XBarS chart

XBarS chart (X-Bar and Standard Deviation chart) is a type of control chart used in statistical process control to monitor the mean and variation of a process using the standard deviation of subgroups. It is preferred over the XBarR chart when subgroup sizes are larger (typically n > 8), as the standard deviation provides a more efficient estimate of process variation.

Map the data fields

• Subgroup/Lot: This field represents the subgroup identifier. All observations sharing the same subgroup value are grouped together.
• Process Value: This field contains the individual measurement value for each observation within a subgroup.
• UCL (SPC Variable): Optional field for custom upper control limits for the SPC Variable (X̄) series.
• LCL (SPC Variable): Optional field for custom lower control limits for the SPC Variable (X̄) series.
• Central line (SPC Variable): Optional field for custom center line values for the SPC Variable (X̄) series.
• UCL (SPC Range): Optional field for custom upper control limits for the SPC Range (S) series.
• LCL (SPC Range): Optional field for custom lower control limits for the SPC Range (S) series.
• Central line (SPC Range): Optional field for custom center line values for the SPC Range (S) series.

Calculation Methodology

The XBarS chart requires subgroup sizes between 2 and 25. Calculations are performed as follows:

Subgroup Mean (x̄ᵢ)

x̄ᵢ = Sum of all values in subgroup i / Number of observations in subgroup i

Subgroup Standard Deviation (Sᵢ)

Sᵢ = sqrt( Σ(xⱼ - x̄ᵢ)² / (nᵢ - 1) )

Grand Mean (X̿)

X̿ = Sum of all subgroup means / Number of subgroups (k)

Average Standard Deviation (S̄)

S̄ = Sum of all subgroup standard deviations / Number of subgroups (k)

Control Limits — X-Bar Chart

Upper Control Limit (UCL):

UCL = X̿ + A3 × S̄

Lower Control Limit (LCL):

LCL = X̿ - A3 × S̄

Center Line (CL):

CL = X̿

A3 Constants Table

n	A3	n	A3
2	2.659	14	0.817
3	1.954	15	0.789
4	1.628	16	0.763
5	1.427	17	0.739
6	1.287	18	0.718
7	1.182	19	0.698
8	1.099	20	0.680
9	1.032	21	0.663
10	0.975	22	0.647
11	0.927	23	0.633
12	0.886	24	0.619
13	0.850	25	0.606

Notes:

• If custom limits are provided, they override the calculated values
• Subgroup size (n) must be between 2 and 25
• Subgroup standard deviation uses the sample formula (divides by n-1)
• The plotted value on the X-Bar chart is the subgroup mean, not individual observations
• XBarS is preferred over XBarR when subgroup sizes are larger (typically n > 8), as standard deviation is a more statistically efficient measure of spread