Xbar chart

An Xbar-chart is a type of control chart used to monitor the process mean when measuring subgroups at regular intervals from a process.

Each point on the chart represents the value of a subgroup mean.

The center line is the process mean. If unspecified, the process mean is the weighted mean of the subgroup means.

The control limits are either:
  • A multiple (k) of sigma above and below the center line. Default k=3.
  • Probability limits, defined as the probability (alpha) of a point exceeding the limits. Default alpha=0.27%.
If unspecified, the process sigma is the pooled standard deviation of the subgroups, unless the chart is combined with an R- or S- chart where it is estimated as described for the respective chart.

The observations are assumed to be independent, and the means normally distributed. Individual observations need not be normally distributed. Due to the central limit theorem, the subgroup means are often approximately normally distributed for subgroup sizes larger than 4 or 5 regardless of the distribution of the individual observations.

For data with different subgroup sizes, the control limits vary. A standardized version of the control chart plots the points in standard deviation units. Such a control chart has a constant center line at 0, and upper and lower control limits of -3 and +3 respectively making patterns in the data easier to see.

It is important to ensure process variability is in a state of statistical control before using the Xbar-chart to investigate if the process mean is in control. Therefore a Xbar-chart is often combined with an R- or S-chart to monitor process variability. If the variability is not under control, the control limits may be too wide leading to an inability to detect special causes of variation affecting the process mean.