and R chart

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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
Size of shift to detect≥ 1.5σ
Process variation chart
R chart for a paired xbar and R chart.svg
Center line
Upper control limit
Lower control limit
Plotted statisticRi = max(xj) - min(xj)
Process mean chart
Xbar chart for a paired xbar and R chart.svg
Center line
Control limits
Plotted statistic

In statistical process monitoring (SPM), the 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 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— 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 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:

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]

  • (lower) and (upper) for monitoring the process variability
  • for monitoring the process mean
where and 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 and s and individuals control charts, the chart is only valid if the within-sample variability is constant.[5] 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.


For monitoring mean and variance of normal distribution, the combination of and S chart is usually better than that of and R chart.

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

See also[edit]


  1. ^ "Shewhart X-bar and R and S Control Charts". NIST/Sematech Engineering Statistics Handbook]. National Institute of Standards and Technology. Retrieved 2009-01-13.
  2. ^ Woodall, William H. (2017-01-02). "Bridging the gap between theory and practice in basic statistical process monitoring". Quality Engineering. 29 (1): 2–15. doi:10.1080/08982112.2016.1210449. ISSN 0898-2112.
  3. ^ Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 222. ISBN 978-0-471-65631-9. OCLC 56729567.
  4. ^ Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 197. ISBN 978-0-471-65631-9. OCLC 56729567.
  5. ^ Montgomery, Douglas (2005). Introduction to Statistical Quality Control. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 214. ISBN 978-0-471-65631-9. OCLC 56729567.
  6. ^ Chen, Gemai; Cheng, Smiley W. (1998). "MAX CHART: COMBINING X-BAR CHART AND S CHART". Statistica Sinica. 8 (1): 263–271. ISSN 1017-0405. JSTOR 24306354.
  7. ^ McCracken, A. K.; Chakraborti, S.; Mukherjee, A. (2013-10-01). "Control Charts for Simultaneous Monitoring of Unknown Mean and Variance of Normally Distributed Processes". Journal of Quality Technology. 45 (4): 360–376. doi:10.1080/00224065.2013.11917944. ISSN 0022-4065.