The s chart replaces the R chart and provides an increase in sensitivity to variation of the spread of the data.

The s-chart works better with 10 or more items per sample in order to obtain the s (standard deviation) estimate. The use of a spreadsheet or calculator expedites the calculation of the sample standard deviation.

The formula for the sample standard deviation is

$$ \large\displaystyleÂ s=\sqrt{\frac{{{\sum{\left( X-\bar{X} \right)}}^{2}}}{n-1}}$$

Where

Î£ is the summation sign

X are the individual measurements

X-bar is the average of the individual measurements in a sample

n is the sample size

The X-bar chart construction is the same as described above except we use s for the control limit calculations with the following two formulas

$$ \large\displaystyleÂ UC{{L}_{{\bar{X}}}}=\bar{\bar{X}}+{{A}_{3}}\left( {\bar{s}} \right)$$

$$ \large\displaystyleÂ LC{{L}_{{\bar{X}}}}=\bar{\bar{X}}-{{A}_{3}}\left( {\bar{s}} \right)$$

Where

X-double bar is the grand average

A3 is the factor to determine X-bar control limits based on sample size.

s-bar is the average standard deviation

The control limits for the s-chart use the following formulas

$$ \large\displaystyleÂ UC{{L}_{{\bar{X}}}}={{B}_{4}}\left( {\bar{s}} \right)$$

$$ \large\displaystyleÂ LC{{L}_{{\bar{X}}}}={{B}_{3}}\left( {\bar{s}} \right)$$

B3 and B4 are factors to determine the control limits adjusted for sample size.

S-bar is the average sample standard deviation and is the center line of the s-chart.

X-bar and s chart factors

n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 25 |

B_{4} | 3.27 | 2.57 | 2.27 | 2.09 | 1.97 | 1.88 | 1.82 | 1.76 | 1.72 | 1.44 |

B_{3} | 0 | 0 | 0 | 0 | 0.03 | 0.12 | 0.18 | 0.24 | 0.28 | 0.56 |

A_{3} | 2.66 | 1.95 | 1.63 | 1.43 | 1.29 | 1.18 | 1.10 | 1.03 | 0.98 | 0.61 |

Related:

8 Steps to creating an X-bar and s control chartÂ (article)

8 steps to Creating a X-bar and R control chartÂ (article)

Variables Control Chart BackgroundÂ (article)

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