### XBar-S Chart Calculations

The XBar-S chart displays two charts, one for the Xbar values and the other for the Sigma values. The formulas for calculating the centerlines and control limits for the XBar-S control chart are detailed in the following sections.

XBar-S Chart Center Line

Based on the sample size, the mean for a given sample can be calculated as:

where x1 through xn are the values of each observation in the sample and n is the number of observations in the sample.

Because an XBar-S chart displays the average of each sample in the set, its centerline is calculated as:

where through are the averages of each sample, and m is the number of samples. This grand average becomes the centerline of the XBar chart.

XBar-S Chart Control Limits

To construct the upper control limit (UCL) and lower control limit (LCL) for the XBar-S chart, we need to know the standard deviation of the data.

In a sample of size n, the standard deviation is:

If through is the range of m standard deviations, then the average standard deviation is:

The formulas used for calculating the upper and lower control limits for the XBar chart are:

Where A3 is a tabulated value based on the sample size. See the XBar Chart Factors for Control Limits table for values of A3 for sample sizes ranging from 1 to 25.

If the sample size is greater than 25, the formulas are:

where S is the standard deviation for the chart.

S Chart Center Line

For the S chart, the formula for calculating the center line is:

S Chart Control Limits

The formulas used for calculating the upper and lower control limits for the S chart are:

where B3 and B4 are tabulated values based on the sample size. See the Range Chart Factors for Control Limits table for values of B3 and B4 for sample sizes ranging from 1 to 25.

If the sample size is greater than 25, the formulas are:

where S is the standard deviation for the chart.

 XBar-R, XBar-S, R, S and X chart calculations.