X̅ and s chart

In statistical quality control, the and s chart is a type of control chart used to monitor variables data when samples are collected at regular intervals from a business or industrial process. This is connected to traditional statistical quality control and statistical process control. However, Woodall noted that "I believe that the use of control charts and other monitoring methods should be referred to as “statistical process monitoring,” not “statistical process control.”"

Uses

The chart is advantageous in the following situations:

The sample size is relatively large
The sample size is variable
Computers can be used to ease the burden of calculation

The "chart" actually consists of a pair of charts: One to monitor the process standard deviation and another to monitor the process mean, as is done with the and R and individuals control charts. The and s chart plots the mean value for the quality characteristic across all units in the sample,, plus the standard deviation of the quality characteristic across all units in the sample as follows:

Assumptions

The normal distribution is the basis for the charts and requires the following assumptions:

The quality characteristic to be monitored is adequately modeled by a normally-distributed random variable
The parameters μ and σ for the random variable are the same for each unit and each unit is independent of its predecessors or successors
The inspection procedure is same for each sample and is carried out consistently from sample to sample

Control limits

The control limits for this chart type are:

and for monitoring the process variability
for monitoring the process mean

Validity

As with the and R and individuals control charts, the chart is only valid if the within-sample variability is constant. Thus, the s chart is examined before the chart; if the s chart indicates the sample variability is in statistical control, then the chart is examined to determine if the sample mean is also in statistical control. If on the other hand, the sample variability is not in statistical control, then the entire process is judged to be not in statistical control regardless of what the chart indicates.

Unequal samples

When samples collected from the process are of unequal sizes, there are two approaches:

Technique	Description
Use variable-width control limits	Each observation plots against its own control limits as determined by the sample size-specific values, n_i, of A₃, B₃, and B₄
Use control limits based on an average sample size	Control limits are fixed at the modal sample size-specific value of A₃, B₃, and B₄

Limitations and improvements

Effect of estimation of parameters plays a major role. Also a change in variance affects the performance of chart while a shift in mean affects the performance of the S chart.
Therefore, several authors recommend using a single chart that can simultaneously monitor and S. McCracken, Chackrabori and Mukherjee developed one of the most modern and efficient approach for jointly monitoring the Gaussian process parameters, using a set of reference sample in absence of any knowledge of true process parameters.