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{{correct title|<math>\bar x</math> and s chart}}
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{{Infobox control chart
| name = <math>\bar x</math> and s chart
| proposer = [[Walter A. Shewhart]]
| subgroupsize = n > 10
| measurementtype = Average quality characteristic per unit
| qualitycharacteristictype = [[Variable and attribute (research)|Variables data]]
| distribution = [[Normal distribution]]
| sizeofshift = ≥ 1.5σ
| varchart = S chart for a paired xbar and s chart.svg
| varcenter = <math>\bar s = \frac {\sum_{i=1}^m \sqrt \frac{\sum_{j=1}^n \left ( x_{ij} - \bar {\bar x} \right )^2 }{n - 1}}{m}</math>
| varupperlimit = <math>B_4 \bar S</math>
| varlowerlimit = <math>B_3 \bar S</math>
| varstatistic = <math>\bar s_i = \sqrt \frac{\sum_{j=1}^n \left ( x_{ij} - \bar x_i \right )^2 }{n - 1}</math>
| meanchart = Xbar chart for a paired xbar and s chart.svg
| meancenter = <math>\bar x = \frac {\sum_{i=1}^m \sum_{j=1}^n x_{ij}}{mn}</math>
| meanlimits = <math>\bar x \pm A_3 \bar s</math>
| meanstatistic = <math>\bar x_i = \frac {\sum_{j=1}^n x_{ij}}{n}</math>
}}
 
In [[statistical process control|statistical quality control]], the '''<math>\bar x</math> and s chart''' is a type of [[control chart]] used to monitor [[Variable and attribute (research)|variables data]] when samples are collected at regular intervals from a [[Business process|business]] or [[List of industrial processes|industrial process]].<ref>{{Cite web|url=http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm|title=Shewhart X-bar and R and S Control Charts|accessdate=2009-01-13|work=[http://www.itl.nist.gov/div898/handbook/index.htm NIST/Sematech Engineering Statistics Handbook]|publisher=[[National Institute of Standards and Technology]]}}</ref>
 
The chart is advantageous in the following situations:<ref>{{Cite book | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]], Inc. | year = 2005 | location = [[Hoboken, New Jersey]] | pages = 222 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567}}</ref>
#The sample size is relatively large (say, n > 10—[[Xbar and R chart|<math>\bar x</math> and R charts]] are typically used for smaller sample sizes)
#The sample size is variable
#Computers can be used to ease the burden of calculation
 
The "chart" actually consists of a pair of charts: One to monitor the process standard deviation and another to monitor the process mean, as is done with the [[Xbar and R chart|<math>\bar x</math> and R]] and [[Shewhart individuals control chart|individuals control chart]]s. The <math>\bar x</math> and s chart plots the mean value for the quality characteristic across all units in the sample, <math>\bar x_i</math>, plus the standard deviation of the quality characteristic across all units in the sample as follows:
:<math>s = \sqrt{\frac {\sum_{i=1}^n {\left ( x_i - \bar x \right )}^2}{n - 1}}</math>.
 
The [[normal distribution]] is the basis for the charts and requires the following assumptions:
*The quality characteristic to be monitored is adequately modeled by a [[Normal distribution|normally-distributed]] [[random variable]]
*The parameters μ and σ for the random variable are the same for each unit and each unit is independent of its predecessors or successors
*The inspection procedure is same for each sample and is carried out consistently from sample to sample
 
The control limits for this chart type are:<ref>{{Cite book | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]], Inc. | year = 2005 | location = [[Hoboken, New Jersey]] | pages = 225 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567}}</ref>
*<math>B_3 \bar s</math> (lower) and <math>B_4 \bar s</math> (upper) for monitoring the process variability
*<math>\bar x \pm A_3 \bar s</math> for monitoring the process mean
:where <math>\bar x</math> and <math>\bar s = \frac {\sum_{i=1}^m s_i}{m}</math> are the estimates of the long-term process mean and range established during control-chart setup and A<sub>3</sub>, B<sub>3</sub>, and B<sub>4</sub> are sample size-specific [[Unbiased estimation of standard deviation|anti-biasing]] constants. The anti-biasing constants are typically found in the appendices of textbooks on [[statistical process control]].
 
As with the [[Xbar and R chart|<math>\bar x</math> and R]] and [[Shewhart individuals control chart|individuals control chart]]s, the <math>\bar x</math> chart is only valid if the within-sample variability is constant.<ref>{{Cite book | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]], Inc. | year = 2005 | location = [[Hoboken, New Jersey]] | pages = 214 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567}}</ref> Thus, the s chart is examined before the <math>\bar x</math> chart; if the s chart indicates the sample variability is in statistical control, then the <math>\bar x</math> chart is examined to determine if the sample mean is also in statistical control. If on the other hand, the sample variability is ''not'' in statistical control, then the entire process is judged to be not in statistical control regardless of what the <math>\bar x</math> chart indicates.
 
When samples collected from the process are of unequal sizes (arising from a mistake in collecting them, for example), there are two approaches:
 
{| class="wikitable"
! Technique !! Description
|-
| Use variable-width control limits<ref>{{Cite book | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]], Inc. | year = 2005 | location = [[Hoboken, New Jersey]] | pages = 227 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567}}</ref>
| Each observation plots against its own control limits as determined by the sample size-specific values, n<sub>i</sub>, of A<sub>3</sub>, B<sub>3</sub>, and B<sub>4</sub>
|-
| Use control limits based on an average sample size<ref>{{Cite book | last = Montgomery | first = Douglas | title = Introduction to Statistical Quality Control | publisher = [[John Wiley & Sons]], Inc. | year = 2005 | location = [[Hoboken, New Jersey]] | pages = 229 | url = http://www.eas.asu.edu/~masmlab/montgomery/ | isbn = 978-0-471-65631-9 | oclc = 56729567}}</ref>
| Control limits are fixed at the [[Mode (statistics)|modal]] (or most common) sample size-specific value of A<sub>3</sub>, B<sub>3</sub>, and B<sub>4</sub>
|}
 
==See also==
*[[Xbar and R chart|<math>\bar x</math> and R chart]]
*[[Shewhart individuals control chart]]
 
==References==
{{reflist}}
 
[[Category:Quality control tools]]
[[Category:Statistical charts and diagrams]]

Latest revision as of 21:54, 21 September 2014

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