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${\bar {x}}$ and s chart
Originally proposed by	Walter A. Shewhart
Process observations
Rational subgroup size	n > 10
Measurement type	Average quality characteristic per unit
Quality characteristic type	Variables data
Underlying distribution	Normal distribution
Performance
Size of shift to detect	≥ 1.5σ
Process variation chart

Center line	${\bar {s}}={\frac {\sum _{i=1}^{m}{\sqrt {\frac {\sum _{j=1}^{n}\left(x_{ij}-{\bar {\bar {x}}}\right)^{2}}{n-1}}}}{m}}$
Upper control limit	$B_{4}{\bar {S}}$
Lower control limit	$B_{3}{\bar {S}}$
Plotted statistic	${\bar {s}}_{i}={\sqrt {\frac {\sum _{j=1}^{n}\left(x_{ij}-{\bar {x}}_{i}\right)^{2}}{n-1}}}$
Process mean chart

Center line	${\bar {x}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}x_{ij}}{mn}}$
Control limits	${\bar {x}}\pm A_{3}{\bar {s}}$
Plotted statistic	${\bar {x}}_{i}={\frac {\sum _{j=1}^{n}x_{ij}}{n}}$

In statistical quality control, the ${\bar {x}}$ and s chart is a type of control chart used to monitor variables data when samples are collected at regular intervals from a business or industrial process. This is connected to traditional statistical quality control (SQC) and statistical process control (SPC). However, Woodall noted that "I believe that the use of control charts and other monitoring methods should be referred to as “statistical process monitoring,” not “statistical process control (SPC).”"

Uses

The chart is advantageous in the following situations:

The sample size is relatively large (say, n > 10— ${\bar {x}}$ 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 ${\bar {x}}$ and R and individuals control charts. The ${\bar {x}}$ and s chart plots the mean value for the quality characteristic across all units in the sample, ${\bar {x}}_{i}$ , plus the standard deviation of the quality characteristic across all units in the sample as follows:

s={\sqrt {\frac {\sum _{i=1}^{n}{\left(x_{i}-{\bar {x}}\right)}^{2}}{n-1}}}

Assumptions

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 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

Control limits

The control limits for this chart type are:

$B_{3}{\bar {s}}$ (lower) and $B_{4}{\bar {s}}$ (upper) for monitoring the process variability
${\bar {x}}\pm A_{3}{\bar {s}}$ for monitoring the process mean

where

{\bar {x}}

and

{\bar {s}}={\frac {\sum _{i=1}^{m}s_{i}}{m}}

are the estimates of the long-term process mean and range established during control-chart setup and A₃, B₃, and B₄ are sample size-specific anti-biasing constants. The anti-biasing constants are typically found in the appendices of textbooks on statistical process control. NIST provides guidance on manually calculating these constants "6.3.2. What are Variables Control Charts?".

Validity

As with the ${\bar {x}}$ and R and individuals control charts, the ${\bar {x}}$ chart is only valid if the within-sample variability is constant. Thus, the s chart is examined before the ${\bar {x}}$ chart; if the s chart indicates the sample variability is in statistical control, then the ${\bar {x}}$ 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 ${\bar {x}}$ chart indicates.

Unequal samples

When samples collected from the process are of unequal sizes (arising from a mistake in collecting them, for example), there are two approaches:

Technique	Description
Use variable-width control limits	Each observation plots against its own control limits as determined by the sample size-specific values, n_i, of A₃, B₃, and B₄
Use control limits based on an average sample size	Control limits are fixed at the modal (or most common) sample size-specific value of A₃, B₃, and B₄

Limitations and improvements

Effect of estimation of parameters plays a major role. Also a change in variance affects the performance of ${\bar {X}}$ chart while a shift in mean affects the performance of the S chart.

Therefore, several authors recommend using a single chart that can simultaneously monitor ${\bar {X}}$ and S. McCracken, Chackrabori and Mukherjee developed one of the most modern and efficient approach for jointly monitoring the Gaussian process parameters, using a set of reference sample in absence of any knowledge of true process parameters.

References

"Shewhart X-bar and R and S Control Charts". NIST/Sematech Engineering Statistics Handbook. National Institute of Standards and Technology. Retrieved 2009-01-13.
Woodall, William H. (2016-07-19). "Bridging the Gap between Theory and Practice in Basic Statistical Process Monitoring". Quality Engineering: 00. doi:10.1080/08982112.2016.1210449. ISSN 0898-2112. S2CID 113516285.
Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 222. ISBN 978-0-471-65631-9. OCLC 56729567.
Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 225. ISBN 978-0-471-65631-9. OCLC 56729567.
Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 214. ISBN 978-0-471-65631-9. OCLC 56729567.
Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 227. ISBN 978-0-471-65631-9. OCLC 56729567.
Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 229. ISBN 978-0-471-65631-9. OCLC 56729567.
Chen, Gemai; Cheng, Smiley W. (1998). "Max Chart: Combining X-Bar Chart and S Chart". Statistica Sinica. 8 (1): 263–271. ISSN 1017-0405. JSTOR 24306354.
^ McCracken, A. K.; Chakraborti, S.; Mukherjee, A. (October 2013). "Control Charts for Simultaneous Monitoring of Unknown Mean and Variance of Normally Distributed Processes". Journal of Quality Technology. 45 (4): 360–376. doi:10.1080/00224065.2013.11917944. ISSN 0022-4065. S2CID 117307669.

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