Shewhart individuals control chart

In statistical quality control, the individual/moving-range chart is a type of control chart used to monitor variables data from a business or industrial process for which it is impractical to use rational subgroups.

Individuals and moving range control chart
Originally proposed by	Walter A. Shewhart
Process observations
Rational subgroup size	n = 1
Measurement type	Average quality characteristic per unit
Quality characteristic type	Variables data
Underlying distribution	none
Performance
Size of shift to detect	≥ 1.5σ
Process variation chart
Center line	${\displaystyle {\overline {MR}}={\frac {\sum _{i=2}^{m}{\big |}x_{i}-x_{i-1}{\big |}}{m-1}}}$
Upper control limit	${\displaystyle D_{4}{\overline {MR}}}$
Lower control limit	${\displaystyle D_{3}{\overline {MR}}}$
Plotted statistic	${\displaystyle MR_{i}={\big |}x_{i}-x_{i-1}{\big |}}$
Process mean chart
Center line	${\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{m}x_{i}}{m}}}$
Control limits	${\displaystyle {\overline {x}}\pm 3{\frac {\overline {MR}}{d_{2}}}}$
Plotted statistic	xi

The chart is necessary in the following situations:^: 231

1. Where automation allows inspection of each unit, so rational subgrouping has less benefit.
2. Where production is slow so that waiting for enough samples to make a rational subgroup unacceptably delays monitoring
3. For processes that produce homogeneous batches (e.g., chemical) where repeat measurements vary primarily because of measurement error

The "chart" actually consists of a pair of charts: one, the individuals chart, displays the individual measured values; the other, the moving range chart, displays the difference from one point to the next. As with other control charts, these two charts enable the user to monitor a process for shifts in the process that alter the mean or variance of the measured statistic.