Which chart should be used to measure the within sample variability when sample sizes are relatively large?
x
¯
{\displaystyle {\bar {x}}}
and s chartOriginally proposed byWalter A. ShewhartProcess observationsRational subgroup sizen > 10Measurement typeAverage quality characteristic per unitQuality characteristic typeVariables dataUnderlying distributionNormal distributionPerformanceSize of shift to detect≥ 1.5σProcess variation chartCenter line
s
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=
∑
i
=
1
m
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j
=
1
n
(
x
i
j
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x
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2
n
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1
m
{\displaystyle {\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
S
¯
{\displaystyle B_{4}{\bar {S}}}
Lower control limit
B
3
S
¯
{\displaystyle B_{3}{\bar {S}}}
Plotted statistic
s
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i
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j
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1
n
(
x
i
j
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x
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i
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2
n
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1
{\displaystyle {\bar {s}}_{i}={\sqrt {\frac {\sum _{j=1}^{n}\left(x_{ij}-{\bar {x}}_{i}\right)^{2}}{n-1}}}}
Process mean chartCenter line
x
¯
=
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i
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1
m
∑
j
=
1
n
x
i
j
m
n
{\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}x_{ij}}{mn}}}
Control limits
x
¯
±
A
3
s
¯
{\displaystyle {\bar {x}}\pm A_{3}{\bar {s}}}
Plotted statistic
x
¯
i
=
∑
j
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1
n
x
i
j
n
{\displaystyle {\bar {x}}_{i}={\frac {\sum _{j=1}^{n}x_{ij}}{n}}}
In statistical quality control, the x ¯ {\displaystyle {\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.[1] This is connected to traditional statistical quality control (SQC) and statistical process control (SPC). However, Woodall[2] 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).”" UsesThe chart is advantageous in the following situations:[3]
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 x ¯ {\displaystyle {\bar {x}}} and R and individuals control charts. The x ¯ {\displaystyle {\bar {x}}} and s chart plots the mean value for the quality characteristic across all units in the sample, x ¯ i {\displaystyle {\bar {x}}_{i}} , plus the standard deviation of the quality characteristic across all units in the sample as follows: s = ∑ i = 1 n ( x i − x ¯ ) 2 n − 1 {\displaystyle s={\sqrt {\frac {\sum _{i=1}^{n}{\left(x_{i}-{\bar {x}}\right)}^{2}}{n-1}}}} .AssumptionsThe normal distribution is the basis for the charts and requires the following assumptions:
Control limitsThe control limits for this chart type are:[4]
ValidityAs with the x ¯ {\displaystyle {\bar {x}}} and R and individuals control charts, the x ¯ {\displaystyle {\bar {x}}} chart is only valid if the within-sample variability is constant.[5] Thus, the s chart is examined before the x ¯ {\displaystyle {\bar {x}}} chart; if the s chart indicates the sample variability is in statistical control, then the x ¯ {\displaystyle {\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 x ¯ {\displaystyle {\bar {x}}} chart indicates. Unequal samplesWhen samples collected from the process are of unequal sizes (arising from a mistake in collecting them, for example), there are two approaches:
Limitations and improvementsEffect of estimation of parameters plays a major role. Also a change in variance affects the performance of X ¯ {\displaystyle {\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 X ¯ {\displaystyle {\bar {X}}} and S.[8] McCracken, Chackrabori and Mukherjee [9] 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. See also
References
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