apr</span> 2010</span>.</cite></span>

</li>

<li id="cite_note-WheelerSPC-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-WheelerSPC_20-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWheeler" class="citation web cs1">Wheeler, Donald J. <a rel="nofollow" class="external text" href="http://www.qualitydigest.com/jul/spctool.html">"What About Charts for Count Data?"</a>. Quality Digest<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-03-23</span></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFDeming1975" class="citation journal cs1">Deming, W. E. (1975). "On probability as a basis for action". <i>The American Statistician</i>. <b>29</b> (4): <span class="nowrap">146–</span>152. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.470.9636">10.1.1.470.9636</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2683482">10.2307/2683482</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2683482">2683482</a>.</cite></li>

<li><cite id="CITEREFDeming1982" class="citation book cs1">Deming, W. E. (1982). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/outofcrisisquali00demi"><i>Out of the Crisis: Quality, Productivity and Competitive Position</i></a></span>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-30553-2</bdi>.</cite></li>

<li><cite id="CITEREFDengRungerTuv2012" class="citation journal cs1">Deng, H.; Runger, G.; Tuv, Eugene (2012). "System monitoring with real-time contrasts". <i>Journal of Quality Technology</i>. <b>44</b> (1): <span class="nowrap">9–</span>27. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00224065.2012.11917878">10.1080/00224065.2012.11917878</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119835984">119835984</a>.</cite></li>

<li><cite id="CITEREFMandel1969" class="citation journal cs1">Mandel, B. J. (1969). "The Regression Control Chart". <i>Journal of Quality Technology</i>. <b>1</b> (1): <span class="nowrap">1–</span>9. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00224065.1969.11980341">10.1080/00224065.1969.11980341</a>.</cite></li>

<li><cite id="CITEREFOakland2002" class="citation book cs1">Oakland, J. (2002). <i>Statistical Process Control</i>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-5766-2</bdi>.</cite></li>

<li><cite id="CITEREFShewhart1931" class="citation book cs1">Shewhart, W. A. (1931). <i>Economic Control of Quality of Manufactured Product</i>. American Society for Quality Control. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-87389-076-2</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFShewhart1939" class="citation book cs1">Shewhart, W. A. (1939). <i>Statistical Method from the Viewpoint of Quality Control</i>. Courier Corporation. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-65232-0</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFWheeler2000" class="citation book cs1">Wheeler, D. J. (2000). <i>Normality and the Process-Behaviour Chart</i>. SPC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-945320-56-2</bdi>.</cite></li>

<li><cite id="CITEREFWheelerChambers1992" class="citation book cs1">Wheeler, D. J.; Chambers, D. S. (1992). <i>Understanding Statistical Process Control</i>. SPC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-945320-13-5</bdi>.</cite></li>

<li><cite id="CITEREFWheeler1999" class="citation book cs1">Wheeler, Donald J. (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/understandingvar00dona"><i>Understanding Variation: The Key to Managing Chaos</i></a> (2nd ed.). SPC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-945320-53-1</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Control_charts" class="extiw external" title="commons:Category:Control charts">Control charts</a></span>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/index.htm">NIST/SEMATECH e-Handbook of Statistical Methods</a></li>

<li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pmc/pmc.htm">Monitoring and Control with Control Charts</a></li></ul>

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</style><div id="Seven_basic_tools_of_quality54" style="font-size:114%;margin:0 4em"><a href="Seven_basic_tools_of_quality" title="Seven basic tools of quality">Seven basic tools of quality</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ishikawa_diagram" title="Ishikawa diagram">Cause-and-effect diagram</a></li>

<li><a href="Check_sheet" title="Check sheet">Check sheet</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pareto_chart" title="Pareto chart">Pareto chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter diagram</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratification</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="Quality_(business)" title="Quality (business)">Quality (business)</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>

<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Confounding" title="Confounding">Confounding variable</a></li>

<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>

<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>

<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>

<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>

<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>

<li><a href="System_identification" title="System identification">System identification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>

<li><a href="National_accounts" title="National accounts">National accounts</a></li>

<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>

<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>

<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>

<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>

<li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-05-19" href="https://en.wikipedia.org/wiki/?title=Control_chart&oldid=1291178072">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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