RCHART Statement: SHEWHART Procedure

Constructing Range Charts

The following notation is used in this section:

$\sigma $

Process standard deviation (standard deviation of the population of measurements)

$R_{i}$

Range of measurements in ith subgroup

$n_{i}$

Sample size of ith subgroup

$d_{2}(n)$

Expected value of the range of n independent normally distributed variables with unit standard deviation

$d_{3}(n)$

Standard error of the range of n independent observations from a normal population with unit standard deviation

$D_{p}(n)$

100pth percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation

Plotted Points

Each point on an R chart indicates the value of a subgroup range ($R_{i}$). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is $R_{10}=19-12=7$.

Central Line

By default, the central line for the ith subgroup indicates an estimate of the expected value of $R_{i}$, which is computed as $d_{2}(n_{i})\hat{\sigma }$, where $\hat{\sigma }$ is an estimate of $\sigma $. If you specify a known value ($\sigma _{0}$) for $\sigma $, the central line indicates the value of $d_{2}(n_{i})\sigma _{0}$. Note that the central line varies with $n_{i}$.

Control Limits

You can compute the limits in the following ways:

  • as a specified multiple (k) of the standard error of $R_{i}$ above and below the central line. The default limits are computed with k = 3 (these are referred to as $3\sigma $ limits).

  • as probability limits defined in terms of $\alpha $, a specified probability that $R_{i}$ exceeds the limits

The following table provides the formulas for the limits:

Table 18.41: Limits for R Charts

Control Limits

LCL = lower limit = $\mbox{max}(d_{2}(n_{i})\hat{\sigma } - kd_{3}(n_{i})\hat{\sigma },0)$

UCL = upper limit = $d_{2}(n_{i})\hat{\sigma } + kd_{3}(n_{i})\hat{\sigma }$

Probability Limits

LCL = lower limit = $D_{\alpha /2}\hat{\sigma } $

UCL = upper limit = $D_{1-\alpha /2}\hat{\sigma } $


The formulas assume that the data are normally distributed. Note that the control limits vary with $n_{i}$ and that the probability limits for $R_{i}$ are asymmetric around the central line. If a standard value $\sigma _{0}$ is available for $\sigma $, replace $\hat{\sigma }$ with $\sigma _{0}$ in Table 18.41.

You can specify parameters for the limits as follows:

  • Specify k with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.

  • Specify $\alpha $ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.

  • Specify a constant nominal sample size $n_{i} \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.

  • Specify $\sigma _{0}$ with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.