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| {{Infobox control chart
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| | name = p-chart
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| | proposer = [[Walter A. Shewhart]]
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| | subgroupsize = n > 1
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| | measurementtype = Fraction nonconforming in a sample
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| | qualitycharacteristictype = [[Variable and attribute (research)|Attributes data]]
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| | distribution = [[Binomial distribution]]
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| | sizeofshift = ≥ 1.5σ
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| | meanchart = P control chart.svg
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| | meancenter = <math>\bar p = \frac {\sum_{i=1}^m \sum_{j=1}^n \begin{cases} 1 & \mbox{if }x_{ij}\mbox{ defective} \\ 0 & \mbox{otherwise} \end{cases}}{mn}</math>
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| | meanlimits = <math>\bar p \pm 3\sqrt{\frac{\bar p(1-\bar p)}{n}}</math>
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| | meanstatistic = <math>\bar p_i = \frac {\sum_{j=1}^n \begin{cases} 1 & \mbox{if }x_{ij}\mbox{ defective} \\ 0 & \mbox{otherwise} \end{cases}}{n}</math>
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| }}
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| In [[statistical process control|statistical quality control]], the '''p-chart''' is a type of [[control chart]] used to monitor the proportion of [[Nonconformity (quality)|nonconforming units]] in a [[Sample (statistics)|sample]], where the sample proportion nonconforming is defined as the ratio of the number of nonconforming units to the sample size, n.<ref>{{cite web|url=http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm|title=Proportions Control Charts|accessdate=2010-01-05|work=[http://www.itl.nist.gov/div898/handbook/index.htm NIST/Sematech Engineering Statistics Handbook]|publisher=[[National Institute of Standards and Technology]]}}</ref>
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| The p-chart only accommodates "pass"/"fail"-type inspection as determined by one or more [[Go-NoGo gauge|go-no go gauges]] or tests, effectively applying the [[Specification (technical standard)|specifications]] to the data ''before'' they are plotted on the chart. Other types of control charts display the magnitude of the quality characteristic under study, making troubleshooting possible directly from those charts.
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| ==Assumptions==
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| The [[binomial distribution]] is the basis for the p-chart and requires the following assumptions<ref name="Montgomery2005">{{cite book | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]], Inc. | year = 2005 | location = [[Hoboken, New Jersey]] | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567}}</ref>{{rp|267}}:
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| *The probability of nonconformity {{math|<VAR>p</VAR>}} is the same for each unit;
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| *Each unit is independent of its predecessors or successors;
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| *The inspection procedure is same for each sample and is carried out consistently from sample to sample
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| <!-- NOTE: IF THE ABOVE THREE CONDITIONS ARE MET, THEN THE DATA FOLLOW A BINOMIAL DISTRIBUTION--see the definition given in [[Binomial distribution]] -->
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| ==Calculation and plotting==
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| The control limits for this chart type are <math>\bar p \pm 3\sqrt{\frac{\bar p(1-\bar p)}{n}}</math> where <math>\bar p</math> is the estimate of the long-term process mean established during control-chart setup.<ref name="Montgomery2005"/>{{rp|268}} Naturally, if the lower control limit is less than or equal to zero, process observations only need be plotted against the upper control limit. Note that observations of proportion nonconforming below a positive lower control limit are cause for concern as they are more frequently evidence of improperly calibrated test and inspection equipment or inadequately trained inspectors than of sustained quality improvement.<ref name="Montgomery2005"/>{{rp|279}}
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| Some organizations may elect to provide a standard value for p, effectively making it a target value for the proportion nonconforming. This may be useful when simple process adjustments can consistently move the process mean, but in general, this makes it more challenging to judge whether a process is fully out of control or merely off-target (but otherwise in control).<ref name="Montgomery2005"/>{{rp|269}}
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| ==Potential pitfalls==
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| There are two circumstances that merit special attention:
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| *Ensuring enough observations are taken for each sample
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| *Accounting for differences in the number of observations from sample to sample
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| ===Adequate sample size===
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| Sampling requires some careful consideration. If the organization elects to use 100% inspection on a process, the production rate determines an appropriate sampling rate which in turn determines the sample size.<ref name="Montgomery2005"/>{{rp|277}} If the organization elects to only inspect a fraction of units produced, the sample size should be chosen large enough so that the chance of finding at least one nonconforming unit in a sample is high—otherwise the false alarm rate is too high. One technique is to fix sample size so that there is a 50% chance of detecting a process shift of a given amount (for example, from 1% defective to 5% defective). If δ is the size of the shift to detect, then the sample size should be set to <math>n \ge \left ( \frac{3}{\delta} \right )^2 \bar p(1-\bar p)</math>.<ref name="Montgomery2005"/>{{rp|278}} Another technique is to choose the sample size large enough so that the p-chart has a positive lower control limit or <math>n > \frac{3^2 (1 - \bar p)}{\bar p}</math>.
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| ===Varying sample sizes===
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| In the case of 100% inspection, variation in the production rate (e.g., due to maintenance or shift changes) conspires to produce different sample sizes for each observation plotted on the p-chart. There are three ways to deal with this:
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| {| class="wikitable"
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| ! Technique !! Description
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| | Use variable-width control limits<ref name="Montgomery2005"/>{{rp|280}}
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| | Each observation plots against its own control limits: <math>\bar p \pm 3\sqrt{\frac{\bar p(1-\bar p)}{n_i}}</math>, where n<SUB>i</SUB> is the size of the sample that produced the ith observation on the p-chart
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| |-
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| | Use control limits based on an average sample size<ref name="Montgomery2005"/>{{rp|282}}
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| | Control limits are <math>\bar p \pm 3\sqrt{\frac{\bar p(1-\bar p)}{\bar n}}</math>, where <math>\bar n</math> is the average size of all the samples on the p-chart, <math>\frac {\sum_{i=1}^m n_i}{m}</math>
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| |-
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| | Use a standardized control chart<ref name="Montgomery2005"/>{{rp|283}}
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| | Control limits are ±3 and the observations, <math>\hat{p}_i</math>, are [[Standard score|standardized]] using <math>Z_i = \frac {\hat{p}_i - \bar p}{\sqrt {\frac {\bar p (1 - \bar p)}{n_i}}}</math>, where n<SUB>i</SUB> is the size of the sample that produced the ith observation on the p-chart
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| |}
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| ===Sensitivity of control limits===
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| Some practitioners have pointed out that the p-chart is sensitive to the underlying assumptions, using control limits derived from the binomial distribution rather than from the observed sample variance. Due to this sensitivity to the underlying assumptions, p-charts are often implemented incorrectly, with control limits that are either too wide or too narrow, leading to incorrect decisions regarding process stability. A p-chart is a form of the [[Shewhart individuals control chart|Individuals chart]] (also referred to as "XmR" or "ImR"), and these practitioners recommend the individuals chart as a more robust alternative for count-based data.<ref>{{cite book | last = Wheeler | first = Donald | title = Understanding Variation: The Key to Managing Chaos | publisher = [[SPC Press]] | year = 2000 | pages = 140 | url = http://www.spcpress.com/ | isbn = 0-945320-53-1 }}</ref>
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| ==See also==
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| *[[np-chart]]
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| *[[Shewhart individuals control chart|Individuals chart]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:P-Chart}}
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| [[Category:Quality control tools]]
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| [[Category:Statistical charts and diagrams]]
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The name of the writer is Jayson. For years he's been residing in Alaska and he doesn't plan on altering it. To perform lacross is some thing he would by no means give up. My working day job is an info officer but I've already utilized for an additional 1.
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