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
$$ \large\displaystyle s=\sqrt{\frac{{{\sum{\left( X-\bar{X} \right)}}^{2}}}{n-1}}$$
Where
Σ 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
$$ \large\displaystyle UC{{L}_{{\bar{X}}}}=\bar{\bar{X}}+{{A}_{3}}\left( {\bar{s}} \right)$$
$$ \large\displaystyle LC{{L}_{{\bar{X}}}}=\bar{\bar{X}}-{{A}_{3}}\left( {\bar{s}} \right)$$
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
$$ \large\displaystyle UC{{L}_{{\bar{X}}}}={{B}_{4}}\left( {\bar{s}} \right)$$
$$ \large\displaystyle LC{{L}_{{\bar{X}}}}={{B}_{3}}\left( {\bar{s}} \right)$$
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 |
| B4 | 3.27 | 2.57 | 2.27 | 2.09 | 1.97 | 1.88 | 1.82 | 1.76 | 1.72 | 1.44 |
| B3 | 0 | 0 | 0 | 0 | 0.03 | 0.12 | 0.18 | 0.24 | 0.28 | 0.56 |
| A3 | 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 chart (article)
8 steps to Creating a X-bar and R control chart (article)
Variables Control Chart Background (article)
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