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X-bar and Sigma Charts

X-bar and sigma chart

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

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


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

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

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


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

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

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
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