X̅ and R chart


In statistical process control, the and R chart, also known as an averages and range 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. It is often used to monitor the Variable and [attribute (research)|variables data] but the performance of the and R chart may suffer when the normality assumption is not valid.

Properties

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 s and individuals control charts. The and R chart plots the mean value for the quality characteristic across all units in the sample,, plus the range of the quality characteristic across all units in the sample as follows:
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:
  • and for monitoring the process variability
  • for monitoring the process mean

Usage of the chart

The chart is advantageous in the following situations:
  1. The sample size is relatively small
  2. The sample size is constant
  3. Humans must perform the calculations for the chart
As with the and s and individuals control charts, the chart is only valid if the within-sample variability is constant. Thus, the R chart is examined before the chart; if the R 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.

Limitations

For monitoring the mean and variance of a normal distribution, the and s chart is usually better than the and R chart.