# x̅ and R chart

${\displaystyle {\bar {x}}}$ and R chart
Originally proposed byWalter A. Shewhart
Process observations
Rational subgroup size1 < n ≤ 10
Measurement typeAverage quality characteristic per unit
Quality characteristic typeVariables data
Underlying distributionNormal distribution
Performance
Size of shift to detect≥ 1.5σ
Process variation chart
Center line${\displaystyle {\bar {R}}={\frac {\sum _{i=1}^{m}max(x_{ij})-min(x_{ij})}{m}}}$
Upper control limit${\displaystyle D_{4}{\bar {R}}}$
Lower control limit${\displaystyle D_{3}{\bar {R}}}$
Plotted statisticRi = max(xj) - min(xj)
Process mean chart
Center line${\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{m}\sum _{j=1}^{n}x_{ij}}{mn}}}$
Control limits${\displaystyle {\bar {x}}\pm A_{2}{\bar {R}}}$
Plotted statistic${\displaystyle {\bar {x}}_{i}={\frac {\sum _{j=1}^{n}x_{ij}}{n}}}$

In statistical process monitoring (SPM), the ${\displaystyle {\bar {X}}}$ and R chart is a type of scheme, popularly known as control chart, used to monitor the mean and range of a normally distributed variables simultaneously, when samples are collected at regular intervals from a business or industrial process.[1]. It is often used to monitor the variables data but the performance of the ${\displaystyle {\bar {X}}}$ and R chart may suffer when the normality assumption is not valid. This is connected traditional statistical quality control (SQC) and statistical process control (SPC). However, Woodall[2] 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 (SPC).”"

The chart is advantageous in the following situations:[3]

1. The sample size is relatively small (say, n ≤ 10—${\displaystyle {\bar {x}}}$ and s charts are typically used for larger sample sizes)
2. The sample size is constant
3. Humans must perform the calculations for the chart

The "chart" actually consists of a pair of charts: One to monitor the process standard deviation (as approximated by the sample moving range) and another to monitor the process mean, as is done with the ${\displaystyle {\bar {x}}}$ and s and individuals control charts. The ${\displaystyle {\bar {x}}}$ and R chart plots the mean value for the quality characteristic across all units in the sample, ${\displaystyle {\bar {x}}_{i}}$, plus the range of the quality characteristic across all units in the sample as follows:

R = xmax - xmin.

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

The control limits for this chart type are:[4]

• ${\displaystyle D_{3}{\bar {R}}}$ (lower) and ${\displaystyle D_{4}{\bar {R}}}$ (upper) for monitoring the process variability
• ${\displaystyle {\bar {x}}\pm A_{2}{\bar {R}}}$ for monitoring the process mean
where ${\displaystyle {\bar {x}}}$ and ${\displaystyle {\bar {R}}={\frac {\sum _{i=1}^{m}\left(R_{max}-R_{min}\right)}{m}}}$ are the estimates of the long-term process mean and range established during control-chart setup and A2, D3, and D4 are sample size-specific anti-biasing constants. The anti-biasing constants are typically found in the appendices of textbooks on statistical process control.

As with the ${\displaystyle {\bar {x}}}$ and s and individuals control charts, the ${\displaystyle {\bar {x}}}$ chart is only valid if the within-sample variability is constant.[5] Thus, the R chart is examined before the ${\displaystyle {\bar {x}}}$ chart; if the R chart indicates the sample variability is in statistical control, then the ${\displaystyle {\bar {x}}}$ 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 ${\displaystyle {\bar {x}}}$ chart indicates.

## Limitations

For monitoring mean and variance of normal distribution, the combination of ${\displaystyle {\bar {X}}}$and S chart is usually better than that of ${\displaystyle {\bar {X}}}$and R chart.

Moreover, changes in mean affect R or S chart and changes in standard deviation affects ${\displaystyle {\bar {X}}}$chart. Noting this, several authors recommend using a single chart that can simultaneously monitor ${\displaystyle {\bar {X}}}$and S[6].