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. The chart is necessary in the following situations:
Where automation allows inspection of each unit, so rational subgrouping has less benefit. Where production is slow so that waiting for enough samples to make a rational subgroup unacceptably delays monitoring 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.
Interpretation As with other control charts, the individuals and moving range charts consist of points plotted with the control limits, or natural process limits. These limits reflect what the process will deliver without fundamental changes. Points outside of these control limits are signals indicating that the process is not operating as consistently as possible; that some assignable cause has resulted in a change in the process. Similarly, runs of points on one side of the average line should also be interpreted as a signal of some change in the process. When such signals exist, action should be taken to identify and eliminate them. When no such signals are present, no changes to the process control variables (i.e. "tampering") are necessary or desirable.
Assumptions The normal distribution is NOT assumed nor required in the calculation of control limits. Thus making the IndX/mR chart a very robust tool. This is demonstrated by Wheeler using real-world data, and for a number of highly non-normal probability distributions.
Calculation and plotting
Calculation of moving range The difference between data point, x i {\displaystyle x_{i}} , and its predecessor, x i − 1 {\displaystyle x_{i-1}} , is calculated as M R i = | x i − x i − 1 | {\displaystyle {MR}_{i}={\big |}x_{i}-x_{i-1}{\big |}} . For m {\displaystyle m} individual values, there are m − 1 {\displaystyle m-1} ranges. Next, the arithmetic mean of these values is calculated as
M R ¯ = ∑ i = 2 m M R i m − 1 {\displaystyle {\overline {MR}}={\frac {\sum _{i=2}^{m}{MR_{i}}}{m-1}}}
If the data are normally distributed with standard deviation σ {\displaystyle \sigma } then the expected value of M R ¯ {\displaystyle {\overline {MR}}} is d 2 σ = 2 σ / π {\displaystyle d_{2}\sigma =2\sigma /{\sqrt {\pi }}} , the mean absolute difference of the normal distribution.
Calculation of moving range control limit The upper control limit for the range (or upper range limit) is calculated by multiplying the average of the moving range by 3.267:
U C L r = 3.267 M R ¯ {\displaystyle UCL_{r}=3.267{\overline {MR}}} . The value 3.267 is taken from the sample size-specific D4 anti-biasing constant for n=2, as given in most textbooks on statistical process control (see, for example, Montgomery).
Calculation of individuals control limits First, the average of the individual values is calculated:
x ¯ = ∑ i = 1 m x i m {\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{m}{x_{i}}}{m}}} . Next, the upper control limit (UCL) and lower control limit (LCL) for the individual values (or upper and lower natural process limits) are calculated by adding or subtracting 2.66 times the average moving range to the process average:
U C L = x ¯ + 2.66 M R ¯ {\displaystyle UCL={\overline {x}}+2.66{\overline {MR}}} .
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